tac_metadata.json

[
{"url": "http://www.tac.mta.ca/tac/volumes/35/1/35-01abs.html", "title": "The word problem for double categories", "authors": "Antonin Delpeuch", "keywords": ["double categories", "word problem", "string diagrams"], "abstract": "We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation is of interest in its own right since and can for instance be used to reason about double categories with the language of 2-categories, sidestepping the pinwheel problem. It also shows that although double categories are formally more general than 2-categories, they are not actually more expressive, explaining the rarity of applications of  this notion."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/19/35-19abs.html", "title": "Multiple vector bundles: cores, splittings and decompositions", "authors": "Malte Heuer and Madeleine Jotz Lean", "keywords": ["n-fold vector bundle atlas", "linear decomposition"], "abstract": "This paper introduces ∞- and n-fold vector bundles as   special functors from the ∞- and n-cube categories to the   category of smooth manifolds. We study the cores and \"n-pullbacks\" of   n-fold vector bundles and we prove that any n-fold vector bundle   admits a non-canonical isomorphism to a decomposed n-fold vector   bundle. A colimit argument then shows that ∞-fold vector   bundles admit as well non-canonical decompositions. For the   convenience of the reader, the case of triple vector bundles is   discussed in detail."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/15/35-15abs.html", "title": "Orbifolds of Reshetikhin-Turaev TQFTs", "authors": "Nils Carqueville, Ingo Runkel, Gregor Schaumann", "keywords": ["topological quantum field theory", "orbifold construction", "Reshetikhin-Turaev theory", "modular tensor categories"], "abstract": "We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category C, using the language of defect TQFT: (i) spherical fusion categories give orbifolds for the \"trivial\" defect TQFT associated to Vect, (ii) G-crossed extensions of C give group orbifolds for any finite group G, and (iii) we construct orbifolds from commutative Δ-separable Frobenius algebras in C. We also explain how the Turaev-Viro state sum construction fits into our framework by proving that it is isomorphic to the orbifold of case (i). Moreover, we treat the cases (ii) and (iii) in the more general setting of ribbon tensor categories. For case (ii) we show how Morita equivalence leads to isomorphic orbifolds, and we discuss Tambara-Yamagami categories as particular examples."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/16/35-16abs.html", "title": "Fibrations of AU-contexts beget fibrations of toposes", "authors": "Sina Hazratpour and Steven Vickers", "keywords": ["internal fibration", "2-fibration", "context", "bicategory", "elementary topos", "Grothendieck topos", "arithmetic universe"], "abstract": "Suppose an extension map U: T_1 -> T_0 in the 2-category Con of contexts for arithmetic universes satisfies a Chevalley criterion for being an (op)fibration in Con. If M is a model of T_0 in an elementary topos S with nno, then the classifier p: S[T_1/M] -> S satisfies the representable definition of being an (op)fibration in the 2-category ETop of elementary toposes (with nno) and geometric morphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/14/35-14abs.html", "title": "Symmetric monoidal categories and Γ-categories", "authors": "Amit Sharma", "keywords": ["Segal's Nerve functor", "Theory of Bicycles", "Leinster construction"], "abstract": "In this paper we construct a symmetric monoidal closed model category of coherently commutative monoidal categories.  The main aim of this paper is to establish a Quillen equivalence  between a model category of coherently commutative monoidal categories  and a natural model category of Permutative (or strict symmetric monoidal) categories, Perm, which is not a symmetric monoidal closed model category.  The right adjoint of this Quillen equivalence is the classical Segal's Nerve functor."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/12/35-12abs.html", "title": "Exact sequences in the enchilada category", "authors": "M. Eryüzlü, S. Kaliszewski, and John Quigg", "keywords": ["short exact sequence", "C*-correspondence", "exact functor", "crossed product"], "abstract": "We define exact sequences in the enchilada category of C*-algebras and correspondences, and prove that the reduced-crossed-product functor is not exact for the enchilada categories. Our motivation was to determine whether we can have a better understanding of the Baum-Connes conjecture by using enchilada categories. Along the way we prove numerous results showing that the enchilada category is rather strange."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/13/35-13abs.html", "title": "Homotopy theory with marked additive categories", "authors": "Ulrich Bunke, Alexander Engel, Daniel Kasprowski, and Christoph Winges", "keywords": ["Additive categories", "marked categories", "model categories"], "abstract": "We construct combinatorial model category structures on the categories of   (marked) categories and  (marked) preadditive categories, and we characterize  (marked) additive categories as fibrant objects in a Bousfield localization of preadditive categories. These model category structures  are used to present the corresponding  infinity-categories obtained by inverting equivalences. We apply these results to explicitly  calculate   limits and colimits in these infinity-categories. The motivating application is a systematic construction of the equivariant coarse algebraic K-homology with coefficients in an additive category from its non-equivariant version."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/17/35-17abs.html", "title": "Shifted double Lie-Rinehart algebras", "authors": "Johan Leray", "keywords": ["Noncommutative geometry", "Double Poisson algebra", "Double Lie-Rinehart algebra"], "abstract": "We generalize the notions of shifted double Poisson and shifted double Lie-Rinehart structures to monoids in a symmetric monoidal abelian category. The main result is that an n-shifted double Lie-Rinehart structure on a pair (A,M) is equivalent to a non-shifted double Lie-Rinehart structure on the pair (A,M[-n])."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/18/35-18abs.html", "title": "Cubical model categories and quasi-categories", "authors": "Brice Le Grignou", "keywords": ["Enriched categories", "cubical sets"], "abstract": "The goal of this article is to emphasize the role of cubical sets in enriched category theory and infinity-category theory. We show in particular that categories enriched in cubical sets provide a convenient way to describe many infinity-categories appearing in the context of homological algebra."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/11/35-11abs.html", "title": "A new Galois structure in the category of internal preorders", "authors": "Alberto Facchini, Carmelo Finocchiaro and Marino Gran", "keywords": ["Internal preorders", "partial orders", "Galois theory", "monotone-light factorization system", "Alexandroff-discrete spaces"], "abstract": "Let PreOrd(C) be the category of internal preorders in an exact category C. We show that the pair (Eq(C),ParOrd(C)) is a pretorsion theory in PreOrd(C), where Eq(C) and ParOrd(C) are the full subcategories of internal equivalence relations and of internal partial orders in C, respectively. We observe that ParOrd(C) is a reflective subcategory of PreOrd(C) such that each component of the unit of the adjunction is a pullback-stable regular epimorphism. The reflector F: PreOrd(C) -> PardOrd(C) turns out to have stable units in the sense of Cassidy, Hébert and Kelly, thus inducing an admissible categorical Galois structure. In particular, when C is the category Set of sets, we show that this reflection induces a monotone-light factorization system (in the sense of Carboni, Janelidze, Kelly and Paré) in PreOrd(Set). A topological interpretation of our results in the category of Alexandroff-discrete spaces is also given, via the well-known isomorphism between this latter category and PreOrd(Set)."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/10/35-10abs.html", "title": "Augmented virtual double categories", "authors": "Seerp Roald Koudenburg", "keywords": ["augmented virtual double category", "multicategory", "yoneda structure"], "abstract": "\nIn this article the notion of virtual double category (also known as fc-multicategory) is extended as follows. While cells in a virtual double category classically have a horizontal multi-source and single horizontal target, the notion of augmented virtual double category introduced here extends the latter notion by including cells with empty horizontal target as well.\nAny augmented virtual double category comes with a built-in notion of \"locally small object\" and we describe advantages of using augmented virtual double categories as a setting for formal category rather than 2-categories, which are classically equipped with a notion of \"admissible object\" by means of a yoneda structure in the sense of Street and Walters.\nAn object is locally small precisely if it admits a horizontal unit, and we show that the notions of augmented virtual double category and virtual double category coincide in the presence of all horizontal units. Without assuming the existence of horizontal units we show that most of the basic theory for virtual double categories, such as that of restriction and composition of horizontal morphisms, extends to augmented virtual double categories. We introduce and study in augmented virtual double categories the notion of \"pointwise\" composition of horizontal morphisms, which formalises the classical composition of profunctors given by the coend formula."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/6/35-06abs.html", "title": "Spec(Z) and the Gromov Norm", "authors": "Alain Connes and Caterina Consani", "keywords": ["Gamma spaces", "Gamma rings", "Site", "Gromov norm", "Arakelov geometry", "Homology theory"], "abstract": "We define the homology of a simplicial set with coefficients in a Segal's Gamma-set (s-module). We show the relevance of this new homology with values in s-modules by proving that taking as coefficients the s-modules at the archimedean place over the structure sheaf on Spec(Z), one obtains on  the singular homology with real coefficients of a topological space X, a norm equivalent to the Gromov norm. Moreover, we prove that the two norms agree when X is an oriented compact Riemann surface."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/5/35-05abs.html", "title": "Small categories of homological dimension one", "authors": "Karimah Sweet and Charles Ching-An Cheng", "keywords": ["homological dimension", "cohomological dimension", "small category", "cancellative category", "cyclic system", "crown", "supported crown", "DCC category", "Malcev sequence", "embeddability into a group"], "abstract": "We derive three equivalent necessary conditions for a small category to have homological dimension one, generalizing a result of Novikov.  As a consequence, any small cancellative category of homological dimension one is embeddable in a groupoid."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/10/29-10abs.html", "title": "Finite products in partial morphism categories", "authors": "S.N. Hosseini, A.R. Shir Ali Nasab", "keywords": ["partial morphism category", "partial morphism classifier", "binary product", "terminal object"], "abstract": "In this article we give necessary and sufficient conditions for a binary product to exist in a partial morphism category. We also give necessary and sufficient conditions for the existence of a productive terminal in such categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/9/35-09abs.html", "title": "Rank-based persistence", "authors": "Mattia G. Bergomi and Pietro Vertechi", "keywords": ["rank", "persistence", "categorification", "regular category", "abelian category", "semisimple category", "classification", "group action", "point cloud", "poset", "bottleneck", "interleaving"], "abstract": "Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, others combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we define colorable ranks, persistence diagrams and prove the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on filtered topological spaces with a group action and labeled point cloud data."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/8/35-08abs.html", "title": "L'-localization in an ∞-topos", "authors": "Marco Vergura", "keywords": ["reflective subfibration", "separated map", "higher topos theory", "localization theory", "homotopy type theory"], "abstract": "We prove that, given any reflective subfibration L on an ∞-topos E, there exists a reflective subfibration L' on E whose local maps are the L-separated maps, that is, the maps whose diagonals are L-local."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/7/35-07abs.html", "title": "Morita invariance of equivariant Lusternik-Schnirelmann category and  invariant topological complexity", "authors": "A. Angel, H. Colman, M. Grant and J. Oprea", "keywords": ["Topological complexity", "Lusternik-Schnirelmann category"], "abstract": "We use the homotopy invariance of equivariant principal bundles to prove that the equivariant A-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action  and the  invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/18/18-18abs.html", "title": "On the axiomatisation of Boolean categories  with and without medial", "authors": "Lutz Strassburger", "keywords": ["Boolean category", "*-autonomous category", "proof theory", "classical logic", "proof nets"], "abstract": "The term ``Boolean category'' should be used for describing an   object that is to categories what a Boolean algebra is to   posets. More specifically, a Boolean category should provide the   abstract algebraic structure underlying the proofs in Boolean Logic,   in the same sense as a Cartesian closed category captures the proofs   in intuitionistic logic and a *-autonomous category captures the   proofs in linear logic. However, recent work has shown that there is   no canonical axiomatisation of a Boolean category. In this work, we   will see a series (with increasing strength) of possible such   axiomatisations, all based on the notion of *-autonomous   category. We will particularly focus on the medial map, which has   its origin in an inference rule in KS, a cut-free deductive system   for Boolean logic in the calculus of structures. Finally, we will   present a category of proof nets as a particularly well-behaved   example of a Boolean category."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/7/16-07abs.html", "title": "Preserving homology", "authors": "Michael Barr", "keywords": ["preservation of homology"], "abstract": "We raise the question of saying what it means for a functor between abelian categories to preserve homology.  We give a kind of answer and explore the reasons it is unsatisfactory in general (although fine for left or right exact functors)."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/6/16-06abs.html", "title": "Points of affine categories and additivity", "authors": "A. Carboni and G. Janelidze", "keywords": ["Algebraic categories", "affine spaces"], "abstract": "A category C is additive if and only if, for every object B of C, the category Pt(C,B) of pointed objects in the comma category (C,B) is canonically equivalent to C. We reformulate the proof of this known result in order to obtain a stronger one that uses not all objects of B of C, but only a conveniently defined generating class S. If C is a variety of universal algebras, then one can take S to be the class consisting of any single free algebra on a non-empty set."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/5/16-05abs.html", "title": "Every Grothendieck topos has a one-way site", "authors": "Colin McLarty", "keywords": ["Grothendieck topos", "petit topos", "locale", "one way site"], "abstract": "Lawvere has urged a project of characterizing petit toposes which have the character of generalized spaces and gros toposes which have the character of categories of spaces. Etendues and locally decidable toposes are seemingly petit and have a natural common generalization in sites with all idempotents identities. This note shows every Grothendieck topos has such a site. More, it defines slanted products which take any site to an equivalent one way site, a site where all endomorphisms are identities. On the other hand subcanonical one-way sites are very special. A site criterion for petit toposes will probably require subcanonical sites."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html", "title": "Frobenius algebras and ambidextrous adjunctions", "authors": "Aaron D. Lauda", "keywords": ["pseudoadjunction", "pseudomonad", "Frobenius pseudomonoid", "Eilenberg-Moore completion"], "abstract": "In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category $M$ arises from an ambijunction (simultaneous left and right adjoints) in some 2-category $\\mathcal{D}$ into which $M$ fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2-category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2-category and utilizing the free completion under Eilenberg-Moore objects.  We then categorify this theorem by replacing the monoidal category $M$ with a semistrict monoidal 2-category $M$, and replacing the 2-category $\\mathcal{D}$ into which it embeds by a semistrict 3-category.  To state this more powerful result, we must first define the notion of a `Frobenius pseudomonoid', which categorifies that of a Frobenius object. We then define the notion of a `pseudo ambijunction', categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism.  Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2-category arises from a pseudo ambijunction in some semistrict 3-category."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/3/16-03abs.html", "title": "Inverting weak dihomotopy equivalence using homotopy continuous flow", "authors": "Philippe Gaucher", "keywords": ["concurrency", "homotopy", "Whitehead theorem", "directed homotopy", "weak  factorization system", "model category", "localization"], "abstract": "A flow is homotopy continuous if it is indefinitely divisible up to   S-homotopy. The full subcategory of cofibrant homotopy continuous   flows has nice features. Not only it is big enough to contain all   dihomotopy types, but also a morphism between them is a weak   dihomotopy equivalence if and only if it is invertible up to   dihomotopy. Thus, the category of cofibrant homotopy continuous   flows provides an implementation of Whitehead's theorem for the full   dihomotopy relation, and not only for S-homotopy as in previous   works of the author. This fact is not the consequence of the   existence of a model structure on the category of flows because it   is known that there does not exist any model structure on it whose   weak equivalences are exactly the weak dihomotopy equivalences. This   fact is an application of a general result for the localization of a   model category with respect to a weak factorization system."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/2/16-02abs.html", "title": "Action groupoid in protomodular categories", "authors": "Dominique Bourn", "keywords": ["Protomodular categories", "representation of actions", "internal groupoids", "abelian objects", "central relations and center"], "abstract": "We give here some examples of non pointed protomodular categories $\\mathbb C$ satisfying a property similar to the property of representation of actions which holds for the pointed protomodular category $Gp$ of groups: any slice category of $Gp$, any category of groupoids with a fixed set of objects, any essentially affine category. This property gives rise to an internal construction of the center of any object $X$, and consequently to a specific characterization of the abelian objects in $\\mathbb C$."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n11/5-11abs.html", "title": "Generalized congruences -- Epimorphisms in Cat", "authors": "Marek A. Bednarczyk,         Andrzej M. Borzyszkowski,         Wieslaw Pawlowski", "keywords": ["congruence", "epimorphic functor", "coequalizer", "category of small categories."], "abstract": "The paper generalizes the notion of a congruence on a category   and pursues some of its applications.  In particular, generalized   congruences are used to provide a concrete construction of coequalizers   in ${\\cal C}at$.  Extremal, regular and various other classes of epimorphic   functors are characterized and inter-related."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n10/5-10abs.html", "title": "Aspects of fractional exponent functors", "authors": "Anders Kock and Gonzalo E. Reyes", "keywords": ["atom", "coalgebra", "enrichment."], "abstract": "We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the  base topos; there are in fact often two distinct enrichments."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html", "title": "On property-like structures", "authors": "G. M. Kelly and Stephen Lack", "keywords": ["2-category", "monad", "structure", "property."], "abstract": "A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of `category with finite products'. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of `essentially unique' and investigating its consequences. We call such 2-monads  property-like. We further consider the more restricted class of  fully property-like 2-monads, consisting of  those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which `structure is adjoint to unit', and which we now call  lax-idempotent 2-monads: both these and their  colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n8/3-08abs.html", "title": "Monads and interpolads in bicategories", "authors": "Jurgen Koslowski", "keywords": ["bicategory", "closed bicategory", "Cauchy-complete bicategory", "lax functor", "monad", "module", "interpolation property", "taxonomy."], "abstract": "Given a bicategory, 2, with stable local coequalizers, we construct a bicategory of monads Y-mnd by using lax functors from the generic 0-cell, 1-cell and 2-cell, respectively, into Y.  Any lax functor into Y factors through Y-mnd and the 1-cells turn out to be the familiar bimodules.  The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchy-complete, but have a well-known Cauchy-completion in common.  This prompts us to formulate a concept of Cauchy-completeness for bicategories that are not locally ordered and suggests a weakening of the notion of monad.  For this purpose, we develop a calculus of general modules between unstructured endo-1-cells.  These behave well with respect  to composition, but in general fail to have identities.  To overcome this problem, we do not need to impose the full structure of a monad on endo-1-cells.  We show that associative coequalizing multiplications suffice and call the resulting structures    interpolads.  Together with structure-preserving i-modules these form a bicategory Y-int that is indeed Cauchy-complete, in our sense, and contains the bicategory of monads as a not necessarily full sub-bicategory.  Interpolads over rel are idempotent relations, over the suspension of set they correspond to interpolative semi-groups, and over spn they lead to a notion of ``category without identities'' also known as ``taxonomy''.  If Y locally has equalizers, then modules in general, and the bicategories Y-mnd and Y-int in particular, inherit the property of being closed with respect to 1-cell composition."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n10/3-10abs.html", "title": "Closed model categories for $[n,m]$-types", "authors": "J. Ignacio Extremiana Aldana, Luis J. Hernandez Paricio, Maria T. Rivas Rodriguez", "keywords": ["Closed model category", "Homotopy category", "[n;m]-types."], "abstract": "For m >= n > 0, a map f between pointed spaces is said to be a weak [n,m]-equivalence if f induces isomorphisms of the homotopy groups \\pi_k for n <= k <= m~. Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures have the same class of weak equivalences but different classes of fibrations and therefore of cofibrations. Using one of these structures, one obtains that the localized category is equivalent to the category of n-reduced CW-complexes with dimension less than or equal to m+1 and m-homotopy classes of cellular pointed maps. Using the other structure we see that the localized category is also equivalent to the homotopy category of (n-1)-connected (m+1)-coconnected CW-complexes."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n7/3-07abs.html", "title": "Crossed squares and 2-crossed modules of commutative algebras", "authors": "Zekeriya Arvasi", "keywords": ["Simplicial Algebras", "Crossed n-Cubes", "Crossed Squares", "2-Crossed Modules."], "abstract": "In this paper, we construct a neat description of the passage from crossed squares of commutative algebras to 2-crossed modules analogous to that given by Conduche in the group case. We also give an analogue, for commutative algebra, of T. Porter's  simplicial groups to n-cubes of groups which implies an inverse functor to Conduche's one."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n11/3-11abs.html", "title": "Multilinearity of Sketches", "authors": "David B. Benson", "keywords": ["categorical model theory", "Ehresmann sketches", "data structures."], "abstract": "We give a precise characterization for when the models of the tensor product of sketches are structurally isomorphic to the models of either sketch in the models of the other. For each base category K call the just mentioned property (sketch) K-multilinearity. Say that two sketches are K-compatible with respect to base category K just in case in each K-model, the limits for each limit specification in each sketch commute with the colimits for each colimit specification in the other sketch and all limits and colimits are pointwise. Two sketches are K-multilinear if and only if the two sketches are K-compatible. This property then extends to strong Colimits of sketches.\nWe shall use the technically useful property of limited completeness and completeness of every category of models of sketches. That is, categories of sketch models have all limits commuting with the sketched colimits and and all colimits commuting with the sketched limits.    Often used implicitly, the precise statement of this property and its proof appears here."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n6/3-06abs.html", "title": "The reflectiveness of covering morphisms in algebra and geometry", "authors": "G. Janelidze and G. M. Kelly", "keywords": ["factorization system", "refective subcategory", "covering space", "Galois theory", "central extension."], "abstract": "Each full reflective subcategory X of a finitely-complete category C gives rise to a factorization system (E, M) on C, where E consists of the morphisms of C inverted by the reflexion I : C --> X. Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A --> B lies in M precisely when it is the pullback along the unit \\etaB : B --> IB of its reflexion If : IA --> IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A --> B is said to be a covering of B if, for some effective descent morphism p : E --> B, the pullback p^*f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \\Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass -- or rather a subcategory -- of the category C \\downarrow B \\subset C/B whose objects are those f : A --> B with f in \\Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is reflective in C \\downarrow B; and we give a number of disparate conditions, each sufficient for this to be so. In this way we recapture old results and establish new ones on the reflexion of local homeomorphisms into coverings, on the Galois theory of commutative rings, and on generalized central extensions of universal algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n8/2-08abs.html", "title": "Remarks on Quintessential and Persistent Localizations", "authors": "P.T. Johnstone", "keywords": [], "abstract": "We define a localization L of a category E to be quintessential if the left adjoint to the inclusion functor is also right adjoint to it, and persistent if L is closed under subobjects in E. We show that quintessential localizations of an arbitrary Cauchy-complete category correspond to idempotent natural endomorphisms of its identity functor, and that they are necessarily persistent. Our investigation of persistent localizations is largely restricted to the case when E is a topos: we show that persistence is equivalence to the closure of L under finite coproducts and quotients, and that it implies that L is coreflective as well as reflective, at least provided E admits a geometric morphism to a Boolean topos. However, we provide examples to show that the reflector and coreflector need not coincide."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n7/2-07abs.html", "title": "Combinatorics of curvature, and the Bianchi identity", "authors": "Anders Kock", "keywords": ["Connection", "curvature", "groupoid", "first neighbourhood of the diagonal."], "abstract": "We analyze the Bianchi Identity as an instance of a  basic fact of combinatorial groupoid theory, related to  the Homotopy Addition Lemma. Here it becomes formulated  in terms of 2-forms with values in the gauge group bundle  of a groupoid, and leads in particular to the   (Chern-Weil) construction of characteristic classes. The method is that of synthetic differential geometry,  using \"the first neighbourhood of the diagonal\" of a  manifold as its basic combinatorial structure. We introduce  as a tool a new and simple description of wedge (= exterior)  products of differential forms in this context."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n3/1-03abs.html", "title": "On finite induced crossed modules and the homotopy 2-type of mapping cones,", "authors": "Ronald Brown and Christopher D. Wensley", "keywords": ["crossed modules", "homotopy 2�types", "generalised van Kampen Theo�  rem", "crossed resolution", "Postnikov invariant."], "abstract": "Results on the finiteness of induced crossed  modules are proved both algebraically and topologically.   Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2-types of mapping cones of  classifying spaces of groups. Calculations of the cohomology  classes of some finite crossed  modules are given,  using crossed complex methods."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n5/1-05abs.html", "title": "Symmetric monoidal categories model all connective spectra", "authors": "R. W. Thomason", "keywords": ["club", "connective spectrum", "E1�space", "operad", "lax algebra", "spectrum", "stable homotopy", "symmetric monoidal."], "abstract": "The classical infinite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of -1-connective spectra."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n4/1-04abs.html", "title": "Kan extensions along promonoidal functors", "authors": "Brian Day and Ross Street", "keywords": ["monoidal functor", "promonoidal category", "Kan extension", "Fourier transform", "convolution tensor product."], "abstract": "Strong promonoidal functors are defined. Left Kan extension (also  called \"existential quantification\") along a strong promonoidal  functor is shown to be a strong monoidal functor. A construction for the  free monoidal category on a promonoidal category is provided. A  Fourier-like transform of presheaves is defined and shown to take  convolution product to cartesian product."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n6/2-06abs.html", "title": "On quantic conuclei on orthomodular lattices", "authors": "Leopoldo Roman and Rita E. Zuazua", "keywords": [], "abstract": "In this paper we study the lattice of quantic conuclei for orthomudular lattices. We show that under certain condition we can get a complete characterization of all quantic conuclei. The thing to note is we use a non commutative, non associative disjunction operation which can be thought of as non commutative, non associative linear logic."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n3/2-03abs.html", "title": "A Counterexample to a Conjecture of Barr", "authors": "Sarah Whitehouse", "keywords": ["Hochschild homology", "Harrison homology", "Andr'e/Quillen homology."], "abstract": "We discuss two versions of a conjecture attributed to M. Barr.  The Harrison cohomology of a commutative algebra is known to coincide with the Andre/Quillen cohomology over a field of characteristic zero but not in prime characteristics. The conjecture is that a modified version of Harrison cohomology, taking into account torsion, always agrees with Andre/Quillen cohomology. We give a counterexample."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n9/1-09abs.html", "title": "On the Size of Categories", "authors": "Peter Freyd and Ross Street", "keywords": ["small", "locally small", "small homsets", "idempotent", "presheaf category."], "abstract": "The purpose is to give a simple proof that a category is equivalent  to a small category if and only if both it and its presheaf category  are locally small."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n8/1-08abs.html", "title": "Categorical Data-Specifications", "authors": "F. Piessens and E. Steegmans", "keywords": ["data�specifications", "sketches."], "abstract": "We introduce MD-sketches, which are a particular kind of Finite Sum sketches. Two interesting results about MD-sketches are proved. First, we show that, given two MD-sketches, it is algorithmically decidable whether their  model categories are equivalent. Next we show that data-specifications, as used in database-design and software engineering, can be translated to MD-sketches. As a corollary, we obtain that equivalence of  data-specifications is decidable."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/17/18-17abs.html", "title": "Exploring the gap between linear and classical logic", "authors": "Francois Lamarche", "keywords": ["*-autonomous categories", "denotational semantics", "linear logic", "classical logic", "deep inference", "Medial rule"], "abstract": "The Medial rule was first devised as a deduction rule in the   Calculus of Structures. In this paper we explore it from the point   of view of category theory, as additional structure on a   *-autonomous category. This gives us some insights on the   denotational semantics of classical propositional logic, and allows   us to construct new models for it, based on suitable generalizations   of the theory of coherence spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n7/1-07abs.html", "title": "A forbidden-suborder characterization of binarily-composable diagrams in double categories", "authors": "Robert Dawson", "keywords": ["Double categories", "tileorders", "binary composition", "Hasse diagrams", "forbidden  suborders."], "abstract": "Tilings of rectangles with rectangles, and tileorders (the associated double order structures) are useful as ``templates'' for composition in double categories. In this context, it is particularly relevant to ask which tilings may be joined together, two rectangles at a time, to form one large rectangle. We characterize such tilings via forbidden suborders, in a manner analogous to Kuratowski's characterization of planar graphs."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n1/1-01abs.html", "title": "Oriented Singular Homology", "authors": "Michael Barr", "keywords": ["Oriented singular homology", "acyclic models."], "abstract": "We formulate three slightly different notions of oriented singular chain complexes and show that all three are naturally homotopic to ordinary singular chain complexes."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n2/1-02abs.html", "title": "Functorial and algebraic properties of Browns P functor", "authors": "Luis-Javier Hernandez-Paricio", "keywords": ["Category of fractions", "Pro�category", "Monoid", "M �set", "Brown's P functor", "Tower", "Pro�object", "Near�ring", "Near�module", "Generator", "Pro�set", "Pro�group", "Pro�abelian group."], "abstract": "In 1975 E. M. Brown constructed a functor $\\cal P$ which carries  the tower of fundamental groups of the end of a (nice) space to  the Brown-Grossman fundamental group. In this work, we study this  functor and its extensions and analogues defined for pro-sets,  pro-pointed sets, pro-groups and pro-abelian groups. The new versions of the $\\cal P$ functor are provided with more algebraic  structure. Examples given in the paper prove that in general the  $\\cal P$ functors  are not faithful, however, one of our main  results establishes that the restrictions of the corresponding  $\\cal P$ functors to the full subcategories of towers are faithful.  We also prove that the restrictions of the $\\cal P$ functors to  the corresponding full subcategories of finitely generated towers  are also full. Consequently, in these cases, the towers of objects  in the categories of sets, pointed sets, groups and abelian groups,  can be replaced by adequate algebraic models ($M$-sets, $M$-pointed  sets, near-modules and modules.) The article also contains the  construction of left adjoints for the $\\cal P$ functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n10/2-10abs.html", "title": "A note on free regular and exact completions and their infinitary generalizations", "authors": "Hongde Hu and Walter Tholen", "keywords": ["regular category", "exact category", "regular completion", "exact completion", "flat  functor", "accessible category."], "abstract": "Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html", "title": "Distributive Adjoint Strings", "authors": "R. Rosebrugh and R. J. Wood", "keywords": ["adjoint functor", "distributivity", "simplicial 2�category."], "abstract": "For an adjoint string V -| W -| X -| Y : B --> C, with Y fully faithful, it is frequently, but not always, the case that the composite VY underlies an idempotent monad. When it does, we call the string distributive. We also study shorter and longer `distributive' adjoint  strings and how to generate them. These provide a new construction of  the simplicial 2-category, Delta."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n9/2-09abs.html", "title": "Glueing Analysis for Complemented Subtoposes", "authors": "Anders Kock and Till Plewe", "keywords": ["Artin glueing", "complemented subtoposes", "complemented sublocale", "locally  closed subtoposes", "locally closed sublocale", "prolongation by 0", "extension by 0."], "abstract": "We prove how any (elementary) topos may be reconstructed from  the data of two complemented subtoposes together with a pair  of left exact ``glueing functors''. This generalizes the  classical glueing theorem for toposes, which deals with the  special case of an open subtopos and its closed complement.\nOur glueing analysis applies in a particularly simple form to  a locally closed subtopos and its complement, and one of the  important properties (prolongation by zero for abelian groups)  can be succinctly described in terms of it."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n4/2-04abs.html", "title": "An algebraic description of locally multipresentable categories", "authors": "Jiri Adamek, Jiri Rosicky", "keywords": ["locally multipresentable category", "sketch."], "abstract": "Locally finitely presentable categories are known to be precisely the categories of models of essentially algebraic theories, i.e., categories  of partial algebras whose domains of definition are determined by equations in total operations. Here we show an analogous description of locally finitely multipresentable categories. We also prove that locally finitely multipresentable categories are precisely categories of models of sketches with finite limit and countable coproduct specifications, and we present an example of a locally finitely multipresentable category not sketchable by a sketch with finite limit and finite colimit specifications."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n5/2-05abs.html", "title": "Finiteness of a Non-abelian Tensor Product of Groups", "authors": "Nick Inassaridze", "keywords": ["Non�abelian tensor product of groups", "Comp�subgroup", "Comp�pairs", "compatibility resolution", "half compatible actions."], "abstract": "Some sufficient conditions for finiteness of a generalized non-abelian tensor product of groups are established  extending Ellis' result for compatible actions."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n2/2-02abs.html", "title": "The Chu construction", "authors": "Michael Barr", "keywords": ["Chu category", "bimodules", "cofree coalgebras."], "abstract": "We take another look at the Chu construction and show how to simplify it by looking at it as a module category in a trivial Chu category.  This simplifies the construction substantially, especially in the case of a non-symmetric biclosed monoidal category.  We also show that if the original category is accessible, then for any of  a large class of ``polynomial-like'' functors, the category of coalgebras has cofree objects."},
{"url": "http://www.tac.mta.ca/tac/volumes/1996/n1/2-01abs.html", "title": "Computing crossed modules induced by an inclusion of a normal subgroup,  with applications to homotopy 2-types", "authors": "Ronald Brown and Christopher D. Wensley", "keywords": ["crossed modules", "homotopy 2�types", "generalized Van Kampen theorem", "crossed resolution", "Postnikov invariant", "classifying spaces of discrete groups."], "abstract": "We obtain some explicit calculations of  crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this  enables the  computation of the homotopy 2-types and  second homotopy modules of certain homotopy pushouts of maps of  classifying spaces of discrete groups."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n5/3-05abs.html", "title": "Proof theory for full intuitionistic linear logic,  bilinear logic, and MIX categories", "authors": "J.R.B. Cockett and  R.A.G. Seely", "keywords": ["monoidal closed categories", "tensorial strength", "coherence", "categorical proof theory."], "abstract": "This note applies techniques we have developed to study coherence in monoidal categories with two tensors, corresponding to the tensor-par fragment of linear logic, to several new situations, including Hyland and de Paiva's Full Intuitionistic Linear Logic (FILL), and Lambek's Bilinear Logic (BILL). Note that the latter is a noncommutative logic; we also consider the noncommutative version of FILL.  The essential difference between FILL and BILL lies in requiring that a certain tensorial strength be an isomorphism. In any FILL category, it is possible to isolate a full subcategory of objects (the ``nucleus'') for which this transformation is an isomorphism. In addition, we define and study the appropriate categorical structure underlying the MIX rule.  For all these structures, we do not restrict consideration to the ``pure'' logic as we allow non-logical axioms. We define the appropriate notion of proof nets for these logics, and use them to describe coherence results for the corresponding categorical structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n4/3-04abs.html", "title": "Lax Operad Actions and Coherence for Monoidal  n-Categories, A_{\\infty} Rings  and Modules", "authors": "Gerald Dunn", "keywords": ["braided monoidal n-category", "operad", "ring spectrum", "A8 ring", "Morita equivalence."], "abstract": "We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also obtain coherence theorems for A_{\\infty} and E_{\\infty} rings and for lax modules over such rings. Using these results we give an extension of Morita equivalence to A_{\\infty} rings and some applications to infinite loop spaces and algebraic K-theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n3/3-03abs.html", "title": "Note on a  theorem of Putnam's", "authors": "Michael Barr", "keywords": ["model", "relational theory", "back and forth lemma."], "abstract": "In a 1981 book, H. Putnam claimed that in a pure relational language without equality, for any model of a relation that was neither empty nor full, there was another model that satisfies the same first order sentences.  Ed Keenan observed that this was false for finite models since equality is a definable predicate in such cases.  This note shows that Putnam's claim is true for infinite models, although it requires a more sophisticated proof than the one outlined by Putnam."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n2/3-02abs.html", "title": "Doctrines whose structure forms a fully faithful adjoint string", "authors": "F. Marmolejo", "keywords": ["KZ-doctrines", "Pseudomonads", "Algebras", "Gray-categories."], "abstract": "We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd -| m -| dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZ-doctrine."},
{"url": "http://www.tac.mta.ca/tac/volumes/1997/n1/3-01abs.html", "title": "Higher Dimensional Peiffer Elements in Simplicial Commutative Algebras", "authors": "Z. Arvasi and T. Porter", "keywords": ["Simplicial commutative algebra", "boundaries", "Moore complex ."], "abstract": "Let E be a simplicial commutative algebra such that E_n is generated by   degenerate elements.  It is shown that in this case the n^th term of the  Moore complex of E is generated by images of certain pairings from lower  dimensions.  This is then used to give a description of the boundaries in dimension n-1 for n = 2, 3, and 4."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n10/4-10abs.html", "title": "On Generic Separable Objects", "authors": "Robbie Gates", "keywords": ["extensive category", "separable object", "generic solution", "direct image."], "abstract": "The notion of {\\em separable} (alternatively {\\em unramified}, or {\\em decidable}) objects and their place in a categorical theory of space have been described by Lawvere (see \\cite{lawvere:como}), drawing on notions of separable from algebra and unramified from geometry. In \\cite{schanuel:halifax}, Schanuel constructed the generic separable object in an extensive category with products as an object of the free category with finite sums on the dual of the category of finite sets and injections.\nWe present here a generalization of the work of \\cite{schanuel:halifax}, replacing the category of finite sets and injections by a category $\\cat A$ with a suitable factorization system.  We describe the analogous construction, and identify and prove a universal property of the constructed category for both extensive categories and extensive categories with products (in the case $\\cat A$ admits sums).\nIn constructing the machinery for proving the required universal property, we recall briefly the boolean algebra structure of the summands of an object in an extensive category.  We further present a notion of direct image for certain maps in an extensive category, to allow construction of left adjoints to the inverse image maps obtained from pullbacks.\nPlease note the electronically available References at  http://www.tac.mta.ca/tac/volumes/1998/n10/reference.html"},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n9/4-09abs.html", "title": "Geometric Construction of the Levi-Civita Parallelism", "authors": "Anders Kock", "keywords": ["quadratic differential form", "affine connection", "synthetic differential geometry."], "abstract": "In terms of synthetic differential geometry, we give a variational characterization of the connection (parallelism) associated to a pseudo-Riemannian metric on a manifold."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n8/4-08abs.html", "title": "Freeness Conditions for 2-Crossed Modules and Complexes", "authors": "A. Mutlu, T. Porter", "keywords": ["Simplicial groups", "construction data", "2-crossed module", "2-crossed complex", "homotopy 3-types."], "abstract": "Using free simplicial groups, it is shown how to construct a free or totally free 2-crossed module on suitable construction data.  2-crossed complexes are introduced and similar freeness results for these are discussed.\nPlease note the electronically available References at  http://www.tac.mta.ca/tac/volumes/1998/n8/reference.html"},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n7/4-07abs.html", "title": "Applications of Peiffer pairings in the Moore complex of a simplicial group", "authors": "A. Mutlu, T. Porter", "keywords": ["Simplicial groups", "Brown-Loday lemma", "Peiffer elements", "2-crossed modules."], "abstract": "Generalising a result of Brown and Loday, we give for n=3 and 4, a  decomposition of the group, d_nNG_n, of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2-crossed modules and quadratic modules are discussed.\nPlease note the electronically available References at  http://www.tac.mta.ca/tac/volumes/1998/n7/reference.html"},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n6/4-06abs.html", "title": "The separated extensional Chu category", "authors": "Michael Barr", "keywords": ["*-autonomous categories", "Chu construction", "separated", "extensional."], "abstract": "This paper shows that, given a factorization system, E/M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also star-autonomous under weaker conditions than had been given previously ([Barr, 1991]).  In the process we find conditions under which the intersection of a full reflective subcategory and its coreflective dual in a Chu category is star-autonomous."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n5/4-05abs.html", "title": "A 2-Categorical Approach To Change Of Base And Geometric Morphisms II", "authors": "A.Carboni, G.M.Kelly, D.Verity and R.J.Wood", "keywords": ["equipment", "adjunction", "span."], "abstract": "We introduce a notion of  equipment  which generalizes the earlier notion of  pro-arrow equipment  and includes such familiar constructs as $\\rel\\K$, $\\spn\\K$, $\\par\\K$, and $\\pro\\K$ for a suitable category $\\K$, along with related constructs such as the $\\V$-$\\pro$ arising from a suitable monoidal category $\\V$. We further exhibit the equipments as the objects of a 2-category, in such a way that  arbitrary  functors $F:\\eL ---> \\K$ induce equipment arrows $\\rel F:\\rel\\eL --->\\rel\\K$, $\\spn F:\\spn\\eL ---> \\spn\\K$, and so on, and similarly for arbitrary monoidal functors $\\V ---> \\W$. The article I with the title above dealt with those equipments $\\M$ having each $\\M(A,B)$ only an ordered set, and contained a detailed analysis of the case $\\M =\\rel\\K$; in the present article we allow the $\\M(A,B)$ to be general categories, and illustrate our results by a detailed study of the case $\\M=\\spn\\K$. We show in particular that $\\spn$ is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those $\\spn G$ which arise from a  geometric morphism $G$."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html", "title": "A theory of enriched sketches", "authors": "F. Borceux,  C. Quinteiro,  J. Rosicky", "keywords": [], "abstract": "The theory of enriched accessible categories over a suitable base category V is developed. It is proved that these enriched accessible categories coincide with the categories of flat functors, but also with the categories of models of enriched sketches. A particular attention is devoted to enriched locally presentable categories and enriched functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n4/4-04abs.html", "title": "Simplicial and categorical diagrams, and their equivariant applications", "authors": "Rudolf Fritsch and Marek Golasinski", "keywords": ["comma category", "Grothendieck construction", "homotopy colimit", "pull-back", "simplicial set", "small category."], "abstract": "We show that the homotopy category of simplicial diagrams $I-SS$ indexed by a small category $I$ is equivalent to a homotopy category of $SS\\downarrow NI$ simplicial sets over the nerve $NI$. Then their equivalences, by means of the nerve functor N : Cat --> SS$ from the category $Cat$ of small categories, with respective homotopy categories associated to $Cat$ are established. Consequently, an equivariant simplicial version of the Whitehead Theorem is derived."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n2/4-02abs.html", "title": "Protomodularity, descent, and semidirect products", "authors": "D. Bourn and G. Janelidze", "keywords": [], "abstract": "Using descent theory we give various forms of short five-lemma in protomodular categories, known in the case of exact protomodular categories.  We also describe the situation where the notion of a semidirect product can be defined categorically."},
{"url": "http://www.tac.mta.ca/tac/volumes/1998/n1/4-01abs.html", "title": "Pasting in multiple categories", "authors": "Richard Steiner", "keywords": ["pasting diagram", "n-category", ".omega-category", "infinite-category", "partial omega-category", "parity complex", "omega-complex", "directed complex."], "abstract": "In the literature there are several kinds of concrete and abstract cell  complexes representing composition in n-categories, \\omega-categories or \\infty-categories, and the slightly more general partial \\omega-categories. Some examples are parity c omplexes, pasting schemes and directed complexes. In this paper we give an axiomatic treatment: that is to say, we study the class of `\\omega-complexes' which consists of all complexes representing partial \\omega-categories. We show that \\omega-complexes can be given geometric structures and that in most important examples they become well-behaved CW complexes; we characterise \\omega-complexes by conditions on their cells; we show that a product of \\omega-complexes is again an \\omega-complex;  and we describe some products in detail."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n9/5-09abs.html", "title": "When projective does not imply flat, and other homological anomalies", "authors": "L. Gaunce Lewis, Jr.", "keywords": ["closed category", "symmetric monoidal category", "Mackey functor", "projective", "flat", "compact Lie group."], "abstract": "If $\\cal M$ is both an abelian category and a symmetric monoidal closed category, then it is natural to ask whether projective objects in $\\cal M$ are flat, and whether the tensor product of two projective objects is projective.  In the most familiar such categories, the answer to these questions is obviously yes.  However, the category $\\cal M_G$ of Mackey functors for a compact Lie group $G$ is a category of this type in which projective objects need not be so well-behaved.  This category is of interest since good equivariant cohomology theories are Mackey functor valued.  The tensor product on $\\cal M_G$ is important in this context because of the role it plays in the not yet fully understood universal coefficient and K\\\"{u}nneth formulae.  This role makes the relationship between projective objects and the tensor product especially critical. Unfortunately, if $G$ is, for example, $O(n)$, then projectives need not be flat in $\\cal M_G$ and the tensor product of projective objects need not be projective.  This misbe haviorcomplicates the search for full strength equivariant universal coefficient and K\\\"{u}nneth formulae.\nThe primary purpose of this article is to investigate these questions about the interaction of the tensor product with projective objects in   symmetric monoidal abelian categories.  Our focus is on functor categories whose monoidal structures arise in a fashion described by Day. Conditions are given under which such a structure interacts appropriately  with projective objects.  Further, examples are given to show that, when these conditions aren't met, this interaction can be quite bad.  These examples were not fabricated to illustrate the abstract possibility of misbehavior.  Rather, they are drawn from the literature.  In particular, $\\cal M_G$ is badly behaved not only for the groups $O(n)$, but also for the groups $SO(n)$, $U(n)$, $SU(n)$, $Sp(n)$, and $ Spin(n)$.  Similar misbehavior occurs in two categories of global Mackey functors which are   widely used in the study of classifying spaces of finite groups.  Given  the extent of the homological misbehavior in Mackey functor categories    described here, it is reasonable to expect that similar problems occur in other functor categories carrying symmetric monoidal closed structures provided by Day's machinery."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n12/5-12abs.html", "title": "Localizations of Maltsev varieties", "authors": "Marino Gran and Enrico Maria Vitale", "keywords": [], "abstract": "We give an abstract characterization of categories which are localizations of Maltsev varieties. These results can be applied to characterize localizations of naturally Maltsev varieties."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n8/5-08abs.html", "title": "Chu-spaces, a group algebra and induced representations", "authors": "Eva Schläpfer", "keywords": ["chu-spaces", "group algebra", "induced representation."], "abstract": "Using the Chu-construction, we define a group algebra for topological Hausdorff groups. Furthermore, for isometric, weakly continuous representations of a subgroup $H$ of a Hausdorff group $G$ induced representations are constructed."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n6/5-06abs.html", "title": "Convergence in exponentiable spaces", "authors": "Claudio Pisani", "keywords": ["exponentiable spaces", "quasi-local-compactness", "convergence", "ultrafilter monad", "lax monads and algebras", "continuous lattices."], "abstract": "Exponentiable spaces are characterized in terms of convergence.  More precisely, we prove that a relation $R:{\\cal U}X \\rightharpoonup X$  between  ultrafilters and elements of a set $X$ is the convergence relation for a  quasi-locally-compact (that is, exponentiable)  topology on $X$ if and only if the following conditions are satisfied:\n1. $id \\subseteq R\\circ\\eta  $  2.$R\\circ {\\cal U}R = R\\circ\\mu $\nwhere $\\eta : X \\to {\\cal U}X$ and $\\mu : {\\cal U}({\\cal U}X) \\to  {\\cal U}X$ are the unit and the multiplication of the ultrafilter monad, and ${\\cal U} : \\bi{Rel} \\to \\bi{Rel}$ extends the ultrafilter functor ${\\cal U} : \\bi{Set} \\to \\bi{Set}$ to the category of sets and  relations. $({\\cal U},\\eta,\\mu)$ fails to be a monad on $\\bi{Rel}$ only because  $\\eta$ is not a strict natural transformation. So, exponentiable spaces are the lax (with respect to the unit law) algebras for a lax monad on $\\bi{Rel}$. Strict algebras are exponentiable and $T_1$ spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n5/5-05abs.html", "title": "Distributive laws for pseudomonads", "authors": "Francisco Marmolejo", "keywords": ["Pseudomonads", "distributive laws", "KZ-doctrines", "Gray-categories."], "abstract": "We define distributive laws between pseudomonads in a Gray-category A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the \\gray-category PSM(A) of pseudomonads in A, and define a lifting to be a pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co-)KZ-doctrine and the other a KZ-doctrine."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n7/5-07abs.html", "title": "Double categories, 2-categories, thin structures and connections", "authors": "Ronald Brown and Ghafar H. Mosa", "keywords": ["Double category", "2-category", "thin structure", "connection."], "abstract": "The main result is that two possible structures which may be imposed on an edge symmetric double category, namely  a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2-categories is equivalent to the category of edge symmetric    double categories with thin structure."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n4/5-04abs.html", "title": "A useful category for mixed Abelian groups", "authors": "Grigore Calugareanu", "keywords": ["additive category", "quotient category", "splitting mixed Abelian groups."], "abstract": "All the useful categories in the study of the mixed abelian groups (e.g. {\\bf Warf} and {\\bf Walk}) ignore the torsion. We introduce a new category denoted ${\\cal A}$ which ignores the torsion-freeness and could characterize some classes of nonsplitting mixed groups with the aid of {\\bf Walk.}"},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n3/5-03abs.html", "title": "A note on the exact completion of a regular category, and its infinitary generalizations", "authors": "Stephen Lack", "keywords": ["regular category", "exact category", "exact completion", "category of sheaves."], "abstract": "A new description of the exact completion $\\cal C_{ex/reg}$ of a regular category $\\cal C$ is given, using a certain topos $Shv(\\cal C)$ of sheaves on $\\cal C$; the exact completion is then constructed as the closure of $\\cal C$ in $Shv(\\cal C)$ under finite limits and coequalizers of equivalence relations. An infinitary generalization is proved, and the classical description of the exact completion is derived."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n1/5-01abs.html", "title": "A note on discrete Conduché fibrations", "authors": "Peter Johnstone", "keywords": [], "abstract": "The class of functors known as discrete Conduché fibrations forms a common generalization of discrete fibrations and discrete opfibrations, and shares many of the formal properties of these two classes. F. Lamarche conjectured that, for any small category $\\cal B$, the category ${\\bf DCF}/{\\cal B}$ of discrete Conduché fibrations over $\\cal B$ should be a topos. In this note we show that, although for suitable categories $\\cal B$ the discrete Conduch&eacute fibrations over $\\cal B$ may be presented as the `sheaves' for a family of coverings on a category ${\\cal B}_{tw}$ constructed from $\\cal B$, they are in general very far from forming a topos."},
{"url": "http://www.tac.mta.ca/tac/volumes/1999/n2/5-02abs.html", "title": "A tensor product for Gray-categories", "authors": "Sjoerd Crans", "keywords": [], "abstract": "In this paper I extend Gray's tensor product of 2-categories to a new  tensor product of Gray-categories. I give a description in terms of  generators and relations, one of the relations being an ``interchange'' relation, and a description similar to Gray's description of his tensor product of  2-categories. I show that this tensor product of Gray-categories  satisfies a universal property with respect to quasi-functors of two variables,  which are defined in terms of lax-natural transformations between  Gray-categories. The main result is that this tensor product is part of a monoidal structure  on Gray-Cat, the proof requiring interchange in an essential way.  However,  this does not give a monoidal {(bi)closed} structure, precisely because of interchange. And although I define composition of lax-natural  transformations, this composite need not be a lax-natural transformation again, making Gray-Cat only a partial Gray-Cat$_\\otimes$-CATegory."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n9/6-09abs.html", "title": "Natural deduction and coherence  for non-symmetric linearly distributive categories", "authors": "Robert R. Schneck", "keywords": ["categorical proof theory", "linear logic", "monoidal categories."], "abstract": "In this paper certain proof-theoretic techniques of [BCST] are applied to non-symmetric linearly distributive categories, corresponding to non-commutative negation-free multiplicative linear logic (mLL).  First, the correctness criterion for the two-sided proof nets developed in [BCST] is adjusted to work in the non-commutative setting.  Second, these proof nets are used to represent morphisms in a (non-symmetric) linearly distributive category; a notion of proof-net equivalence is developed which permits a considerable sharpening of the previous coherence results concerning these categories, including a decision procedure for the equality of maps when there is a certain restriction on the units.  In particular a decision procedure is obtained for the equivalence of proofs in non-commutative negation-free mLL without non-logical axioms."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n6/6-06abs.html", "title": "Comparing coequalizer and exact completions", "authors": "M. C. Pedicchio and J. Rosicky", "keywords": ["exact completion", "coequalizer completion", "variety."], "abstract": "We characterize when the coequalizer and the exact completion of a category $\\cal C$  with finite sums and weak finite limits coincide."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n8/6-08abs.html", "title": "Epimorphic regular contexts", "authors": "Robert Raphael", "keywords": ["epimorphism", "von Neumann regular."], "abstract": "A von Neumann regular extension of a semiprime ring naturally defines a epimorphic extension in the category of rings.  These are studied, and four natural examples are considered, two in commutative ring theory, and two in rings of continuous functions."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n7/6-07abs.html", "title": "Enriched Lawvere theories", "authors": "John Power", "keywords": ["Lawvere theory", "monad."], "abstract": "We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category $V$ that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on $V$. Moreover, the $V$-category of models of a Lawvere $V$-theory is equivalent to the $V$-category of algebras for the corresponding $V$-monad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where $V$ is $Cat$, and explain how the correspondence extends to pseudo maps of algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n5/6-05abs.html", "title": "Contravariant Functors on Finite Sets and Stirling Numbers", "authors": "Robert Paré", "keywords": ["Functor", "cardinality", "Stirling numbers."], "abstract": "We characterize the numerical functions which arise as the cardinalities of contravariant functors on finite sets, as those which have a series expansion in terms of Stirling functions. We give a procedure for calculating the coefficients in such series and a concrete test for determining whether a function is of this type. A number of examples are considered."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n4/6-04abs.html", "title": "A Note on Rewriting Theory for Uniqueness of Iteration", "authors": "M. Okada and P. J. Scott", "keywords": ["typed lambda calculus", "rewriting theory", "strong normalization", "Mal'cev operations."], "abstract": "Uniqueness for higher type term constructors in lambda calculi (e.g. surjective pairing for product types, or uniqueness of iterators on the natural numbers) is easily expressed using universally quantified conditional equations.  We use a technique of Lambek [18] involving Mal'cev operators to equationally express uniqueness of iteration (more generally, higher-order primitive recursion) in a simply typed lambda calculus, essentially Godel's T [29,13]. We prove the following facts about typed lambda calculus with uniqueness for primitive recursors:  (i) It is undecidable, (ii) Church-Rosser fails, although ground Church-Rosser holds, (iii) strong normalization (termination) is still valid.  This entails the undecidability of the coherence problem for cartesian closed categories with strong natural numbers objects, as well as providing a natural example of the following computational paradigm:  a non-CR, ground CR, undecidable, terminating rewriting system."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n3/6-03abs.html", "title": "The categorical theory of self-similarity", "authors": "Peter Hines", "keywords": ["Monoidal Categories", "Categorical Trace", "Compact Closure", "Linear Logic", "Inverse Semigroups."], "abstract": "We demonstrate how the identity $N\\otimes N \\cong N$ in a monoidal  category allows us to construct a functor from the full subcategory generated by  $N$ and $\\otimes$ to the endomorphism monoid of the object $N$. This provides a  categorical foundation for one-object analogues of the symmetric monoidal categories used by J.-Y. Girard in his Geometry  of Interaction series of papers, and explicitly described in terms of inverse  semigroup theory in [6,11].\nThis functor also allows the construction of one-object analogues  of other categorical structures. We give the example of one-object  analogues of the categorical trace, and   compact closedness.  Finally, we demonstrate how the categorical theory of self-similarity can be related to the algebraic theory (as presented in  [11]), and Girard's dynamical algebra,  by considering one-object analogues of projections and inclusions."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n1/6-01abs.html", "title": "\\star-Autonomous categories: once more around the track", "authors": "Michael Barr", "keywords": ["duality", "topological algebras", "Chu categories."], "abstract": "This represents a new and more comprehensive approach to the \\star-autonomous categories constructed in the monograph [Barr, 1979]. The main tool in the new approach is the Chu construction.  The main conclusion is that the category of separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n15/7-15abs.html", "title": "Geometric and Higher Order Logic in terms of Abstract Stone Duality", "authors": "Paul Taylor", "keywords": ["locally compact spaces", "compact Hausdor. spaces", "open spaces", "overt spaces", "open maps", "proper maps", "higher order logic", "contravariant powerset", "elementary topos", "subobject classifier", "Euclidean principle", "synthetic domain theory", "monadic adjunction", "pretopos", "quantifiers."], "abstract": "The contravariant powerset, and its generalisations $\\Sigma^X$ to the  lattices of open subsets of a locally compact topological space and  of recursively enumerable subsets of numbers, satisfy the  Euclidean principle that $\\phi\\meet F(\\phi)=\\phi\\meet F(\\top)$.\nConversely, when the adjunction $\\Sigma^{(-)}\\dashv\\Sigma^{(-)}$ is   monadic, this equation implies that $\\Sigma$ classifies some class   of monos, and the Frobenius law   $\\exists x.(\\phi(x)\\meet\\psi)=(\\exists x.\\phi(x))\\meet\\psi)$   for the existential quantifier.\nIn topology, the lattice duals of these equations also hold,  and are related to the Phoa principle in synthetic domain theory.\nThe natural definitions of discrete and Hausdorff spaces correspond  to equality and inequality, whilst the quantifiers considered as  adjoints characterise open (or, as we call them, overt) and  compact spaces.  Our treatment of overt discrete spaces and open  maps is precisely dual to that of compact Hausdorff spaces and  proper maps.\nThe category of overt discrete spaces forms a pretopos and the paper  concludes with a converse of Paré's theorem (that the  contravariant powerset functor is monadic) that characterises  elementary toposes by means of the monadic and Euclidean properties  together with all quantifiers, making no reference to subsets."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n14/7-14abs.html", "title": "A simplicial description of the homotopy category of simplicial groupoids", "authors": "A. R. Garzon, J. G. Miranda and R. Osorio", "keywords": ["closed model category", "path object", "cylinder object", "homotopy relation."], "abstract": "In this paper we use Quillen's model structure given by Dwyer-Kan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case."},
{"url": "http://www.tac.mta.ca/tac/volumes/6/n2/6-02abs.html", "title": "A bicategorical approach to static modules", "authors": "Renato Betti", "keywords": ["Bicategory", "module", "Clifford theory."], "abstract": "The purpose of this paper is to indicate some bicategorical  properties of ring theory. In this interaction, static modules are analyzed."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n13/7-13abs.html", "title": "Solution manifolds for systems of differential equations", "authors": "John F. Kennison", "keywords": ["smooth topos", "differential equation."], "abstract": "This paper defines a solution manifold and a stable submanifold for a system of differential equations. Although we eventually work in the smooth topos, the first two sections do not mention topos theory and should be of interest to non-topos theorists. The paper characterizes solutions in terms of barriers to growth and defines solutions in what are called filter rings (characterized as $C^{\\infty}$-reduced rings in a paper of Moerdijk and Reyes). We examine standardization, stabilization, perturbation, change of variables, non-standard solutions, strange attractors and cycles at infinity."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n12/7-12abs.html", "title": "Quasi-varieties of presheaves", "authors": "Enrico M. Vitale", "keywords": [], "abstract": "In analogy with the varietal case, we give an abstract characterization of those categories occurring as regular epireflective subcategories of presheaf categories such that the inclusion functor preserves small sums."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n11/7-11abs.html", "title": "On the object-wise tensor product of functors to modules", "authors": "Marek Golasinski", "keywords": ["category of canonical orbits", "injective (projective) RC-module", "linearly compact k-module", "tensorpr oduct."], "abstract": "We investigate preserving of projectivity and injectivity by the object-wise tensor product of $R\\Bbb{C}$-modules, where $\\Bbb{C}$ is a small category. In particular, let ${\\cal O}(G,X)$ be the category of canonical orbits of a discrete group $G$, over a $G$-set $X$. We show that projectivity of $R{\\cal O}(G,X)$-modules is preserved by this tensor product. Moreover, if $G$ is a finite group, $X$ a finite $G$-set and $R$ is an integral domain then such a tensor product of two injective $R{\\cal O}(G,X)$-modules is again injective."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n10/7-10abs.html", "title": "Central extensions in Mal'tsev varieties", "authors": "G. Janelidze and G.M. Kelly", "keywords": [], "abstract": "We show that every algebraically-central extension in a Mal'tsev variety - that is, every surjective homomorphism $f : A \\longrightarrow B$ whose kernel-congruence is contained in the centre of $A$, as defined using the theory of commutators - is also a central extension in the sense of categorical Galois theory; this was previously known only for varieties of $\\Omega$-groups, while its converse is easily seen to hold for any congruence-modular variety."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n9/7-09abs.html", "title": "Normal functors and strong protomodularity", "authors": "Dominique Bourn", "keywords": ["abstract normal subobject", "preservation and reflection of normal subobject", "Mal�cev and protomodular categories."], "abstract": "The notion of normal subobject having an intrinsic meaning in any protomodular category, we introduce the notion of normal functor, namely left exact conservative functor which reflects normal subobjects. The point is that for the category {\\bf Gp} of groups the change of base functors, with respect to the fibration of pointed objects, are not only conservative (this is the definition of a protomodular category), but also normal. This leads to the notion of strongly protomodular category. Some of their properties are given, the main one being that this notion is inherited by the slice categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n8/7-08abs.html", "title": "${\\cal M}$-Completeness is seldom monadic over graphs", "authors": "Jiri Adamek and G. M. Kelly", "keywords": ["category", "graph", "limit", "adjunction."], "abstract": "For a set ${\\cal M}$ of graphs the category ${\\bf Cat}_{\\cal M}$ of all ${\\cal M}$-complete categories and all strictly ${\\cal M}$-continuous functors is known to be monadic over ${\\bf Cat}$.  The question of monadicity of ${\\bf Cat}_{\\cal M}$ over the category of graphs is known to have an affirmative answer when ${\\cal M}$ specifies either (i) all finite limits, or (ii) all finite products, or (iii) equalizers and terminal objects, or (iv) just terminal objects.  We prove that, conversely, these four cases are (essentially) the only cases of monadicity of $\\Cat_\\M$ over the category of graphs, provided that ${\\cal M}$ is a set of finite graphs containing the empty graph."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n7/7-07abs.html", "title": "On the monadicity of categories with chosen colimits", "authors": "G. M. Kelly and Stephen Lack", "keywords": ["monadicity", "categories with limits", "weighted limits", "enriched categories."], "abstract": "There is a 2-category {\\cal J}{\\bf-Colim} of small categories equipped with a choice of colimit for each diagram whose domain $J$ lies in a given small class {\\cal J} of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2-functor from {\\cal J}{\\bf-Colim} to the 2-category {\\bf Cat} of small categories is known to be monadic. We extend this result by considering not just conical colimits, but general weighted colimits; not just ordinary categories but enriched ones; and not just small classes of colimits but large ones; in this last case we are forced to move from the 2-category {\\cal V}{\\bf-Cat} of small {\\cal V}-categories to {\\cal V}-categories with object-set in some larger universe. In each case, the functors preserving the colimits in the usual ``up-to-isomorphism'' sense are recovered as the {\\em pseudomorphisms} between algebras for the 2-monad in question."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n6/7-06abs.html", "title": "Balanced Coalgebroids", "authors": "Paddy McCrudden", "keywords": ["Symmetric monoidal bicategories", "balanced Vop-categories", "coalgebras", "quantum groups."], "abstract": "A balanced coalgebroid is a ${\\cal V}^{op}$-category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of representations. This includes the reconstruction of dual quasi-bialgebras, quasi-triangular dual quasi-bialgebras, and balanced quasi-triangular dual quasi-bialgebras; the latter of which is a quantum group when equipped with a compatible antipode.  As an application we construct a balanced coalgebroid whose category of representations is equivalent to the symmetric monoidal category of chain complexes. The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n5/7-05abs.html", "title": "Factorization systems for symmetric cat-groups", "authors": "S. Kasangian and E.M. Vitale", "keywords": ["Cat-groups", "factorization systems in a 2-category."], "abstract": "This paper is a first step in the study of symmetric cat-groups as the 2-dimensional analogue of abelian groups. We show that a morphism of symmetric cat-groups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2-categorical sense, in the 2-category of symmetric cat-groups. An application to exact sequences is given."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n4/7-04abs.html", "title": "On saturated classes of morphisms", "authors": "Carles Casacuberta and Armin Frei", "keywords": ["saturation", "orthogonality", "monad", "shape", "category of fractions."], "abstract": "The term ``saturated,'' referring to a class of morphisms  in a category, is used in the literature for two nonequivalent concepts. We make precise the relationship between these two concepts and show that the class of equivalences associated with any monad is saturated in both senses."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n3/7-03abs.html", "title": "Pure morphisms of commutative rings are  effective descent morphisms for modules -- a new proof", "authors": "Bachuki Mesablishvili", "keywords": ["Pure morphisms", "(effective) Descent morphisms", "Split coequalizers."], "abstract": "The purpose of this paper is to give a new proof of the Joyal-Tierney theorem (unpublished), which asserts that a morphism $f:R\\rightarrow S$ of commutative rings is an effective descent morphism for modules if and only if $f$ is pure as a morphism of $R$-modules.\n"},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n2/7-02abs.html", "title": "A Pseudo Representation Theorem for Various Categories of Relations", "authors": "M. Winter", "keywords": ["Relation Algebra", "Dedekind category", "Allegory", "Representability", "Matrix Algebra."], "abstract": "It is well-known that, given a Dedekind category  {\\cal R} the category of (typed) matrices with coefficients from  {\\cal R} is a Dedekind category with arbitrary relational sums. In this paper we show that under slightly stronger assumptions the converse is also true. Every atomic Dedekind category {\\cal R} with relational sums and subobjects is equivalent to a category of matrices over a suitable basis. This basis is the full proper subcategory induced by the integral objects of {\\cal R}. Furthermore, we use our concept of a basis to extend a known result from the theory of heterogeneous relation algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n23/8-23abs.html", "title": "$V$-Cat is locally presentable or locally bounded if $V$ is so", "authors": "G. M. Kelly and Stephen Lack", "keywords": ["enriched category", "locally presentable category", "locally bounded category."], "abstract": "We show, for a monoidal closed category $V = (V_0,\\otimes,I)$, that the category $V$-Cat of small $V$-categories is locally $\\lambda$-presentable if $V_0$ is so, and that it is locally $\\lambda$-bounded if the closed category $V$ is so, meaning that $V_0$   is locally $\\lambda$-bounded and that a side condition involving the monoidal structure is satisfied."},
{"url": "http://www.tac.mta.ca/tac/volumes/7/n1/7-01abs.html", "title": "On Branched Covers in Topos Theory", "authors": "Jonathon Funk", "keywords": [], "abstract": "We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small    coproducts in the including topos. Our main result is a description of the covers of this subtopos as a category of fractions of branched covers, in the sense of Fox, of the including topos. We also have some new  results concerning the general theory of KZ-doctrines, such as the closure under composition of discrete fibrations for a KZ-doctrine, in the sense of Bunge and Funk."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n22/8-22abs.html", "title": "The extensive completion of a distributive category", "authors": "J.R.B. Cockett and Stephen Lack", "keywords": ["distributive category", "extensive category", "free construction."], "abstract": "A category with finite products and finite coproducts is said to be  distributive if the canonical map  $A \\times B + A \\times C \\to A \\times (B + C)$ is invertible for all objects $A$, $B$, and $C$. Given a distributive category $\\cal D$, we describe a universal functor $\\cal D \\to \\cal D_{ex}$ preserving finite products and finite coproducts, for which $\\cal D_{ex}$ is  extensive; that is, for all objects $A$ and $B$ the functor $\\cal D_{ex}/A \\times \\cal D_{ex}/B \\to \\cal D_{ex}/(A + B)$ is an equivalence of categories.\nAs an application, we show that a distributive category $\\cal D$ has a full distributive embedding into the product of an extensive category with products and a distributive preorder."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n21/8-21abs.html", "title": "Closure operators in exact completions", "authors": "Matias Menni", "keywords": ["Exact completions", "closure operators", "toposes."], "abstract": "In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions.  The study of separated objects in exact completions will then allow us to give conceptual proofs of local cartesian closure of different categories of pseudo equivalence relations. Finally, we characterize when certain categories of sheaves are toposes."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n20/8-20abs.html", "title": "On Functors Which Are Lax Epimorphisms", "authors": "Jiri Adamek, Robert El Bashir, Manuela Sobral, Jiri Velebil", "keywords": ["lax epimorphism."], "abstract": "We show that lax epimorphisms in the category Cat are precisely the functors $P : {\\cal E} \\to {\\cal B}$ for which the functor $P^{*}: [{\\cal B}, Set] \\to [{\\cal E}, Set]$ of composition with $P$ is fully faithful. We present two other characterizations. Firstly, lax epimorphisms are precisely the ``absolutely dense'' functors, i.e., functors $P$ such that every object $B$ of ${\\cal B}$ is an absolute colimit of all arrows $P(E)\\to B$ for $E$ in ${\\cal E}$. Secondly, lax epimorphisms are precisely the functors $P$ such that for every morphism $f$ of ${\\cal B}$ the category of all factorizations through objects of $P[{\\cal E}]$ is connected.\nA relationship between pseudoepimorphisms and lax epimorphisms is discussed."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n19/8-19abs.html", "title": "A sheaf-theoretic view of loop spaces", "authors": "Mark W. Johnson", "keywords": ["loop spaces", "spectra", "Quillen closed model categories", "enriched sheaves."], "abstract": "The context of enriched sheaf theory introduced in the author's thesis     provides a convenient viewpoint for models of the stable homotopy     category as well as categories of finite loop spaces.     Also, the languages     of algebraic geometry and algebraic topology have been interacting     quite heavily in recent years, primarily due to the work of     Voevodsky and that of Hopkins.  Thus, the language of Grothendieck     topologies is becoming a necessary tool for the algebraic topologist.     The current article is intended to give a somewhat relaxed     introduction to this language of sheaves in a topological     context, using familiar examples such as n-fold loop spaces and     pointed G-spaces.  This language also includes the     diagram categories of spectra      as well as spectra in the sense     of Lewis, which will be discussed in some detail."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n18/8-18abs.html", "title": "Cartesian closed topological hull of the construct of closure spaces", "authors": "V. Claes, E. Lowen-Colebunders and G. Sonck", "keywords": ["closure space", "cartesian closedness", "function space", "cartesian closed topological hull."], "abstract": "A cartesian closed topological hull of the construct CLS of closure spaces and continuous maps is constructed. The construction is performed in two steps. First a cartesian closed extension L of CLS is obtained. We apply a method worked out by J. Adamek and J. Reiterman  for constructing extensions of constructs that in some sense ``resemble'' the construct of uniform spaces. Secondly, within this extension L the cartesian closed topological hull L* of CLS is characterized as a full subconstruct. In order to find the internal characterization of the objects of L* we produce a concrete functor to the category of power closed collections based on CLS as introduced by J. Adamek, J. Reiterman and G.E. Strecker."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n17/8-17abs.html", "title": "Essential localizations and infinitary exact completion", "authors": "Enrico M. Vitale", "keywords": [], "abstract": "We prove the universal property of the infinitary exact completion of a category with weak small limits. As an application, we slightly weaken the conditions characterizing essential localizations of varieties (in particular, of module categories) and of presheaf categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n16/8-16abs.html", "title": "Perfect Maps are Exponentiable - Categorically", "authors": "Gunther Richter and Walter Tholen", "keywords": ["proper map", "separated map", "perfect map", "quotient map", "exponentiable map."], "abstract": "A categorical proof of the statement given by the title is provided, in generalization of a result for topological spaces proved recently by Clementino, Hofmann and Tholen."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n14/8-14abs.html", "title": "A categorical genealogy for the congruence distributive property", "authors": "Dominique Bourn", "keywords": ["congruence distributivity", "Mal�cev", "arithmetical and protomodular categories."], "abstract": "In the context of Mal'cev categories, a left exact root for the congruence distributive property is given and investigated, namely the property that there is no non trivial internal group inside the fibres of the fibration of pointed objects. Indeed, when moreover the basic category $\\mathbb{C}$ is Barr exact, the two previous properties are shown to be equivalent."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n13/8-13abs.html", "title": "How large are left exact functors?", "authors": "J. Adamek, V. Koubek  and V. Trnkova", "keywords": ["left exact functor", "small functor", "regular ultrafilter."], "abstract": "For a broad collection of categories $\\cal K$, including all presheaf categories, the following statement is proved to be consistent: every left exact (i.e. finite-limits preserving) functor from $\\cal K$ to $\\Set$ is small, that is, a small colimit of representables. In contrast, for the (presheaf) category ${\\cal K}=\\Alg(1,1)$ of unary algebras we construct a functor from $\\Alg(1,1)$ to $\\Set$ which preserves finite products and is not small. We also describe all left exact set-valued functors as directed unions of ``reduced representables'', generalizing reduced products."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n15/8-15abs.html", "title": "Pseudogroupoids and commutators", "authors": "George Janelidze and M.Cristina Pedicchio", "keywords": ["variety", "pregroupoid", "commutator."], "abstract": "We develop a new approach to Commutator theory based on the theory of internal categorical structures, especially of so called pseudogroupoids. It is motivated by our previous work on internal categories and groupoids in congruence modular varieties."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n12/8-12abs.html", "title": "Combinatorics of branchings in higher dimensional automata", "authors": "Philippe Gaucher", "keywords": ["cubical set", "thin element", "globular higher dimensional category", "branching", "higher dimensional automata", "concurrency", "homology theory."], "abstract": "We explore the combinatorial properties of the branching areas of   execution paths in higher dimensional automata. Mathematically, this   means that we investigate the combinatorics of the negative corner    (or branching) homology of a globular $\\omega$-category and the        combinatorics of a new homology theory called the reduced branching   homology. The latter is the homology of the quotient of the   branching complex by the sub-complex generated by its thin elements.   Conjecturally it coincides with the non reduced theory for higher   dimensional automata, that is $\\omega$-categories freely generated   by precubical sets. As application, we calculate the branching   homology of some $\\omega$-categories and we give some invariance   results for the reduced branching homology. We only treat the   branching side. The merging side, that is the case of merging areas   of execution paths is similar and can be easily deduced from the   branching side."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n11/8-11abs.html", "title": "Limites inductives point par point dans les catégories accessibles", "authors": "Pierre Ageron", "keywords": ["sketches", "accessible categories", "connected objects", "pseudofiltered colimits."], "abstract": "We give an abstract characterization of the categories of models of sketches all of whose distinguished cones are based on  connected (resp. non empty) categories.\nNous donnons une caractérisation abstraite des catégories de modèles des esquisses dont tous les cônes projectifs distingués sont d'indexation connexe (resp. non vide)."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n9/8-09abs.html", "title": "How algebraic is algebra?", "authors": "J. Adamek, F. W. Lawvere and J. Rosicky", "keywords": ["variety", "exact category", "pseudomonad."], "abstract": "The 2-category  VAR of finitary varieties is not varietal over CAT. We introduce the concept of an algebraically exact category and prove that the 2-category ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all operations with rank. Every algebraically exact category $\\cal K$ is complete, exact, and has filtered colimits which (a) commute with finite limits and (b) distribute over products; besides (c) regular epimorphisms in $\\cal K$ are product-stable. It is not known whether (a) - (c)  characterize algebraic exactness. An equational hull of VAR w.r.t. all operations is also discussed."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n4/8-04abs.html", "title": "On Mackey topologies in topological abelian groups", "authors": "Michael Barr and Heinrich Kleisli", "keywords": ["Mackey topologies", "duality", "topological abelian groups."], "abstract": "Let $\\cal C$ be a full subcategory of the category of topological abelian groups and SP$\\cal C$ denote the full subcategory of subobjects of products of objects of $\\cal C$.  We say that SP$\\cal C$ has Mackey coreflections if there is a functor that assigns to each object $A$ of SP$\\cal C$ an object $\\tau A$ that has the same group of characters as $A$ and is the finest topology with that property.  We show that the existence of Mackey coreflections in SP$\\cal C$ is equivalent to the injectivity of the circle with respect to topological subgroups of groups in $\\cal C$.\n"},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n8/8-08abs.html", "title": "Finite sets and symmetric simplicial sets", "authors": "Marco Grandis", "keywords": ["Simplicial sets", "monoidal categories", "generators and relations."], "abstract": "The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the  augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite ordinals, by the existence of a universal symmetric monoid, or by generators and relations. The latter provides a definition of symmetric simplicial sets by faces, degeneracies and transpositions, under suitable relations."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n7/8-07abs.html", "title": "Duality for Simple $\\omega$-Categories and Disks", "authors": "Mihaly Makkai and Marek Zawadowski", "keywords": ["omega-category", "globular set", "omega-graph", "disk", "schizophrenic object", "duality", "theta-category."], "abstract": "A. Joyal has introduced the category $\\cal D$ of the so-called finite disks, and used it to define the concept of $\\theta$-category, a notion of weak $\\omega$-category. We introduce the notion of an $\\omega$-graph being  composable  (meaning roughly that 'it has a unique composite'), and call an $\\omega$-category simple if it is freely generated by a composable $\\omega$-graph.  The category $\\cal S$ of simple $\\omega$-categories is a full subcategory of the category, with strict $\\omega$-functors as morphisms, of all $\\omega$-categories. The category $\\cal S$ is a key ingredient in another concept of weak $\\omega$-category, called protocategory. We prove that $\\cal D$ and $\\cal S$ are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of $\\theta$-category and that of protocategory are equivalent in a suitable sense. We also prove that composable $\\omega$-graphs coincide with the $\\omega$-graphs of the form $T^*$ considered by M.Batanin, which were characterized by R. Street and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free $\\omega$-category on a globular set plays an important role in our paper. We give a self-contained presentation of Batanin's construction that suits our purposes."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n6/8-06abs.html", "title": "On the pullback stability of a quotient map with respect to a closure  operator", "authors": "Lurdes Sousa", "keywords": ["closure operator", "quotient", "pullback", "closed morphism", "open morphism", "final morphism."], "abstract": "There are well-known characterizations of the hereditary quotient maps in the category of topological spaces, (that is, of quotient maps stable under pullback along embeddings), as well as of universal quotient maps (that is, of quotient maps stable under pullback). These are precisely the so-called pseudo-open maps, as shown by Arhangel'skii, and the bi-quotient maps of Michael, as shown by Day and Kelly, respectively. In this paper hereditary and stable quotient maps are characterized in the broader context given by a category equipped with a closure operator. To this end, we derive explicit formulae and conditions for the closure in the codomain of such a quotient map in terms of the closure in its domain."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html", "title": "On sifted colimits and generalized varieties", "authors": "J. Adamek and J. Rosicky", "keywords": [], "abstract": "Filtered colimits, i.e., colimits over schemes $\\cal D$ such that $\\cal D$-colimits in $\\Set$ commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes $\\cal D$ such that $\\cal D$-colimits in $\\Set$ commute with finite products. An important example: reflexive coequalizers are sifted colimits. Generalized varieties are defined as free completions of small categories under sifted-colimits (analogously to finitely accessible categories which are free filtered-colimit completions of small categories). Among complete categories, generalized varieties are precisely the varieties. Further examples: category of fields, category of linearly ordered sets, category of nonempty sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n2/8-02abs.html", "title": "Exponentiable Morphisms: Posets, Spaces, Locales, and Grothendieck Toposes", "authors": "Susan Niefield", "keywords": ["presheaf topos", "poset", "locale", "exponentiable", "factorization lifting", "metastably locally compact", "discrete opfibration."], "abstract": "In this paper, we consider those morphisms $p : P\\to B$ of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes.  In particular, we show that a necessary condition is that the induced map $p^{\\downarrow} : P^{\\downarrow}\\to B^{\\downarrow}$ is exponentiable in the category of topological spaces, where $P^{\\downarrow}$ is the space whose points are elements of $P$ and open sets are downward closed subsets of $P$.  Along the way, we show that $p^{\\downarrow} : P^{\\downarrow}\\to B^{\\downarrow}$ is exponentiable if and only if $p : P\\to B$ is exponentiable in the category of posets and satisfies an additional compactness condition.  The criteria for exponentiability of morphisms of posets is related to (but weaker than) the factorization-lifting property for exponentiability of morphisms in the category of small categories (considered independently by Giraud and Conduché)."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n1/8-01abs.html", "title": "$n$-Permutable locally finitely presentable categories", "authors": "Marino Gran and Maria Cristina Pedicchio", "keywords": ["Locally finitely presentable categories", "n-permutable and Maltsev varieties", "quasivarieties."], "abstract": "We characterize $n$-permutable locally finitely presentable categories  $Lex[{\\mathcal C}^{op}, Set]$ by a condition on the dual of the essentially algebraic theory $\\mathcal C^{op}$. We apply these results to exact Maltsev categories as well as to $n$-permutable quasivarieties and varieties."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html", "title": "Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories", "authors": "John W. Duskin", "keywords": ["bicategory", "simplicial set", "nerve of a bicategory."], "abstract": "To a bicategory B  (in the sense of Bénabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we call a 2-dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain  2-dimensional Postnikov complexes which satisfy certain  restricted `exact horn-lifting' conditions whose satisfaction is controlled by (and here defines)  subsets of (abstractly) invertible 2 and 1-simplices. Those complexes which have, at minimum, their degenerate 2-simplices always invertible and have an invertible 2-simplex $\\chi_2^1(x_{12}, x_{01})$ present for each `composable pair' $(x_{12}, \\_ , x_{01}) \\in \\mhorn_2^1$ are exactly the nerves of bicategories. At the other extreme, where all 2 and 1-simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions >2. These are exactly the nerves of bigroupoids - all 2-cells are isomorphisms and all 1-cells are equivalences."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n9/9-09abs.html", "title": "The alternation hierarchy for the theory of $\\mu$-lattices", "authors": "Luigi Santocanale", "keywords": ["free lattices", "free �-lattices", "fixed points", "parity games."], "abstract": "The alternation hierarchy problem asks whether every $\\mu$-term   $\\phi$, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a $\\mu$-term where the number of nested fixed points of a different type is bounded by a constant independent of $\\phi$.\nIn this paper we give a proof that the alternation hierarchy for the theory of $\\mu$-lattices is strict, meaning that such a constant does not exist if $\\mu$-terms are built up from the basic lattice operations and are interpreted as expected.  The proof relies on the explicit characterization of free $\\mu$-lattices by means of games and strategies."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n8/9-08abs.html", "title": "Centrality and normality in protomodular categories", "authors": "Dominique Bourn and Marino Gran", "keywords": ["Maltsev and protomodular categories", "abstract normal subobject", "centrality of equivalence relations."], "abstract": "We analyse the classical property of centrality of  equivalence relations in terms of normal monomorphisms. For this purpose, the internal structure of connector is introduced, allowing to clarify classical results in Maltsev categories and to prove new ones in protomodular categories. This approach allows to work in the general context of finitely complete categories, without requiring the usual Barr exactness assumption."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n7/9-07abs.html", "title": "The Hurwitz action and braid group orderings", "authors": "Jonathon Funk", "keywords": [], "abstract": "In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2-sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points. Our approach to ramified covers is based on cosheaf spaces, which are closely related to Fox's complete spreads. A feature of a ramification groupoid is that it carries a certain order structure. The Artin group of braids of $n$ strands has an order-invariant action in the ramification groupoid of the sphere with $n+1$ marked-points. Left-invariant linear orderings of the braid group such as the Dehornoy ordering may be retrieved. Our work extends naturally to the braid group on countably many generators. In particular, we show that the underlying set of a free group on countably many generators (minus the identity element) can be linearly ordered in such a way that the classical Artin representation of a braid as an automorphism of the free group is an order-preserving action."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n6/9-06abs.html", "title": "Categorical domain theory: Scott topology, powercategories, coherent categories", "authors": "Panagis Karazeris", "keywords": ["accessible category", "Scott complete category", "classifying topos", "powerdomain", "coherent domain", "perfect topos", "free cocompletion."], "abstract": "In the present article we continue recent work in the direction of domain theory were certain (accessible) categories are used as generalized domains. We discuss the possibility of using certain presheaf toposes as generalizations of the Scott topology at this level. We show that the toposes associated with Scott complete categories are injective with respect to dense inclusions of toposes. We propose analogues of the upper and lower powerdomain in terms of the Scott topology at the level of categories. We show that the class of finitely accessible categories is closed under this generalized upper powerdomain construction (the respective result about the lower powerdomain construction is essentially known). We also treat the notion of ``coherent domain'' by introducing two possible notions of coherence for a finitely accessible category (qua generalized domain). The one of them imitates the stability of the compact saturated sets under intersection and the other one imitates the so-called ``2/3 SFP'' property. We show that the two notions are equivalent. This amounts to characterizing the small categories whose free cocompletion under finite colimits has finite limits."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n5/9-05abs.html", "title": "Classifying spaces of categories and term rewriting", "authors": "Maurizio G. Citterio", "keywords": ["Collapsing scheme", "simplicial set", "classifying space", "rewrite system."], "abstract": "In this paper we show how collapsing schemes can give us information on the homotopy type of the classifying space of a small category, when this category is presented by a complete rewrite system."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n4/9-04abs.html", "title": "A Note on Actions of a Monoidal Category", "authors": "G. Janelidze and G.M. Kelly", "keywords": ["monoidal category", "action", "enriched category", "monoid", "monad", "adjunction."], "abstract": "An action $* : \\cal V \\times \\cal A \\to \\cal A$ of a monoidal category  $\\cal V$ on a category $\\cal A$ corresponds to a strong monoidal functor  $F : \\cal V \\to [\\cal A,\\cal A]$ into the monoidal category of endofunctors of $\\cal A$.  In many practical cases, the ordinary functor  $f : \\cal V \\to [cal \\A, \\cal A]$ underlying the monoidal $F$ has a right adjoint $g$; and when this is so, $F$ itself has a right adjoint $G$ as a monoidal functor - so that, passing to the categories of monoids (also called ``algebras'') in $\\cal V$ and in $[\\cal A, \\cal A]$, we have an adjunction $Mon F$ left adjoint to  $Mon G$  between the category $Mon \\cal V$ of monoids in $\\cal V$ and the category  $Mon [\\cal A, \\cal A] = Mnd \\cal A$ of monads on $\\cal A$.  We give  sufficient conditions for the   existence of the right adjoint $g$, which involve the existence of right adjoints for the functors $X * - $ and $ * A$, and make  $\\cal A$ (at least when $\\cal V$ is symmetric and closed) into a tensored and cotensored $cal \\V$-category ${\\bf A}$.  We give explicit formulae, as large ends, for the right adjoints $g$ and $Mon G$, and also for some related right adjoints, when they exist; as well as another explicit expression for $Mon G$ as a large limit, which uses a new representation of any (large) limit of monads of two special kinds, and an analogous result for general endofunctors."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n3/9-03abs.html", "title": "Simplicial torsors", "authors": "Tibor Beke", "keywords": ["torsors", "simplicial sheaves", "calculus of fractions", "Grothendieck toposes."], "abstract": "The interpretation by Duskin and Glenn of abelian sheaf cohomology as connected components of a category of torsors is extended to homotopy classes.  This is simultaneously an extension of Verdier's version of Cech cohomology to homotopy."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n2/9-02abs.html", "title": "A note on exactness and stability in homotopical algebra", "authors": "Marco Grandis", "keywords": ["Homotopy theory", "abstract homotopy theory", "2-categories", "cofibrations", "fibre spaces", "chain complexes."], "abstract": "Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of `homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be given for very general `categories with homotopies' having  homotopy kernels and cokernels, but become more interesting under suitable `stability' hypotheses, satisfied - in particular - by chain complexes. It is then possible to measure the default of homotopical exactness of a sequence by the homotopy type of a certain object, a sort of `homotopical homology'."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/20/10-20abs.html", "title": "A duality relative to a limit doctrine", "authors": "C. Centazzo and E.M. Vitale", "keywords": ["limit doctrines", "locally D-presentable categories", "duality."], "abstract": "We give a unified proof of Gabriel-Ulmer duality for locally finitely presentable categories, Adamek-Lawvere-Rosicky duality for varieties and Morita duality for presheaf categories. As an application, we compare presheaf categories and varieties."},
{"url": "http://www.tac.mta.ca/tac/volumes/9/n1/9-01abs.html", "title": "Infinitesimal aspects of the Laplace operator", "authors": "Anders Kock", "keywords": ["Laplacian", "harmonic", "conformal", "synthetic dfferential geometry."], "abstract": "In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the  second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a function is affine on it if and only if it is harmonic."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n10/8-10abs.html", "title": "Localization of V-categories", "authors": "Bjorn Ian Dundas", "keywords": [], "abstract": "Let $V$ be a symmetric monoidal closed category with a suitably compatible simplicial model category structure. We show how to extend Dwyer and Kan's notion of simplicial localization to $V$-categories.  This may for instance be applied to the case where our categories are enriched in suitable models for spectra."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/19/10-19abs.html", "title": "Opmonoidal monads", "authors": "Paddy McCrudden", "keywords": ["Hopf Monad", "Eilenberg Moore Algebras", "Multicategories", "Structure and Semantics."], "abstract": "Hopf monads are identified with monads in the 2-category Opmon of monoidal categories, opmonoidal functors and transformations. Using Eilenberg-Moore objects, it is shown that  for a Hopf monad $S$, the categories Alg(Coalg($S$)) and Coalg(Alg($S$)) are canonically isomorphic.  The monadic arrows Opmon are then characterized. Finally, the theory of multicategories  and a generalization of structure and semantics are used to identify the categories of algebras of Hopf monads."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/18/10-18abs.html", "title": "HSP subcategories of Eilenberg-Moore algebras", "authors": "Michael Barr", "keywords": ["Birkhoff subcategories", "factorizations", "reflective subcategories."], "abstract": "Given a triple T on a complete category C and a factorization system E/M on the category of algebras, we show there is a 1-1 correspondence between full subcategories of the category of algebras that are closed under U-split epimorphisms, products, and M-subobjects and triple morphisms T -> S for which the  induced natural transformation between free functors belongs to E."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/17/10-17abs.html", "title": "Entropic Hopf algebras and models of non-commutative logic", "authors": "Richard F. Blute, Francois Lamarche, Paul Ruet", "keywords": ["Linear logic", "monoidal categories", "Hopf algebras."], "abstract": "We give a definition of categorical model for the multiplicative fragment of non-commutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a  partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting.\nIt has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the Tannaka-Krein reconstruction theorem is extended to the entropic setting."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/13/10-13abs.html", "title": "Subspaces in abstract Stone duality", "authors": "Paul Taylor", "keywords": ["axiom of comprehension", "subtype", "typed lambda calculus", "Stone duality", "subspace topology", "locally compact spaces", "nucleus of a locale", "injective object", "monadic adjunction", "Beck�s theorem."], "abstract": "By abstract Stone duality we mean that the    topology or contravariant powerset functor, seen as a self-adjoint    exponential $\\Sigma^{(-)}$ on some category,    is monadic.    Using Beck's theorem, this means that certain equalisers exist      and carry the subspace topology.    These subspaces are encoded by idempotents that play    a role similar to that of nuclei in locale theory.\nParé showed that any elementary topos has this duality,    and we prove it intuitionistically for   the category of locally compact locales.\nThe paper is largely concerned with the construction of such a category    out of one that merely has powers of some fixed object $\\Sigma$.    It builds on Sober Spaces and Continuations,    where the related but weaker notion of abstract sobriety was considered.    The construction is done first by formally adjoining certain equalisers    that $\\Sigma^{(-)}$ takes to coequalisers,       then using Eilenberg-Moore algebras,    and finally presented as a lambda calculus similar to    the axiom of comprehension in set theory.\nThe comprehension calculus has a normalisation theorem,    by which every type can be embedded as a subspace of a type formed    without comprehension, and terms also normalise in a simple way.    The symbolic and categorical structures are thereby shown to be    equivalent.\nFinally, sums and certain quotients are constructed    using the comprehension calculus, giving an extensive category."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/16/10-16abs.html", "title": "Simultaneously Reflective And Coreflective Subcategories of Presheaves", "authors": "Robert El Bashir and Jiri Velebil", "keywords": ["monoidal category", "reflection", "coreflection", "Morita equivalence."], "abstract": "It is proved that any category $\\cal{K}$ which is equivalent to a simultaneously reflective and coreflective full subcategory of presheaves $[\\cal{A}^{op},Set]$, is itself equivalent to the category of the form $[\\cal{B}^{op},Set]$ and the inclusion is induced by a functor $\\cal{A} \\to \\cal{B}$ which is surjective on objects. We obtain a characterization of such functors.\nMoreover, the base category $Set$ can be replaced with any symmetric monoidal closed category $V$ which is complete and cocomplete, and then analogy of the above result holds   if we replace categories by $V$-categories and functors by $V$-functors.\nAs a consequence we are able to derive well-known results on simultaneously reflective and coreflective categories of sets, Abelian groups, etc."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/15/10-15abs.html", "title": "The cyclic spectrum of a Boolean flow", "authors": "John F. Kennison", "keywords": ["flow", "discrete dynamical system", "topos", "Cole spectrum", "strange attractor."], "abstract": "This paper defines flows (or discrete dynamical systems) and cyclic flows in a category and investigates how the trajectories of a point might approach a cycle. The paper considers cyclic flows in the categories of Sets and of Boolean algebras and their duals and characterizes the Stone representation of a cyclic flow in Boolean algebras. A cyclic spectrum is constructed for Boolean flows. Examples include attractive fixpoints, repulsive fixpoints, strange attractors and the logistic equation."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/14/10-14abs.html", "title": "Directed homotopy theory, II. Homotopy constructs", "authors": "Marco Grandis", "keywords": ["homotopy theory", "homotopical algebra", "directed homotopy", "homotopy pushouts", "homotopy pullbacks", "mapping cones", "homotopy fibres."], "abstract": "Directed Algebraic Topology studies phenomena where privileged directions appear, derived from the analysis of concurrency, traffic networks, space-time models, etc.\nThis is the sequel of a paper, `Directed homotopy theory, I. The fundamental category', where we introduced  directed spaces, their non reversible homotopies and their fundamental category. Here we study some basic constructs of homotopy, like homotopy pushouts and pullbacks, mapping cones and homotopy fibres, suspensions and loops, cofibre and fibre sequences."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/12/10-12abs.html", "title": "Sober spaces and continuations", "authors": "Paul Taylor", "keywords": ["observable type", "predicate transformer", "continuation passing style", "sober space", "locally compact space", "locally quasi-compact space", "locale", "prime filter", "Sierpinski space", "schizophrenic object", "Stone duality", "strong monad", "Kleisli category", "premonoidal category", "prime lambda term", "sober lambda calculus", "theory of descriptions", "general recursive function."], "abstract": "A topological space is sober if     it has exactly the points that are dictated by its open sets.     We explain the analogy with the way in which computational values     are determined by the observations that can be made of them.     A new definition of sobriety is formulated in terms of     lambda calculus and elementary category theory,     with no reference to lattice structure,     but, for topological spaces, this coincides with the     standard lattice-theoretic definition.     The primitive symbolic and categorical structures are extended     to make their types sober.     For the natural numbers, the additional structure provides     definition by description and general recursion.\nWe use the same basic categorical construction     that Thielecke, Fuhrmann and Selinger use to study continuations,     but our emphasis is completely different:     we concentrate on the fragment of their calculus     that excludes computational effects,     but show how it nevertheless defines new denotational values.     Nor is this ``denotational semantics of continuations     using sober spaces'', though that could easily be derived.\nOn the contrary, this paper provides the underlying $\\lambda$-calculus     on the basis of which abstract Stone duality will     re-axiomatise general topology.     The leading model of the new axioms is the category of locally compact     locales and continuous maps."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/11/10-11abs.html", "title": "Derived Operations in Goguen Categories", "authors": "Michael Winter", "keywords": ["Goguen category", "Fuzzy Relation", "Dedekind category."], "abstract": "Goguen categories were introduced in as a suitable categorical description of ${\\mathcal L}$-fuzzy relations, i.e., of relations taking values from an arbitrary complete Brouwerian lattice ${\\mathcal L}$ instead of the unit interval $[0,1]$ of the real numbers. In this paper we want to study operations on morphisms of a Goguen category which are derived from suitable binary functions on the underlying lattice of scalar elements, i.e., on the abstract counterpart of ${\\mathcal L}$."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/7/10-07abs.html", "title": "More on injectivity in locally presentable categories", "authors": "J. Rosicky, J. Adamek and F. Borceux", "keywords": ["locally presentable category", "injectivity class", "geometric logic."], "abstract": "a\nInjectivity with respect to morphisms having $\\lambda$-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, $\\lambda$-directed colimits, and $\\lambda$-pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 785-804). In contrast, for geometric logic an example is found of a class closed under directed colimits and pure subobjects, but not axiomatizable by a geometric theory. A more technical characterization of axiomatizable classes in geometric logic is presented."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/9/10-09abs.html", "title": "On some properties of pure morphisms of commutative rings", "authors": "Bachuki Mesablishvili", "keywords": ["Indexed categories", "effective descent morphisms", "pure morphisms."], "abstract": "We prove that pure morphisms of commutative rings are effective $A$-descent morphisms where $A$ is a  (COMMUTATIVE RINGS)$^op$-indexed category given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective modules.\n"},
{"url": "http://www.tac.mta.ca/tac/volumes/10/10/10-10abs.html", "title": "Change of base, Cauchy completeness and reversibility", "authors": "Anna Labella and Vincent Schmitt", "keywords": ["Enriched categories", "two-sided enrichments", "change of base", "reversibility", "Cauchy completion", "sheaves."], "abstract": "We investigate the effect on Cauchy complete objects of the change of base 2-functor ${\\cal V}-Cat \\rightarrow {\\cal W}-Cat$ induced by a two-sided  enrichment ${\\cal V} \\rightarrow {\\cal W}$. We restrict our study to the case of locally partially ordered bases. The reversibility notion introduced  by Walters is extended to two-sided enrichments and Cauchy completion. We show that a reversible left adjoint two-sided enrichment $F: {\\cal V} \\rightarrow {\\cal W}$ between locally partially ordered reversible bicategories induces an adjunction $F_{\\sim} \\dashv F^{\\sim}: \\VSkCRcCat \\rightharpoonup \\WSkCRcCat$ between sub-categories of skeletal and Cauchy-reversible complete enrichments. We give two applications: sheaves over locales and group actions.\n"},
{"url": "http://www.tac.mta.ca/tac/volumes/10/8/10-08abs.html", "title": "Colocalizations and their realizations as spectra", "authors": "Friedrich W. Bauer", "keywords": ["Colocalizations", "chain functors", "realizations of chain functors as spectra."], "abstract": "Every chain functor $A_*$, admits a $L$-colocalization  $A^L_*$ which (in contrast to the case of $L$-localizations) in general does not allow a realization as a spectrum (even if $A_*$ stems from a spectrum itself).  The $[E, ]_*$- colocalization of A. K. Bousfield is retrieved as a special case of a general colocalization process for chain functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/6/10-06abs.html", "title": "Coherence for Factorization Algebras", "authors": "Robert Rosebrugh and R.J. Wood", "keywords": ["coherence", "factorization algebra."], "abstract": "For the 2-monad $((-)^2,I,C)$ on CAT, with unit $I$ described by identities and multiplication $C$ described by composition, we show that a functor $F : {\\cal K}^2 \\rightarrow \\cal K$ satisfying $FI_{\\cal K} = 1_{\\cal K}$  admits a unique, normal, pseudo-algebra structure for $(-)^2$ if and only if there is a mere natural isomorphism $F F^2 \\rightarrow F C_{\\cal K}$. We show that when this is the case the set of all natural transformations $F F^2 \\rightarrow F C_{\\cal K}$ forms a commutative monoid isomorphic to the centre of $\\cal K$."},
{"url": "http://www.tac.mta.ca/tac/volumes/8/n5/8-05abs.html", "title": "Finite sum - product logic", "authors": "J. R. B. Cockett and R. A. G. Seely", "keywords": ["categories", "categorical proof theory", "finite coproducts", "finite products", "deductive systems."], "abstract": "In this paper we describe a deductive system for categories with finite products and coproducts, prove decidability of equality of morphisms  via cut elimination, and prove a ``Whitman theorem'' for the free such categories over arbitrary base categories.  This result provides a nice illustration of some basic techniques in categorical proof theory, and also seems to have slipped past unproved in previous work in this field.  Furthermore, it suggests a type-theoretic approach to 2-player input-output games."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/5/10-05abs.html", "title": "Exponentiability of perfect maps: four approaches", "authors": "Susan Niefield", "keywords": ["exponentiable", "perfect", "proper", "separated", "function space."], "abstract": "Two proofs of the exponentiability of perfect maps are presented and compared to two other recent approaches.  One of the proofs is an  elementaryapproach including a direct construction of the exponentials.   The other, implicit in the literature, uses internal locales in the topos of set-valued sheaves on a topological space.\n"},
{"url": "http://www.tac.mta.ca/tac/volumes/10/4/10-04abs.html", "title": "Homology of Lie algebras with  $\\Lambda/q\\Lambda$ coefficients and exact sequences", "authors": "Emzar Khmaladze", "keywords": ["Lie algebra", "nonabelian derived functor", "exact sequence", "homology group."], "abstract": "Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie algebra homology with $\\Lambda/q\\Lambda$ coefficients is obtained, where $\\Lambda$ is a ground ring and $q$ is a nonnegative integer. Hopf formulas for the second and third homology of a Lie algebra are proved. The condition for the existence and the description of the universal $q$-central relative extension of a Lie epimorphism in terms of relative homologies are given."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/3/10-03abs.html", "title": "Entity-relationship-attribute designs and sketches", "authors": "Michael Johnson, Robert Rosebrugh and R.J. Wood", "keywords": ["sketch", "model", "database", "update."], "abstract": "Entity-Relationship-Attribute ideas are commonly used to    specify and design information systems. They use a graphical technique for displaying the objects of the system and relationships among them. The design process can be enhanced by specifying constraints of the system and the natural environment for these is the categorical notion of sketch. Here we argue that the finite-limit, finite-sum sketches with a terminal node are the appropriate class and call them EA sketches. A model for an EA sketch in a lextensive category is a `snapshot' of a database with values in that category. The category of models of an EA sketch is an object of models of the sketch in a 2-category of lextensive categories.  Moreover, modelling the same sketch in certain objects in other 2-categories defines both the query language for the database and the updates (the dynamics) for the database."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/2/10-02abs.html", "title": "A homotopy double groupoid of a Hausdorff space", "authors": "Ronald Brown, Keith A. Hardie, Klaus Heiner Kamps, Timothy Porter", "keywords": ["double groupoid", "connection", "thin structure", "2-groupoid", "double track", "2- track", "thin square", "homotopy addition lemma."], "abstract": "We associate to a Hausdorff space, $ X $, a double groupoid, $ \\mbox{\\boldmath $ \\rho $}^{\\square}_{2} (X) $, the homotopy double groupoid of $ X $. The construction is based on the geometric notion of thin square.  Under the equivalence of categories between small $ 2 $-categories and double categories with connection the homotopy double groupoid corresponds to the homotopy 2- groupoid, $ {\\bf G}_{2} (X) $.  The cubical nature of $ \\mbox{\\boldmath $ \\rho $}^{\\square}_{2} (X) $ as opposed to the globular nature of $ {\\bf G}_{2} (X) $ should  provide a convenient tool when handling `local-to-global'    problems as encountered in a generalised van Kampen theorem and  dealing with tensor products and enrichments of the category of compactly generated Hausdorff spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html", "title": "A survey of definitions of n-category", "authors": "Tom Leinster", "keywords": ["n-category", "higher-dimensional category", "higher categorical structure."], "abstract": "Many people have proposed definitions of `weak n-category'.  Ten of them are presented here.  Each definition is given in two pages, with a further two pages on what happens when $n\\leq 2$.  The definitions can be read independently.  Chatty bibliography follows."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/20/11-20abs.html", "title": "The strong amalgamation property and (effective) codescent morphisms", "authors": "Dali Zangurashvili", "keywords": ["Strong amalgamation property", "(effective) codescent morphism", "group", "variety of universal algebras"], "abstract": "Codescent morphisms are described in regular categories which satisfy the so-called strong amalgamation property. Among varieties of universal algebras possessing this property are, as is known, categories of groups, not necessarily associative rings, M-sets (for a monoid M), Lie algebras (over a field), quasi-groups, commutative quasi-groups, Steiner quasi-groups, medial quasi-groups, semilattice$lattices, weakly associative lattices, Boolean algebras, Heyting algebras. It is shown that every codescent morphism of groups is effective."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/18/11-18abs.html", "title": "Symmetric monoidal completions and the exponential principle among labeled combinatorial structures", "authors": "Matias Menni", "keywords": ["symmetric monoidal categories", "combinatorics"], "abstract": "We generalize Dress and Müller's main result in Decomposable functors and the exponential principle.   We observe that their result can be seen as a characterization of free algebras for certain monad on the category of species.  This perspective allows to formulate a general  exponential principle in a symmetric monoidal category.  We show that for any groupoid G, the category of presheaves on the symmetric monoidal completion !G of G satisfies the exponential principle.  The main result in Dress and Müller reduces to the case G = 1.  We discuss two notions of functor between categories satisfying the exponential principle and express some well known combinatorial identities as instances of the preservation properties of these functors.  Finally, we give a characterization of G as a subcategory of presheaves on !G."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/19/11-19abs.html", "title": "Composition-Representative Subsets", "authors": "Gary Griffing", "keywords": ["recognizable language", "recognizable forest", "representative functional", "syntactic congruence", "algebraic theory", "translates"], "abstract": "A specific property applicable to subsets of a hom-set in any small category is defined. Subsets with this property are called composition-representative. The notion of composition-representability is motivated both by the representability of a linear functional on an associative algebra, and, by the recognizability of a subset of a monoid. Various characterizations are provided which therefore may be regarded as analogs of certain characterizations for representability and recognizablity. As an application, the special case of an algebraic theory T is considered and simple characterizations for a recognizable forest are given. In particular, it is shown that the composition-representative subsets of the hom-set T([1],[0]), the set of all trees, are the recognizable forests and that they, in turn, are characterized by a corresponding finite `syntactic congruence.' Using a decomposition result (proved here), the composition-representative subsets of the hom-set T([m],[0]),  (0 \\leq m) are shown to be finite unions of m-fold (cartesian) products of recognizable forests."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/17/11-17abs.html", "title": "Modules", "authors": "J.R.B. Cockett, J. Koslowski, R.A.G. Seely, and R.J. Wood", "keywords": ["module", "bicategory", "transformation"], "abstract": "This paper studies lax higher dimensional structure over bicategories. The general notion of a module between two morphisms of bicategories is described. These modules together with their (multi-)2-cells, which we call modulations, organize themselves into a multi-bicategory.  The usual notion of a module can be recovered from this general notion by simply choosing the domain bicategory to be the terminal or final bicategory.\nThe composite of two such modules need not exist.  However, when the domain bicategory is small and the codomain bicategory is locally cocomplete then the composite of any two modules does exist and has a simple construction using the local colimits.  These modules and their modulations then give rise to a bicategory.\nRecall that neither transformations nor optransformations (respectively lax natural transformations and oplax natural transformations) between morphisms of bicategories give rise to a smooth 3-dimensional structure. However, there is a smooth 3-dimensional structure for modules, and both transformations and optransformations give rise to associated modules.  Furthermore, the modulations between two modules associated with transformations can then be described directly as a new sort of modification between the transformations.  This provides a locally full and faithful homomorphism from transformations and modifications into the bicategory of modules.\nFinally, if each 1-cell component of a transformation is a left-adjoint then the right-adjoints provide an optransformation.  In the module bicategory the module associated with this optransformation is right-adjoint to the module associated with the transformation. Therefore the inclusion of transformations whose 1-cells have left adjoints into the (multi-)bicategory of modules provides a source of proarrow equipment."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/16/11-16abs.html", "title": "The category of opetopes and the category of opetopic sets", "authors": "Eugenia Cheng", "keywords": ["n-category", "higher-dimensional category", "opetope", "opetopic set"], "abstract": "We give an explicit construction of the category Opetope of opetopes.  We prove that the category of opetopic sets is equivalent to the category of presheaves over Opetope."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/15/11-15abs.html", "title": "Exponentiability in categories of lax algebras", "authors": "Maria Manuel Clementino, Dirk Hofmann and Walter Tholen", "keywords": ["lax algebra", "partial product", "locally cartesian-closed category", "quasitopos"], "abstract": "For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a  quasitopos. This result encompasses many known and new examples of quasitopoi."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/14/11-14abs.html", "title": "Some calculus with extensive quantities:", "authors": "Anders Kock and Gonzalo E. Reyes", "keywords": ["distribution", "wave equation", "synthetic differential calculus", "extensive quantity"], "abstract": "We take some first steps in providing a synthetic theory of distributions.  In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/13/11-13abs.html", "title": "Partial Toposes", "authors": "Jean Bénabou and Thomas Streicher", "keywords": ["fibred categories", "partial toposes"], "abstract": "We introduce various notions of partial topos, i.e. `topos without terminal object'. The strongest one, called local topos, is motivated by the key examples of finite trees and sheaves with compact support. Local toposes satisfy all the usual exactness properties of toposes but are neither cartesian closed nor have a subobject classifier. Examples for the weaker notions are local homeomorphisms and discrete fibrations. Finally, for partial toposes with supports we show how they can be completed to toposes via an inverse limit construction."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/12/11-12abs.html", "title": "Ring epimorphisms and", "authors": "Michael Barr, W.D. Burgess and R. Raphael", "keywords": ["epimorphism", "ring of continuous functions", "category of rings"], "abstract": "This paper studies the homomorphism of rings of continuous functions $\\rho : C(X)\\to C(Y)$, $Y$ a subspace of a Tychonoff space $X$, induced by restriction.  We ask when $\\rho$ is an epimorphism in the categorical sense.  There are several appropriate categories:  we look at CR, all commutative rings, and R/N, all reduced commutative rings.  When $X$ is first countable and perfectly normal (e.g., a metric space), $\\rho$ is a CR -epimorphism if and only if it is a R/N-epimorphism if and  only if $Y$ is locally closed in $X$.  It is also shown that the restriction of $\\rho$ to $C^*(X)\\to C^*(Y)$, when $X$ is normal, is a CR-epimorphism if and only if it is a surjection.\nIn general spaces the picture is more complicated, as is shown by various examples.  Information about $Spec \\rho$ and $Spec \\rho$ restricted to the proconstructible set of prime z-ideals is given."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/11/11-11abs.html", "title": "Continuous categories revisited", "authors": "J. Adamek, F. W. Lawvere, J. Rosicky", "keywords": ["locally finitely presentable category", "precontinuous category", "continuous lattice", "pseudomonad"], "abstract": "Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a set-theoretical hypothesis this hull is formed by the category of all precontinuous categories, i.e., categories in which limits and filtered colimits distribute. This concept is closely related to the continuous categories of P. T. Johnstone and A. Joyal."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/10/11-10abs.html", "title": "Some algebraic applications of  graded categorical group theory", "authors": "A.M. Cegarra and A.R. Garzon", "keywords": ["graded categorical group", "graded monoidal functor", "equivariant group cohomology", "crossed product", "factor set", "monoid extension", "group  extension", "ring-group extension", "Clifford system", "Azumaya algebra", "graded bialgebra", "graded Hopf  algebra"], "abstract": "The homotopy classification of graded categorical groups and their homomorphisms is applied, in this paper, to obtain appropriate treatments for diverse crossed product constructions with operators which appear in several algebraic contexts.  Precise classification theorems are therefore stated for equivariant extensions by groups either of monoids, or groups, or rings, or rings-groups or algebras as well as for graded Clifford systems with operators, equivariant Azumaya algebras over Galois extensions of commutative rings and for strongly graded bialgebras and Hopf algebras with operators. These specialized classifications follow from the theory of graded categorical groups after identifying, in each case, adequate systems of factor sets with graded monoidal functors to suitable graded categorical groups associated to the structure dealt with."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/9/11-09abs.html", "title": "Characterization of Pointed Varieties of Universal Algebras with Normal Projections", "authors": "Zurab Janelidze", "keywords": [], "abstract": "We characterize pointed varieties of universal algebras in which  $(A\\times B)/A \\approx B$, i.e. all product projections are normal epimorphisms.\n-->"},
{"url": "http://www.tac.mta.ca/tac/volumes/11/7/11-07abs.html", "title": "Resolutions by Polygraphs", "authors": "François Métayer", "keywords": ["$n$-category", "polygraph", "resolution", "homotopy", "homology"], "abstract": "A notion of resolution for higher-dimensional categories is defined, by using polygraphs, and basic invariance theorems are proved."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/8/11-08abs.html", "title": "Cubical sets and their site", "authors": "Marco Grandis and Luca Mauri", "keywords": ["Simplicial sets", "cubical sets", "monoidal categories", "algebraic theories", "generators and relations", "word problem", "classifying categories"], "abstract": "Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid; in fact, K is the classifying category of a monoidal  algebraic theory of such monoids. Analogous results are given for the  restricted cubical  site} I, of  ordinary  cubical sets (just faces and degeneracies) and for the  intermediate  site J   (including connections). We also consider briefly the reversible analogue,  !K."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/6/11-06abs.html", "title": "Characterization of protomodular varieties of universal algebras", "authors": "Dominique Bourn and George Janelidze", "keywords": ["Maltsev and protomodular varieties", "ideal determination"], "abstract": "Protomodular categories were introduced by the first author more than ten years ago.  We show that a variety $\\mathcal V$ of universal algebras is protomodular if and only if it has 0-ary terms $e_1, ..., e_n$, binary terms $t_1, ..., t_n$, and (n+1)-ary term  $t$ satisfying the identities $t(x,t_1(x,y), ...,t_n(x,y)) = y$ and  $t_i(x,x) = e_i$ for each $i = 1, ..., n$."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/3/11-03abs.html", "title": "The branching nerve of HDA and the Kan condition", "authors": "Philippe Gaucher", "keywords": ["cubical set", "thin element", "Kan complex", "branching", "higher dimensional automata", "concurrency", "homology theory"], "abstract": "One can associate to any strict globular $\\omega$-category three augmented simplicial nerves called the globular nerve, the branching and the merging semi-cubical nerves. If this strict globular $\\omega$-category is freely generated by a precubical set, then the corresponding homology theories contain different informations about the geometry of the higher dimensional automaton modeled by the precubical set. Adding inverses in this $\\omega$-category to any morphism of dimension greater than 2 and with respect to any composition laws of dimension greater than 1 does not change these homology theories. In such a framework, the globular nerve always satisfies the Kan condition. On the other hand, both branching and merging nerves never satisfy it, except in some very particular and uninteresting situations. In this paper, we introduce two new nerves (the branching and merging semi-globular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semi-cubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/5/11-05abs.html", "title": "The Wirthmuller isomorphism revisited", "authors": "J. Peter May", "keywords": ["Equivariant stable homotopy category", "$G$-spectra", "Wirthmuller isomorphism"], "abstract": "We show how the formal Wirthmuller isomorphism theorem  simplifies the proof of the Wirthmuller isomorphism in equivariant  stable homotopy theory. Other examples from equivariant stable homotopy theory  show that the hypotheses of the formal Wirthmuller and formal Grothendieck isomorphism theorems cannot be weakened."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html", "title": "Isomorphisms between left and right adjoints", "authors": "H. Fausk, P. Hu, and J.P. May", "keywords": ["Grothendieck duality", "Verdier duality", "Wirthmuller isomorphism", "symmetric monoidal category", "triangulated category"], "abstract": "There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate `dualizing object'. Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to differentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely different, framework that arises in algebraic topology. Another is to make clear which parts of the proofs of such results are formal. The analysis significantly simplifies the proofs of particular cases, as we illustrate in a sequel discussing applications to equivariant stable homotopy theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/2/11-02abs.html", "title": "Morphisms and modules for poly-bicategories", "authors": "J.R.B. Cockett, J. Koslowski, and  R.A.G. Seely", "keywords": ["Bicategories", "polycategories", "multicategories", "modules", "representation theorems"], "abstract": "Linear bicategories are a generalization of ordinary bicategories in which there are two horizontal (1-cell) compositions corresponding to the ``tensor'' and ``par'' of linear logic. Benabou's notion of a morphism (lax 2-functor) of bicategories  may be generalized to linear bicategories, where they are called linear functors.  Unfortunately, as for the bicategorical case, it is not obvious how to organize linear functors smoothly into a higher dimensional structure. Not only do linear functors seem to lack the two compositions expected for a linear bicategory but, even worse, they inherit from the bicategorical level the failure to combine well with the obvious notion of transformation. As we shall see, there are also problems with lifting the notion of lax transformation to the linear setting.\nOne possible resolution is to step up one dimension, taking morphisms as the 0-cell level.  In the linear setting, this suggests making linear functors 0-cells, but what structure should sit above them?  Lax transformations in a suitable sense just do not seem to work very well for this purpose (Section \\ref{S:linnattran}). Modules provide a more promising direction, but raise a number of technical issues concerning the composability of both the modules and their transformations.  In general the required composites will not exist in either the linear bicategorical or ordinary bicategorical setting.  However, when these composites do exist modules between linear functors do combine to form a linear bicategory.   In order to better understand the conditions for the existence of composites, we have found it convenient, particularly in the linear setting, to develop the theory of ``poly-bicategories''.  In this setting we can develop the theory so as to extract the answers to these problems not only for linear bicategories but also for ordinary bicategories.  Poly-bicategories are 2-dimensional generalizations of Szabo's poly-categories, consisting of objects, 1-cells, and poly-2-cells. The latter may have several 1-cells as input and as output and can be composed by means of cutting along a single 1-cell.  While a poly-bicategory does not require that there be any compositions for the 1-cells, such composites are determined (up to 1-cell isomorphism) by their universal properties.  We say a poly-bicategory is representable when there is a representing 1-cell for each of the two possible 1-cell compositions geared towards the domains and codomains of the poly 2-cells.  In this case we recover the notion of a linear bicategory. The poly notions of functors, modules and their transformations are introduced as well.  The poly-functors between two given poly-bicategories P and P' together with poly-modules  between poly-functors and their transformations form a new poly-bicategory provided P is representable and closed in the sense that every  1-cell has both a left and a right adjoint (in the appropriate linear sense). Finally we revisit the notion of linear (or lax) natural transformations, which can only be defined for representable poly-bicategories.  These in fact correspond to modules having special properties."},
{"url": "http://www.tac.mta.ca/tac/volumes/11/1/11-01abs.html", "title": "Categorical models and quasigroup homotopies", "authors": "George Voutsadakis", "keywords": ["sketches", "finite product sketches", "sketch models", "quasigroups", "homotopies."], "abstract": "In many applications of quasigroups isotopies and homotopies are more important than isomorphisms and homomorphisms. In this paper, the way homotopies may arise in the context of categorical quasigroup model theory is investigated. In this context,  the algebraic structures are specified by diagram-based logics, such as sketches, and categories of models become functor categories. An idea, pioneered by Gvaramiya and Plotkin, is used to give a construction of a model category naturally equivalent to the category of quasigroups with homotopies between them."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html", "title": "Higher-dimensional algebra V: 2-Groups", "authors": "John C. Baez and Aaron D. Lauda", "keywords": ["2-group", "categorical group", "Chern-Simons theory", "group cohomology"], "abstract": "A 2-group is a `categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m ; G x G -> G has been replaced by a functor.  Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call `weak' and `coherent' 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a `weak inverse': an object y such that x \\tensor y \\iso 1 \\iso y \\tensor x. A coherent 2-group is a weak 2-group in which every object x is equipped with a  specified weak inverse x' and isomorphisms i_x  : 1 -> x \\tensor x',  e_x : x' \\tensor x -> 1 forming an adjunction.  We describe 2-categories of weak and coherent 2-groups and an `improvement' 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories.  We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups.  We give a tour of examples, including the `fundamental 2-group' of a space and various Lie 2-groups.  We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology.  Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G_h (h in Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object.  These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g_h (h in R) described in a companion paper."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html", "title": "Higher-Dimensional Algebra VI: Lie 2-Algebras", "authors": "John C. Baez and Alissa S. Crans", "keywords": ["Lie 2-algebra", "L_\\infty-algebra", "Lie algebra cohomology"], "abstract": "The theory of Lie algebras can be categorified starting from a new notion of `2-vector space', which we define as an internal category in Vect.  There is a 2-category 2Vect having these 2-vector spaces as objects, `linear functors' as morphisms and `linear natural transformations' as 2-morphisms.  We define a `semistrict Lie 2-algebra' to be a 2-vector space L equipped with a skew-symmetric bilinear functor [ . , . ] : L x L -> L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the `Jacobiator', which in turn must satisfy a certain law of its own.  This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang--Baxter equation.  We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L_\\infty-algebras in the sense of Stasheff.  We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology.  This classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g_h which reduces to g at h = 0.  These are closely related to the 2-groups G_h constructed in a companion paper."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/13/12-13abs.html", "title": "The double powerlocale and exponentiation: A case study in geometric logic", "authors": "Steven Vickers", "keywords": ["locale", "powerlocale", "exponentiable", "geometric logic"], "abstract": "If X is a locale, then its double powerlocale PX is defined to be PU(PL(X)) where PU and PL are the upper and lower powerlocale constructions. We prove various results relating it to exponentiation of locales, including the following. First, if X is a locale for which the exponential S^X exists (where S is the Sierpinski locale), then PX is an exponential S^(S^X). Second, if in addition W is a locale for which PW is homeomorphic to S^X, then X is an exponential S^W.\nThe work uses geometric reasoning, i.e. reasoning stable under pullback along geometric morphisms, and this enables the locales to be discussed in terms of their points as though they were spaces. It relies on a number  of geometricity results including those for locale presentations and for powerlocales."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/12/12-12abs.html", "title": "K-purity and orthogonality", "authors": "Michel Hebert", "keywords": ["pure morphism", "othogonality", "injectivity", "locally presentable categories", "accessible categories"], "abstract": "Adamek and Sousa recently solved the problem of characterizing the subcategories K of a locally $\\lambda$-presentable category C which are $\\lambda$-orthogonal in C, using their concept of K$\\lambda$-pure morphism. We strengthen the latter definition, in order to obtain a characterization of the classes defined by orthogonality with respect to $\\lambda$-presentable morphisms (where $f :  A \\rightarrow B is called $\\lambda$-presentable if it is a $\\lambda$-presentable object of the comma category A/C). Those classes are natural examples of reflective subcategories defined by proper classes of morphisms. Adamek and Sousa's result follows from ours. We also prove that $\\lambda$-presentable morphisms are precisely the pushouts of morphisms between $\\lambda$-presentable objects of C."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/11/12-11abs.html", "title": "Several constructions for factorization systems", "authors": "Dali Zangurashvili", "keywords": ["(local) factorization system", "family of adjunctions between slice categories", "semi-left-exact reflection", "fibration", "(co)pointed endofunctor"], "abstract": "The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) `reflects' factorization systems. In particular, a generalization of the well-known Cassidy-Héebert-Kelly  factorization theorem is given. The problem of relating a factorization system to a pointed endofunctor is considered. Some relevant examples in concrete categories are given."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/10/12-10abs.html", "title": "Vertically iterated classical enrichment", "authors": "Stefan Forcey", "keywords": ["enriched categories", "n-categories", "iterated monoidal categories"], "abstract": "Lyubashenko has described enriched 2-categories as categories enriched  over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. This construction is the strict analogue for  V-functors in V-Cat of Brian Day's probicategories for  V-modules in V-Mod. Here I generalize the strict version to enriched n-categories for k-fold monoidal V. The latter is defined  as by Balteanu, Fiedorowicz, Schwanzl and Vogt but with the addition of  making visible the coherent associators. The  symmetric case can easily be recovered. This paper proposes a recursive definition of V-n-categories and their morphisms. We show that  for V k-fold monoidal the structure of a (k-n)-fold monoidal strict (n+1)-category is possessed by V-n-Cat. This article is a  completion of the work begun by the author in the preprint entitled Higher dimensional enrichment (math.CT/0306086), and the initial  sections duplicate the beginning of that paper."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/9/12-09abs.html", "title": "Algebraically closed and existentially closed substructures in  categorical context", "authors": "Michel Hebert", "keywords": ["pure morphism", "algebraically closed", "existentially"], "abstract": "We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir, as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosicky, Adamek and Borceux, characterizing $\\lambda$-injectivity classes in locally $\\lambda$-presentable categories, is given."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/7/12-07abs.html", "title": "Change of base for relational variable sets", "authors": "Susan Niefield", "keywords": ["relational variable set", "specification structure", "dynamic set", "relational presheaf", "change of base", "exponentiable"], "abstract": "Following Ghilardi and Meloni, a relational variable set on a category B is a lax functor B to  Rel, where Rel is the category of sets and relations.  Change-of-base functors and their adjoints are considered for certain categories of relational variable sets and applied to construct the simplification of a dynamic set (in the sense of Stell)."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/6/12-06abs.html", "title": "Moore Categories", "authors": "Diana Rodelo", "keywords": ["short exact sequence", "normal object", "protomodular category", "strongly protomodular category", "Barr-exact category", "Moore category"], "abstract": "In 1970, M. Gerstenhaber introduced a list of axioms defining Moore categories in order to develop the Baer Extension Theory. In this paper, we study some implications between the axioms and compare them with more recent notions, showing that, apart from size restrictions, a Moore category is a pointed, strongly protomodular and Barr-exact category with cokernels."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/5/12-05abs.html", "title": "Notions of flatness relative to a Grothendieck topology", "authors": "Panagis Karazeris", "keywords": ["flat functor", "postulated colimit", "geometric logic", "exact completion", "pretopos  completion", "left exact Kan extension"], "abstract": "Completions of (small) categories under certain kinds of colimits and exactness conditions have been studied extensively in the literature.  When the category that we complete is not left exact but has some weaker kind of limit for finite diagrams, the universal property of the completion is usually stated with respect to functors that enjoy a property reminiscent of flatness.  In this fashion notions like that of a left covering or a multilimit merging functor have appeared in the literature.  We show here that such notions coincide with flatness when the latter is interpreted relative to (the internal logic of) a site structure associated to the target category.  We exploit this in order to show that the left Kan extensions of such functors, along the inclusion of their domain into its completion, are left exact.  This gives in a very economical and uniform manner the universal property of such completions.  Our result relies heavily on some unpublished work of A. Kock from 1989.  We further apply this to give a pretopos completion process for small categories having a weak finite limit property."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/3/12-03abs.html", "title": "Operads in higher-dimensional category theory", "authors": "Tom Leinster", "keywords": ["n-category", "operad", "higher-dimensional category"], "abstract": "The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak n-category.  Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n is less than or equal to 2.  Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/2/12-02abs.html", "title": "Simplicial approximation", "authors": "J.F. Jardine", "keywords": ["simplicial sets", "simplicial approximation", "model structures"], "abstract": "This paper displays an approach to the construction of the homotopy theory of simplicial sets and the corresponding equivalence with the homotopy theory of topological spaces which is based on simplicial approximation techniques. The required simplicial approximation results for simplicial sets and their proofs are given in full. Subdivision behaves like a covering in the context of the techniques displayed here."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/4/12-04abs.html", "title": "Baer invariants in semi-abelian categories II: Homology", "authors": "T. Everaert and T. Van der Linden", "keywords": ["Baer invariant", "semi-abelian category", "cotriple homology"], "abstract": "This article treats the problem of deriving the reflector of a semi-abelian category $\\cal A$ onto a Birkhoff subcategory $\\cal B$ of $\\cal A$. Basing ourselves on Carrasco, Cegarra and Grandjean's homology theory for crossed modules, we establish a connection between our theory of Baer invariants with a generalization---to semi-abelian categories---of Barr and Beck's cotriple homology theory. This results in a semi-abelian version of Hopf's formula and the Stallings-Stammbach sequence from group homology."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/1/12-01abs.html", "title": "Baer invariants in semi-abelian categories I: General theory", "authors": "T. Everaert and T. Van der Linden", "keywords": ["Baer invariant", "exact", "protomodular", "semi-abelian category", "centrality", "nilpotency"], "abstract": "Extending the work of Fröhlich, Lue and Furtado-Coelho, we consider the theory of Baer invariants in the context of semi-abelian categories. Several exact sequences, relative to a subfunctor of the identity functor, are obtained. We consider a notion of commutator which, in the case of abelianization, corresponds to Smith's. The resulting notion of centrality fits into Janelidze and Kelly's theory of central extensions. Finally we propose a notion of nilpotency, relative to a Birkhoff subcategory of a semi-abelian category."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/15/13-15abs.html", "title": "Internal monotone-light factorization for categories via preorders", "authors": "Joao Xarez", "keywords": ["(reflexive) graph", "(reflexive) relation", "category", "preorder", "factorization system", "localization", "stabilization", "descent theory", "Galois theory", "monotone-light factorization"], "abstract": "It is shown that, for a finitely-complete category C with coequalizers of kernel pairs, if every product-regular epi is also stably-regular then there exist the reflections (R)Grphs(C) --> (R)Rel(C), from (reflexive) graphs into (reflexive) relations in C, and Cat(C) -->  Preord(C), from categories into preorders in C. Furthermore, such a sufficient condition ensures as well that these reflections do have stable units. This last property is equivalent to the existence of a monotone-light factorization system, provided there are sufficiently many effective descent morphisms with domain in the respective full subcategory. In this way, we have internalized the monotone-light factorization for small categories via preordered sets, associated with the reflection Cat --> Preord, which is now just the special case C = Set."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/14/13-14abs.html", "title": "Generic morphisms, parametric representations and weakly cartesian monads", "authors": "Mark Weber", "keywords": [], "abstract": "Two notions, generic morphisms and parametric representations, useful for the analysis of endofunctors arising in enumerative combinatorics, higher dimensional category theory, and logic, are defined and examined. Applications to the Batanin approach to higher category theory, Joyal species and operads are provided."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/13/13-13abs.html", "title": "The monoidal centre as a limit", "authors": "Ross Street", "keywords": ["braiding", "centre", "pseudofunctor", "descent"], "abstract": "The centre of a monoidal category is a braided monoidal category. Monoidal categories are monoidal objects (or pseudomonoids) in the monoidal bicategory of categories. This paper provides a universal construction in a braided monoidal bicategory that produces a braided monoidal object from any monoidal object. Some properties and sufficient conditions for existence of the construction are examined."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/12/13-12abs.html", "title": "Split structures", "authors": "Robert Rosebrugh and R.J. Wood", "keywords": ["complete distributivity", "idempotent splitting completion", "K-Z doctrine"], "abstract": "In the early 1990's the authors proved that the full subcategory of `sup-lattices' determined by the constructively completely distributive (CCD) lattices is equivalent to the idempotent splitting completion of the bicategory of sets and relations. Having many corollaries, this was an extremely useful result. Moreover, as the authors soon suspected, it specializes a much more general result.\nLet D be a monad on a category C in which idempotents split. Write kar(C_D) for the idempotent splitting completion of the Kleisli  category. Write spl(C^D) for the category whose objects are pairs ((L,s),t), where (L,s) is an object of the Eilenberg-Moore category for D, and t is a homomorphism  that splits s, with  spl(C^D)(((L,s),t),((L',s'),t'))=C^D((L,s)(L',s')).\nThe main result is that kar(C_D) is isomorphic to spl(C^D). We also show how this implies the CCD lattice characterization theorem and consider a more general context."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/10/13-10abs.html", "title": "Functorial concepts of complexity for finite automata", "authors": "F. William Lawvere", "keywords": ["Automata", "measurable cardinals", "Aufhebung", "finite topos", "axiomatic  arithmetic"], "abstract": "Some unsolved problems about the classifying topos for Boolean algebras, as well as about the axiomatic arithmetic of finite combinatorial toposes, are closely connected with some simple distinctions between finite automata."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/11/13-11abs.html", "title": "Every small Sl-enriched category is Morita equivalent to an Sl-monoid", "authors": "Bachuki Mesablishvili", "keywords": ["Sup-lattices", "Morita equivalence", "separable category"], "abstract": "We show that every small category enriched over Sl - the symmetric  monoidal closed category of sup-lattices and sup-preserving  morphisms - is Morita equivalent to an Sl-monoid. As a  corollary, we obtain a result of Borceux and Vitale  asserting  that every separable Sl-category is Morita equivalent to a  separable Sl-monoid."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html", "title": "Composing PROPs", "authors": "Stephen Lack", "keywords": ["symmetric monoidal category", "PROP", "monad", "distributive law", "algebra", "bialgebra"], "abstract": "A PROP is a way of encoding structure borne by an object of a symmetric monoidal category. We describe a notion of distributive law for PROPs, based on Beck's distributive laws for monads. A distributive law between PROPs allows them to be composed, and an algebra for the composite PROP consists of a single object with an algebra structure for each of the original PROPs, subject to compatibility conditions encoded by the distributive law. An example is the PROP for bialgebras, which is a composite of the PROP for coalgebras and that for algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/8/13-08abs.html", "title": "Coalgebras, braidings, and distributive laws", "authors": "Stefano Kasangian, Stephen Lack, and Enrico M. Vitale", "keywords": ["Descent data", "monads", "distributive laws", "Yang-Baxter equation"], "abstract": "We show, for a monad T, that coalgebra structures on a T-algebra can be described in terms of \"braidings\", provided that the monad is equipped with an invertible distributive law satisfying the Yang-Baxter equation."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/7/13-07abs.html", "title": "Normed combinatorial homology", "authors": "Marco Grandis", "keywords": ["Cubical sets", "noncommutative C*-algebras", "combinatorial homology", "normed abelian groups"], "abstract": "Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of `noncommutative topology', agreeing with some results of noncommutative geometry but lacking the metric aspects of C* -algebras. Here, we make such similarity stricter by introducing normed cubical sets and their normed directed homology, formed of normed preordered abelian groups. The normed cubical sets  NC_\\theta associated with `irrational' rotations have thus the same classification up to isomorphism as the well-known irrational rotation C* -algebras A_\\theta."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/6/13-06abs.html", "title": "Semi-abelian monadic categories", "authors": "Marino Gran and Jiri Rosicky", "keywords": ["Semi-abelian categories", "monadic categories", "varieties", "quasivarieties", "localizations"], "abstract": "We characterize semi-abelian monadic categories and their localizations.  These results are then used to obtain a characterization of pointed protomodular quasimonadic categories, and in particular of protomodular quasivarieties."},
{"url": "http://www.tac.mta.ca/tac/volumes/12/8/12-08abs.html", "title": "On subgroups of the Lambek pregroup", "authors": "Michael Barr", "keywords": ["subpregroups of the Lambek pregroup"], "abstract": "A pregroup is a partially ordered monoid in which every element has a left and a right adjoint.  The main result is that for some well-behaved subgroups of the group of diffeomorphisms of the real numbers, the set of all endofunctions of the integers that are asymptotic at $\\pm\\infty$ to (the restriction to the integers of) a function in the subgroup is a pregroup."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/5/13-05abs.html", "title": "Derivations of categorical groups", "authors": "A.R. Garzon, H. Inassaridze and A. del Rio", "keywords": ["derivation", "categorical group", "cohomology"], "abstract": "In this paper we introduce and study the categorical group of derivations, Der(G, A), from a categorical group G into a braided categorical group (A,c) equipped with a given coherent left action of G.  Categorical groups provide a 2-dimensional vision of groups and so this object is a sort of 0-cohomology at a higher level for categorical groups.  We show that the functor Der(-, A) is  corepresentable by the semidirect product of A with  G and that Der(G,-) preserves homotopy kernels.  Well-known cohomology groups, and exact sequences relating these groups, in several different contexts are then obtained as examples of this general theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/3/13-03abs.html", "title": "On extensions of lax monads", "authors": "Maria Manuel Clementino and Dirk Hofmann", "keywords": ["relation", "lax algebra", "lax monad"], "abstract": "In this paper we construct extensions of Set-monads -- and, more generally, of lax Rel-monads -- into lax monads of the bicategory Mat(V) of generalized V-matrices, whenever  V is a well-behaved lattice equipped with a tensor product. We add some guiding examples."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/2/13-02abs.html", "title": "Commutator theory in strongly protomodular categories", "authors": "Dominique Bourn", "keywords": ["Commutator", "unital", "Mal'cev", "protomodular", "semi-abelian and strongly  protomodular categories", "fibration of points"], "abstract": "We show that strongly protomodular categories (as the category of groups for instance) provide an appropriate framework in which the commutator of two equivalence relations do coincide with the commutator of their associated normal subobjects, whereas it is not the case in any semi-abelian category."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/1/13-01abs.html", "title": "On von Neumann varieties", "authors": "F. Borceux and J. Rosicky", "keywords": ["variety", "flat algebra", "pure monomorphism", "von Neumann ring", "locally finitely presentable category"], "abstract": "We generalize to an arbitrary variety the von Neumann axiom for a ring. We study its implications on the purity of monomorphisms and the flatness of algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/19/14-19abs.html", "title": "Generalized Brown representability in homotopy categories", "authors": "Jiri Rosicky", "keywords": ["Quillen model category", "Brown representability", "triangulated category", "accessible category"], "abstract": "Brown representability approximates the homotopy category of spectra by means of cohomology functors defined on finite spectra. We will show that if a model category $\\cal K$ is suitably determined by $\\lambda$-small objects then its homotopy category $Ho(\\cal K)$ is approximated by cohomology functors defined on those $\\lambda$-small objects. In the case of simplicial sets, we have $\\lambda = \\omega_1$, i.e., $\\lambda$-small means countable."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/18/14-18abs.html", "title": "Birkhoff's variety theorem with and without free algebras", "authors": "Jiri Adamek and Vera Trnkova", "keywords": ["variety", "Birkhoff's Theorem"], "abstract": "For large signatures $\\Sigma$ we prove that Birkhoff's Variety Theorem holds (i.e., equationally presentable collections of $\\Sigma$-algebras are precisely those closed under limits, subalgebras, and quotient algebras) iff the universe of small sets is not measurable. Under that limitation Birkhoff's Variety Theorem holds in fact for $F$-algebras of an arbitrary endofunctor $F$ of the category Class of classes and functions.\nFor endofunctors $F$ of Set, the category of small sets, Jan Reiterman proved that if $F$ is a varietor (i.e., if free $F$-algebras exist) then Birkhoff's Variety Theorem holds for $F$-algebras. We prove the converse, whenever $F$ preserves preimages: if $F$is not a varietor, Birkhoff's Variety Theorem does not hold. However, we also present a non-varietor satisfying Birkhoff's Variety Theorem. Our most surprising example is two varietors whose coproduct does not satisfy Birkhoff's  Variety Theorem."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/17/14-17abs.html", "title": "Notes on enriched categories with colimits of some class", "authors": "G.M. Kelly and V. Schmitt", "keywords": ["limits", "colimits", "flat", "atomic", "small presentable", "Cauchy completion"], "abstract": "The paper is in essence a survey of categories having $\\phi$-weighted colimits for all the weights $\\phi$ in some class $\\Phi$. We introduce the class $\\Phi^+$ of $\\Phi$-flat weights which are those $\\psi$ for which $\\psi$-colimits commute in the base $\\cal V$ with limits having weights in $\\Phi$; and the class $\\Phi^-$ of $\\Phi$-atomic weights, which are those $\\psi$ for which $\\psi$-limits commute in the base $\\cal V$ with colimits having weights in $\\Phi$. We show that both these classes are saturated (that is, what was called closed in the terminology of Albert and Kelly). We prove that for the class $\\cal P$ of all weights, the classes $\\cal P^+$ and $\\cal P^-$ both coincide with the class $\\Q$ of absolute weights. For any class $\\Phi$ and any category $\\cal A$, we have the free $\\Phi$-cocompletion $\\Phi(\\cal A)$ of $\\cal A$; and we recognize $\\cal Q(\\cal A)$ as the Cauchy-completion of $\\cal A$. We study the equivalence between ${(\\cal Q(\\cal A^{op}))}^{op}$ and $\\cal Q(\\cal A)$, which we exhibit as the restriction of the Isbell adjunction between ${[\\cal A,\\cal V]}^{op}$ and $[\\cal A^{op},\\cal V]$ when $\\cal A$ is small; and we give a new Morita theorem for any class $\\Phi$ containing $\\cal Q$. We end with the study of $\\Phi$-continuous weights and their relation to the $\\Phi$-flat weights."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/16/14-16abs.html", "title": "Enlargements of categories", "authors": "Lars Brünjes, Christian Serpé", "keywords": [], "abstract": "In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorical properties are well behaved under enlargements."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/15/14-15abs.html", "title": "Localic completion of generalized metric spaces I", "authors": "Steven Vickers", "keywords": ["topology", "locale", "geometric logic", "metric", "quasimetric", "completion", "enriched category"], "abstract": "Following Lawvere, a generalized metric space (gms) is a set $X$ equipped with a metric map from $X^{2}$ to the interval of upper reals (approximated from above but not from below) from 0 to $\\infty$ inclusive, and satisfying the zero self-distance law and the triangle inequality.\nWe describe a completion of gms's by Cauchy filters of formal balls. In terms of Lawvere's approach using categories enriched over $[0,\\infty]$, the Cauchy filters are equivalent to flat left modules.\nThe completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Kunzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion.\nNon-expansive functions between gms's lift to continuous maps between the completions.\nVarious examples and constructions are given, including finite products.\nThe completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable point-based reasoning for locales."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/14/14-14abs.html", "title": "Classification of concrete geometrical categories", "authors": "Yves Diers", "keywords": ["concrete geometrical category", "classifying geometrical category", "topological geometrical category"], "abstract": "A precise concept of concrete geometrical category is introduced in an axiomatic way. To any algebra L for an many-sorted infinitary algebraic theory T is associated a concrete  geometrical category Geo(L), the so-called classifying concrete geometrical category of L, satisfying a universal property. The terminology  \"geometrical\" is justified firstly for Geo(L) and secondly for any concrete geometrical category by proving that they are all classifying ones. The legitimate category CGC of concrete geometrical categories is  build up and proved to be the dual of the legitimate category TGC of  topological geometrical categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/13/14-13abs.html", "title": "Every group is representable by all natural transformations of some  set-functor", "authors": "Libor Barto and Petr Zima", "keywords": ["set functor", "group universal category"], "abstract": "For every group G, we construct a functor F : SET --> SET (finitary for a finite group G) such that the monoid of all natural endotransformations of F is a group isomorphic to G."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/11/14-11abs.html", "title": "On the representability of actions in a semi-abelian category", "authors": "F. Borceux, G. Janelidze and G.M. Kelly", "keywords": ["semi-abelian category", "variety", "semi-direct product", "action"], "abstract": "We consider a semi-abelian category V  and we write Act(G,X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(-,X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove first that the representability of Act(-,X) reduces to the preservation of binary coproducts. Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary  coproducts by the functor Act(-,X) in a general semi-abelian category. Finally, we exhibit the precise form of the more involved ``if and only  if'' amalgamation property corresponding to the representability of actions: this condition is in particular related to a new notion of ``normalization of a morphism''. We provide also a wide supply of algebraic examples and counter-examples, giving in particular evidence of the relevance of the object representing Act(-,X), when it turns out to exist."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/12/14-12abs.html", "title": "Groupoid enriched categories and natural systems", "authors": "Teimuraz Pirashvili", "keywords": ["Track category", "linear track extension", "natural system"], "abstract": "We generalize the Baues-Jibladze descent theorem to a large class of groupoid enriched categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/10/14-10abs.html", "title": "Canonical and op-canonical lax algebras", "authors": "Gavin J. Seal", "keywords": ["V-matrix", "(T,V)-algebra", "ordered set", "metric space", "topological space", "approach space", "closure space", "closeness space", "topological category"], "abstract": "The definition of a category of (T,V)-algebras, where V is a unital commutative quantale and T is a Set-monad, requires the existence of a certain lax extension of T. In this article, we present a general construction of such an extension. This leads to the formation of two categories of (T,V)-algebras: the category Alg(T,V) of canonical (T,V)-algebras, and the category Alg(T',V) of op-canonical (T,V)-algebras. The usual topological-like  examples of categories of (T,V)-algebras (preordered sets, topological, metric and approach spaces) are obtained in this way, and the category of closure spaces appears as a category of canonical (P,V)-algebras, where P is the powerset monad. This unified presentation allows us to study how these categories are related, and it is shown that under suitable hypotheses both Alg(T,V) and Alg(T',V) embed coreflectively into Alg(P,V)."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/9/14-09abs.html", "title": "A homotopy double groupoid of a Hausdorff space II:  a van Kampen theorem", "authors": "R. Brown, K.H. Kamps, and T. Porter", "keywords": ["double groupoid", "double category", "thin structure", "connections", "commutative cube", "van Kampen theorem"], "abstract": "This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space.\nWe show that this functor  satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed  by Brown and Higgins for similar theorems for other higher homotopy groupoids.\nAn integral part of the proof is a  detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/8/14-08abs.html", "title": "Introduction to coalgebra", "authors": "Jiri Adamek", "keywords": [], "abstract": "A survey of parts of General Coalgebra is presented with applications to the theory of systems. Stress is laid on terminal coalgebras and coinduction as well as iterative algebras and iterative theories."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/7/14-07abs.html", "title": "A monadic approach to polycategories", "authors": "Juergen Koslowski", "keywords": ["cartesian monad", "S-T-span", "(cartesian) distributive law", "multicategory", "(planar) polycategory", "fc-polycategory", "associative double semigroup"], "abstract": "In the quest for an elegant formulation of the notion of   ``polycategory'' we develop a more symmetric counterpart to   Burroni's notion of ``T- category'', where T is a cartesian   monad on a category X with pullbacks.  Our approach involves two   such monads, S and T, that are linked by a suitable   generalization of a distributive law in the sense of Beck.  This   takes the form of a span omega : TS  ST in the functor   category [X,X] and guarantees essential associativity for a   canonical pullback-induced composition of S-T-spans over X,   identifying them as the 1-cells of a bicategory, whose (internal)   monoids then qualify as ``omega-categories''.  In case that   S and T both are the free monoid monad on set, we construct   an omega utilizing an apparently new classical distributive law   linking the free semigroup monad with itself.  Our construction then   gives rise to so-called ``planar polycategories'', which nowadays   seem to be of more intrinsic interest than Szabo's original   polycategories.  Weakly cartesian monads on X may be accommodated   as well by first quotienting the bicategory of X-spans."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/6/14-06abs.html", "title": "Abstract physical traces", "authors": "Samson Abramsky and Bob Coecke", "keywords": [], "abstract": "We revise our `Physical Traces' paper in the light of the results in [Abramsky and Coecke LiCS`04]. The key fact is that the notion of a strongly compact closed category allows abstract notions of adjoint, bipartite projector and inner product to be defined, and their key properties to be proved. In this paper we improve on the definition of strong compact closure as compared to the one presented in [Abramsky and Coecke LiCS`04]. This modification enables an elegant characterization of strong compact closure in terms of adjoints and a Yanking axiom, and a better treatment of bipartite projectors."},
{"url": "http://www.tac.mta.ca/tac/volumes/13/4/13-04abs.html", "title": "Universal properties of Span", "authors": "R.J.MacG. Dawson, R. Pare and D.A. Pronk", "keywords": ["Span", "$\\Pi_2$", "Beck condition", "adjoints", "universal property", "localizations", "sinister morphisms", "jointed oplax morphisms"], "abstract": "We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these `jointed' oplax morphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/4/14-04abs.html", "title": "Thin elements and commutative shells  in cubical omega-categories", "authors": "Philip J. Higgins", "keywords": ["cubical omega-category", "connections", "thin elements", "thin structure", "folding operations", "commutative shells"], "abstract": "The relationships between thin elements, commutative shells and connections in cubical omega-categories are explored by a method which does not involve the use of pasting theory or nerves of omega-categories (both of which were previously needed for this purpose; see Al-Agl, Brown and Steiner, Section 9). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 in Brown, Kamps and Porter and Brown and  Mosa."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/5/14-05abs.html", "title": "Adjunction models for call-by-push-value with stacks", "authors": "Paul Blain Levy", "keywords": ["call-by-push-value", "adjunction", "CK-machine", "monad", "denotational semantics", "indexed category", "continuations", "possible worlds", "game semantics", "call-by-name", "call-by-value"], "abstract": "Call-by-push-value is a \"semantic machine code\", providing a set of simple primitives from which both the call-by-value and call-by-name paradigms are built. We present its operational semantics as a stack machine, suggesting a term judgement of stacks. We then see that CBPV, incorporating these stack terms, has a simple categorical semantics based on an adjunction between values and stacks.  There are no coherence requirements.\nWe describe this semantics incrementally. First, we introduce locally indexed categories and the opGrothendieck construction, and use these to give the basic structure for interpreting the three judgements: values, stacks and computations. Then we look at the universal property required to interpret each type constructor. We define a model to be a strong adjunction with countable coproducts, countable products and exponentials.\nWe see a wide range of instances of this structure: we give examples for divergence, storage, erratic choice, continuations, possible worlds and games (with or without a bracketing condition), in each case resolving the strong monad from the literature into a strong adjunction.  And we give ways of constructing models from other models.\nFinally, we see that call-by-value and call-by-name are interpreted within the Kleisli and co-Kleisli parts, respectively, of a call-by-push-value adjunction."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/3/14-03abs.html", "title": "Absolute homology", "authors": "Michael Barr", "keywords": ["absolutely homologous chain maps"], "abstract": "Call two maps, f,g from  C to C', of chain complexes absolutely homologous if for any additive functor F, the induced Ff and Fg are homologous (induce the same map on homology).  It is known that the identity is absolutely homologous to 0 iff it is homotopic to 0 and tempting to conjecture that f and g are absolutely homologous iff they are homotopic.  This conjecture is false, but there is an equational characterization of absolute homology.  I also characterize left absolute and right absolute (in which F is quantified over left or right exact functors)."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/1/14-01abs.html", "title": "Categorical structures enriched in a quantaloid: categories, distributors and functors", "authors": "Isar Stubbe", "keywords": ["Quantales and quantaloids", "enriched categories"], "abstract": "We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid  Q. In analogy with V-category theory we discuss such things as adjoint functors, (pointwise)  left Kan extensions, weighted (co)limits, presheaves and free (co)completion, Cauchy completion  and Morita equivalence. With an appendix on the universality of the quantaloid Dist(Q) of  Q-enriched categories and distributors."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/7/15-07abs.html", "title": "A Galois theory with stable units for simplicial sets", "authors": "Joao J. Xarez", "keywords": ["simplicial object", "simplicial set", "internal category", "internal preorder", "regular category", "Mal'cev category", "descent theory", "Galois theory", "reflection with stable units", "monotone-light factorization", "Kan extension", "elementary topos", "geometric morphism"], "abstract": "We recall and reformulate certain known constructions, in order to make a convenient setting for obtaining generalized monotone-light factorizations in the sense of A. Carboni, G. Janelidze, G. M. Kelly and R. Paré. This setting is used to study the existence of monotone-light factorizations both in categories of simplicial objects and in categories of internal categories. It is shown that there is a non-trivial monotone-light factorization for simplicial sets, such that the monotone-light factorization for reflexive graphs via reflexive relations is a special case of it, obtained by truncation. More generally, we will show that there exists a monotone-light factorization associated with every full subcategory Mono(F_n), n >= 0, consisting of all simplicial sets whose unit morphisms are monic for the localization $F_n:\\mathbf{Set}^{\\Delta^{op}}\\rightarrow\\mathbf{Set}^{\\Delta^{op}_n}$, which truncates each simplicial set after the object of n-simplices. The monotone-light factorization for categories via preorders is as well derived from the proposed setting. We also show that, for regular Mal'cev categories, the reflection of internal groupoids into internal equivalence relations necessarily produces monotone-light factorizations. It turns out that all these reflections do have stable units, in the sense of C. Cassidy, M. Hébert and G. M. Kelly, giving rise to Galois theories."},
{"url": "http://www.tac.mta.ca/tac/volumes/14/2/14-02abs.html", "title": "On essential ring embeddings and the epimorphic hull of C(X)", "authors": "R. Raphael, R.G. Woods", "keywords": ["essential morphism", "epimorphic hull"], "abstract": "Storrer introduced the epimorphic hull of a commutative semiprime ring R and showed that it is (up to isomorphism) the unique essential epic von Neumann regular extension of R.  In the case when  R = C(X) with X a Tychonoff space, we show that the embedding induced  by a dense subspace of X is always essential. This simplifies the search for spaces whose epimorphic hull is a full ring of continuous functions, and allows us to obtain new examples where this occurs.  The main theorem comes close to a characterisation of this phenomenon."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html", "title": "Generic commutative separable algebras and cospans of graphs", "authors": "R. Rosebrugh, N. Sabadini and R.F.C. Walters", "keywords": ["separable algebra", "cospan category"], "abstract": "We show that the generic symmetric monoidal category with a commutative separable algebra which has a $\\Sigma$-family of actions is the category of cospans of finite $\\Sigma$-labelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet $\\Sigma$. We use this result to produce semantic functors for $\\Sigma$-automata."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/4/15-04abs.html", "title": "The shape of a category up to directed homotopy", "authors": "Marco Grandis", "keywords": ["omotopy theory", "adjunctions", "reflective subcategories", "directed algebraic topology", "fundamental category", "concurrent processes"], "abstract": "This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of `directed structures', e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence  of categories. Here we introduce  past and  future equivalences   of categories - sort of symmetric versions of an adjunction - and use them  and their combinations to get `directed models' of a category; in the simplest case, these are the join of the least full reflective and the  least full coreflective subcategory."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/5/15-05abs.html", "title": "Algebraic models of intuitionistic theories of sets and classes", "authors": "S. Awodey and H. Forssell", "keywords": ["algebraic set theory", "topos theory", "sheaf theory"], "abstract": "This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in  Awodey, Butz, Simpson and Streicher (2003) by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html", "title": "Model structures for homotopy of internal categories", "authors": "T. Everaert, R.W. Kieboom and T. Van der Linden", "keywords": ["internal category", "Quillen model category", "homotopy", "homology"], "abstract": "The aim of this paper is to describe Quillen model category structures on the category CatC of internal categories and functors in a given finitely complete category C. Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C.\nUnder mild conditions on C, the regular epimorphism topology determines a model structure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Pare). For a Grothendieck topos C we get a structure that, though different from Joyal and Tierney's, has an equivalent homotopy category. In case C is semi-abelian, these weak equivalences turn out to be homology isomorphisms, and the model structure on CatC induces a notion of homotopy of internal crossed modules. In case C is the category Gp of groups and homomorphisms, it reduces to the case of crossed modules of groups.\nThe trivial topology on a category C determines a model structure on CatC where we is the class of strong equivalences (homotopy equivalences), fib the class of internal functors with the homotopy lifting property, and cof the class of functors with the homotopy extension property. As a special case, the ``folk'' Quillen model category structure on the category Cat = CatSet of small categories is recovered."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/2/15-02abs.html", "title": "Reflective Kleisli subcategories of the category of Eilenberg-Moore algebras for factorization monads", "authors": "Marcelo Fiore and Matias Menni", "keywords": ["factorization systems", "monads", "Kleisli categories", "Schanuel topos", "Joyal species", "combinatorial structures", "power series"], "abstract": "It is well known that for any monad, the associated Kleisli category is embedded in the category of Eilenberg-Moore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left adjoint.  To understand this phenomenon we introduce and study a class of monads arising from factorization systems, and thereby termed factorization monads.  For them we show that under some simple conditions on the factorization system the free algebras are a full reflective subcategory of the algebras.  We provide various examples of this situation of a combinatorial nature."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/31/16-31abs.html", "title": "Generating families in a topos", "authors": "Toby Kenney", "keywords": ["Topoi", "generating families", "cogenerators", "semiprojective objects"], "abstract": "A  generating family, in a category C is a collection of objects $\\{A_i|i\\in I\\}$ such that if for any subobject Y >-->  X, every $f: A_i \\rightarrow X$ factors through m, then m is an isomorphism -  i.e. the functors $C(A_i, - )$ are collectively conservative.  In this paper, we examine some circumstances under which subobjects of 1 form a generating family. Objects for which subobjects of 1 do form a generating family are called partially well-pointed. For a Grothendieck topos, it is well known that subobjects of 1 form a generating family if and only if the topos is localic. For the elementary case, little more is known. The problem is studied by Borceux, where it is shown that the result is internally true, an equivalent condition is found in the boolean case, and certain preservation properties are shown. We look at two different approaches to the problem, one based on a generalization of projectivity, and the other based on looking at the most extreme sorts of counterexamples."},
{"url": "http://www.tac.mta.ca/tac/volumes/15/1/15-01abs.html", "title": "Predicative algebraic set theory", "authors": "Steve Awodey and Michael A. Warren", "keywords": ["algebraic set theory", "categorical logic", "predicativity", "ideal completion", "dependent type theory", "$\\Pi$-pretopos", "small maps"], "abstract": "In this paper the machinery and results developed in [Awodey et al., 2004] are extended to the study of constructive set theories.  Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure.  Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories.  Finally, models of these theories are constructed in the category of ideals."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/32/16-32abs.html", "title": "Copower objects and their applications to finiteness in topoi", "authors": "Toby Kenney", "keywords": ["Topoi", "finiteness", "copower objects"], "abstract": "In this paper, we examine a new approach to topos theory - rather than considering subobjects, look at quotients.  This leads to the notion of a copower object, which is the object of quotients of a given object. We study some properties of copower objects, many of which are similar to the properties of power objects. Given enough categorical structure (i.e. in a pretopos) it is possible to get power objects from copower objects, and vice versa.\nWe then examine some new definitions of finiteness arising from the notion of a copower object. We will see that the most naturally occurring such notions are equivalent to the standard notions, K-finiteness (at least for well-pointed objects) and $\\tilde{K}$-finiteness, but that this new way of looking at them gives new information, and in fact gives rise to another notion of finiteness, which is related to the classical notion of an amorphous set - i.e. an infinite set that is not the disjoint union of two infinite sets.\nFinally, We look briefly at two similar notions: potency objects and per  objects."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/29/16-29abs.html", "title": "Categorified algebra and quantum mechanics", "authors": "Jeffrey Morton", "keywords": [], "abstract": "The process some call `categorification' consists of interpreting set-theoretic structures in mathematics as derived from category-theoretic structures.  Examples include the interpretation of N as the Burnside rig of the category of finite sets with product and coproduct, and of N[x] in terms the category of combinatorial species.  This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal's `combinatorial species', and a new generalization called `stuff types' described by Baez and Dolan, which are a special case of Kelly's `clubs'.  Operators between stuff types be represented as rudimentary Feynman diagrams for the oscillator.  In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these `stuff operators' can do, and these turn out to be closely related.  We will describe a categorification of the quantum harmonic oscillator in which the group of `phases' - that is, U(1), the circle group - plays a special role.  We describe a general notion of `M-stuff types' for any monoid M, and see that the case M = U(1) provides an interpretation of time evolution in the combinatorial setting, as well as recovering the usual Feynman rules for the quantum harmonic oscillator."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/28/16-28abs.html", "title": "Pullback and finite coproduct preserving functors between categories of permutation representations", "authors": "Elango Panchadcharam and Ross Street", "keywords": ["topos", "permutation representations", "limits and colimits", "adjunction", "Mackey  functors"], "abstract": "Motivated by applications to Mackey functors, Serge Bouc characterized pullback and finite coproduct preserving functors between categories of permutation representations of finite groups. Initially surprising to a category theorist, this result does have a categorical explanation which we provide."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/30/16-30abs.html", "title": "A cohomological description of connections and curvature over posets", "authors": "John E. Roberts and  Giuseppe Ruzzi", "keywords": [], "abstract": "What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group G. Interpreting a 1-cocycle as a principal bundle, a connection turns out to be a 1-cochain associated in a suitable way with this 1-cocycle; the curvature of a connection turns out to be its 2-coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G.  We discuss holonomy and prove an analogue of the Ambrose-Singer theorem."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/27/16-27abs.html", "title": "Categories of components and loop-free categories", "authors": "Emmanuel Haucourt", "keywords": ["category of fractions", "generalized congruence", "quotient category", "scwol", "small category without loop", "Yoneda-morphism", "Yoneda-system", "concurrency"], "abstract": "Given a groupoid G one has, in addition to the equivalence of categories E from G to its skeleton, a fibration F from G to its set of connected components (seen as a discrete category).  From the observation that E and F differ unless G[x,x]=id_x for every object x of G, we prove there is a fibered equivalence from C[\\Sigma^{-1}] to C/\\Sigma when \\Sigma is a Yoneda-system of a loop-free category C.  In fact, all the equivalences from C[\\Sigma^{-1}]$ to C/\\Sigma are fibered. Furthermore, since the quotient C/\\Sigma shrinks as \\Sigma grows, we define the component category of a loop-free category as C/{\\overline{\\Sigma}} where \\overline{\\Sigma} is the greatest Yoneda-system of C."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/26/16-26abs.html", "title": "Quotient models of a category up to directed homotopy", "authors": "Marco Grandis", "keywords": ["homotopy theory", "adjunctions", "reflective subcategories", "directed algebraic  topology", "fundamental category", "concurrent processes"], "abstract": "Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A `directed space', e.g. an ordered topological space, has directed homotopies (which are generally non reversible) and a fundamental category (replacing the fundamental groupoid of the classical case). Finding a simple - possibly finite - model of the latter is a non-trivial problem, whose solution gives relevant information on the given `space'; a problem which is of interest for applications as well as in general Category Theory.  Here we continue the work ``The shape of a category up to directed homotopy\", with a deeper analysis of `surjective models', motivated by studying the singularities of 3-dimensional ordered spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/25/16-25abs.html", "title": "Topological *-autonomous categories", "authors": "Michael Barr", "keywords": ["duality", "Chu construction", "Mackey spaces"], "abstract": "Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing object."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/24/16-24abs.html", "title": "Descent for Monads", "authors": "Pieter Hofstra and Federico De Marchi", "keywords": ["Descent theory", "monads", "globular sets"], "abstract": "Motivated by a desire to gain a better understanding of the ``dimension-by-dimension'' decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C |-> Mnd(C) as an indexed category; on this base category, there is a natural topology.  Then we single out a class of monads which are well-behaved with respect to reindexing.  The main result is now, that such monads form a stack. Using this, we can shed some light on the free strict $\\omega$-category monad on globular sets and the free operad-with-contraction monad on the category of collections."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/23/16-23abs.html", "title": "Equivalence of 2D-multitopic category and ana-bicategory", "authors": "Param Jyothi Reddy R", "keywords": ["Multitopic category", "bicategory", "FOLDS"], "abstract": "In this paper equivalence of the concepts of ana-bicategory and the 2D-multitopic category is proved. The equivalence is FOLDS equivalence of the FOLDS-Specifications of the two concepts. Two constructions for transforming one form of category to another are given and it is shown that we get a structure equivalent to the original one when we compose the two constructions."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/22/16-22abs.html", "title": "On categorical crossed modules", "authors": "P. Carrasco, A.R. Garzon and E.M. Vitale", "keywords": ["crossed module", "categorical group", "categorical crossed module", "derivation", "2-exact sequence", "cohomology categorical group"], "abstract": "The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group.  We also prove a suitable 2-dimensional version of the kernel-cokernel lemma for a diagram of categorical crossed modules. We then study derivations with coefficients in categorical crossed modules and show the existence of a categorical crossed module given by inner derivations. This allows us to define the low-dimensional cohomology categorical groups and, finally, these invariants are connected by a six-term 2-exact sequence obtained by using the kernel-cokernel lemma."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/21/16-21abs.html", "title": "Compactifications, C(X) and ring epimorphisms", "authors": "W.D. Burgess and R. Raphael", "keywords": ["epimorphism", "ring of continuous functions", "category of rings", "compactifications"], "abstract": "Given a topological space $X$, $K(X)$ denotes the upper semi-lattice of its (Hausdorff) compactifications. Recent studies have asked when, for $\\alpha X \\in K(X)$, the restriction homomorphism $\\rho : C(\\alpha X) \\to C(X)$ is an epimorphism in the category of commutative rings. This article continues this study by examining the sub-semilattice, $K_{epi}(X)$, of those compactifications where $\\rho$ is an epimorphism along with two of its subsets, and its complement $K_{nepi}(X)$. The role of $K_z(X)\\subseteq K(X)$ of those $\\alpha X$ where $X$ is $z$-embedded in $\\alpha X$, is also examined. The cases where $X$ is a $P$-space and, more particularly, where $X$ is discrete, receive special attention."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/19/16-19abs.html", "title": "Numerology in topoi", "authors": "Peter Freyd", "keywords": ["numerals", "topoi"], "abstract": "This paper studies numerals, natural numbers objects and, more generally, free actions, in a topos. A pre-numeral is a poset with a constant, 0, and a unary operation, s, such that: x\\leq y implies sx \\leq sy x\\leq sx A numeral is a minimal pre-numeral."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/20/16-20abs.html", "title": "Categorical representations of categorical groups", "authors": "John W. Barrett and Marco Mackaay", "keywords": ["categorical group", "categorical representations", "monoidal 2-categories"], "abstract": "A representation theory for (strict) categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these bicategories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/18/16-18abs.html", "title": "Paths in  double categories", "authors": "R. J. MacG. Dawson, R. Paré, and D. A. Pronk", "keywords": ["double categories", "oplax double categories", "paths", "localisation"], "abstract": "Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an  equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster's fc-multicategories, with representable identities in the  second case."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/17/16-17abs.html", "title": "Spectra of finitely generated Boolean flows", "authors": "John F. Kennison", "keywords": ["Boolean flow", "dynamical systems", "spectrum", "sheaf"], "abstract": "A flow on a compact Hausdorff space X is given by a map  t : X --> X. The general goal of this paper is to find the \"cyclic parts\" of such a flow. To do this, we approximate (X,t) by a flow on a Stone space (that is, a totally disconnected, compact Hausdorff space). Such a flow can be examined by analyzing the resulting flow on the Boolean algebra of clopen subsets, using the spectrum defined in our previous TAC paper, The cyclic spectrum of a Boolean flow.  In this paper, we describe the cyclic spectrum in terms that do not rely on topos theory. We then compute the cyclic spectrum of any finitely generated Boolean flow. We define when a sheaf of Boolean flows can be regarded as cyclic and find necessary conditions for representing a Boolean flow using the global sections of such a sheaf. In the final section, we define and explore a related spectrum based on minimal subflows of Stone spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/16/16-16abs.html", "title": "TFT Construction of RCFT correlators", "authors": "Jens Fjelstad, Jurgen Fuchs, Ingo Runkel and Christoph Schweigert", "keywords": [], "abstract": "The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of Fuchs, Runkel and Schweigert are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints. We present results both for conformal field theories defined on oriented surfaces and for theories defined on unoriented surfaces. In the latter case, in particular the so-called cross cap constraint is included."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/15/16-15abs.html", "title": "Stable meet semilattice fibrations and free restriction categories", "authors": "J.R.B. Cockett and Xiuzhan Guo", "keywords": ["Restriction categories", "fibrations", "semigroups"], "abstract": "The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are `unitary', in a sense which generalizes that from the theory of inverse semigroups.   Characterization theorems for unitary restriction categories are derived.   The paper ends with an explicit description of the free restriction category on a directed graph."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/14/16-14abs.html", "title": "Categorical structures enriched in a quantaloid: tensored and cotensored categories", "authors": "Isar Stubbe", "keywords": ["quantaloid", "enriched category", "weighted (co)limit", "module"], "abstract": "A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid $\\mathcal{Q}$. We show how cocomplete $\\mathcal{Q}$-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and `order-cocomplete'. In fact, tensors and cotensors in a $\\mathcal{Q}$-category determine, and are determined by, certain adjunctions in the category of $\\mathcal{Q}$-categories; some of these adjunctions can be reduced to adjuctions in the category of ordered sets. Bearing this in mind, we explain how tensored $\\mathcal{Q}$-categories are equivalent to order-valued closed pseudofunctors on $\\mathcal{Q}^{op}$; this result is then finetuned to obtain in particular that cocomplete $\\mathcal{Q}$-categories are equivalent to sup-lattice-valued homomorphisms on $\\mathcal{Q}^{op}$ (a.k.a.\\ $\\mathcal{Q}$-modules)."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/13/16-13abs.html", "title": "Closedness properties of internal relations II: Bourn localization", "authors": "Zurab Janelidze", "keywords": ["Mal'tsev", "unital and subtractive categories", "fibration of points"], "abstract": "We say that a class $\\mathbb{D}$ of categories is the Bourn localization of a class $\\mathbb{C}$ of categories, and we write $\\mathbb{D} = \\mathrm{Loc}\\mathbb{C}$, if $\\mathbb{D}$ is the class of all (finitely complete) categories $\\mathcal{D}$ such that for each object $A$ in $\\mathcal{D}$, $\\mathrm{Pt}(\\mathcal{D}\\downarrow A) \\in \\mathbb{C}$, where $\\mathrm{Pt}(\\mathcal{D}\\downarrow A)$ denotes the category of all pointed objects in the comma-category $(\\mathcal{D}\\downarrow A)$. As D. Bourn showed, if we take $\\mathbb{D}$ to be the class of Mal'tsev categories in the sense of A. Carboni, J. Lambek, and M. C. Pedicchio, and $\\mathbb{C}$ to be the class of unital categories in the sense of D. Bourn, which generalize pointed Jónsson-Tarski varieties, then $\\mathbb{D} = \\mathrm{Loc}(\\mathbb{C})$. A similar result was obtained by the author: if $\\mathbb{D}$ is as above and $\\mathbb{C}$ is the class of subtractive categories, which generalize pointed subtractive varieties in the sense of A. Ursini, then $\\mathbb{D} = \\mathrm{Loc}(\\mathbb{C})$. In the present paper we extend these results to abstract classes of categories obtained from classes of varieties. We also show that the Bourn localization of the union of the classes of unital and subtractive categories is still the class of Mal'tsev categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/10/16-10abs.html", "title": "A characterization of quantic quantifiers in orthomodular lattices", "authors": "Leopoldo Román", "keywords": ["Quantum logic", "orthomodular lattice."], "abstract": "Let L be an arbitrary orthomodular lattice. There is a one to one correspondence between orthomodular sublattices of L satisfying an extra condition and quantic quantifiers. The category of orthomodular lattices is equivalent to the category  of posets having two families of endofunctors satisfying six conditions."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/12/16-12abs.html", "title": "Closedness properties of internal relations I:  A unified approach to Mal'tsev, unital and subtractive categories", "authors": "Zurab Janelidze", "keywords": ["Mal'tsev", "unital and subtractive category/variety", "Jónsson-Tarski variety", "term condition", "internal relation"], "abstract": "We study closedness properties of internal relations in finitely complete categories, which leads to developing a unified approach to: Mal'tsev categories, in the sense of A. Carboni, J. Lambek and M. C. Pedicchio, that generalize Mal'tsev varieties of universal algebras; unital categories, in the sense of D. Bourn, that generalize pointed Jónsson-Tarski varieties; and subtractive categories, introduced by the author, that generalize pointed subtractive varieties in the sense of A. Ursini."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/11/16-11abs.html", "title": "Exponentiability in lax slices of", "authors": "Susan Niefield", "keywords": ["exponentiable space", "function space", "lax slice", "specialization order"], "abstract": "We consider exponentiable objects in lax slices of Top with respect to the specialization order (and its opposite) on a base space B.   We begin by showing that the lax slice over B has binary products which are preserved by the forgetful functor to Top if and only if B is a meet (respective, join) semilattice in Top, and go on to characterize exponentiability over a complete Alexandrov space B."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/9/16-09abs.html", "title": "Free A_\\infty-categories", "authors": "Volodymyr Lyubashenko  and Oleksandr Manzyuk", "keywords": ["A_\\infty-categories", "A_\\infty-functors", "A_\\infty-transformations", "2-category", "free A_\\infty-category"], "abstract": "For a differential graded k-quiver Q we define the free A_\\infty-category FQ generated by Q. The main result is that the restriction A_\\infty-functor A_\\infty(FQ, A) \\to A_1(Q,A) is an equivalence, where objects of the last A_\\infty-category are morphisms of differential graded k-quivers Q \\to  A."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/8/16-08abs.html", "title": "Thin fillers in the cubical nerves of omega-categories", "authors": "Richard Steiner", "keywords": ["omega-category", "cubical nerve", "stratified precubical set", "cubical T-complex", "thin filler"], "abstract": "It is shown that the cubical nerve of a strict omega-category is a sequence of sets with cubical face operations and distinguished subclasses of thin elements satisfying certain thin filler conditions. It is also shown that a sequence of this type is the cubical nerve of a strict omega-category unique up to isomorphism; the cubical nerve functor is therefore an equivalence of categories. The sequences of sets involved are the analogues of cubical T-complexes appropriate for strict omega-categories. Degeneracies are not required in the definition of these sequences, but can in fact be constructed as thin fillers. The proof of the thin filler conditions uses chain complexes and chain homotopies."},
{"url": "http://www.tac.mta.ca/tac/volumes/16/1/16-01abs.html", "title": "Monads of effective descent type and comonadicity", "authors": "Bachuki Mesablishvili", "keywords": ["Monad of effective descent type", "(co)monadicity", "separable functor", "coring", "descent data"], "abstract": "We show, for an arbitrary adjunction $F \\dashv U : \\cal B \\to \\cal A$ with $\\cal B$ Cauchy complete, that the functor $F$  is comonadic if and only if the monad $T$ on $\\cal A$ induced by the adjunction is of effective descent type, meaning that the free $T$-algebra functor $F^{T}: \\cal A \\to \\cal A^{T}$ is comonadic. This result is applied to several situations: In Section 4 to give a sufficient condition for an exponential functor on a cartesian closed category to be monadic, in Sections 5 and 6 to settle the question of the comonadicity of those functors whose domain is Set, or  Set$_{\\star}$, or the category of modules over a semisimple ring, in Section 7 to study the effectiveness  of (co)monads on module categories. Our final application is a descent theorem for noncommutative rings from which we deduce an important result of A. Joyal and M. Tierney and of J.-P. Olivier, asserting that the effective descent morphisms in the opposite of the category of commutative unital rings are precisely the pure monomorphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/4/17-04abs.html", "title": "The Dialectica interpretation of first-order classical affine logic", "authors": "Masaru Shirahata", "keywords": ["linear logic", "dialectica interpretation", "categorical logic"], "abstract": "We give a Dialectica-style interpretation of first-order classical affine logic. By moving to a contraction-free logic, the translation (a.k.a.   D-translation) of a first-order formula into a higher-type $\\exists\\forall$-formula can be made symmetric with respect to duality, including exponentials. It turned out that the propositional part of our D-translation uses the same construction as de Paiva's dialectica category GC and we show how our D-translation extends GC to the first-order setting in terms of an indexed category. Furthermore the combination of Girard's ?!-translation and our D-translation results in the essentially equivalent $\\exists\\forall$-formulas as the double-negation translation and Godel's original D-translation."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/7/17-07abs.html", "title": "Dialectica and Chu Constructions: Cousins?", "authors": "Valeria de Paiva", "keywords": ["linear logic", "categorical models", "monoidal     closed categories", "monoidal comonad", "dialectica categories", "chu spaces"], "abstract": "This note investigates two generic constructions used to produce  categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The  constructions have the same objects, but are rather different in other ways. We discuss similarities and differences and prove that the Dialectica construction can be done over a symmetric monoidal closed basis. We also point out interesting open problems concerning the Dialectica  construction."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/6/17-06abs.html", "title": "An extended view of the Chu-construction", "authors": "Jurgen Koslowski", "keywords": ["closed bicategory", "Chu-spans", "games", "bipartite state transition systems"], "abstract": "The cyclic Chu-construction for closed bicategories with   pullbacks, which generalizes the original Chu-construction for   symmetric monoidal closed categories, turns out to have a non-cyclic   counterpart.  Both use so-called Chu-spans as new 1-cells between   1-cells of the underlying bicategory, which form the new objects.   Chu-spans may be seen as a natural generalization of 2-cell-spans   in the base bicategory that no longer are confined to a single   hom-category.  This view helps to clarify the composition of   Chu-spans.\nWe consider various approaches of linking the underlying bicategory   with the newly constructed ones, e.g. by means of two-dimensional   generalizations of bifibrations.  In the quest for a better   connection, we investigate, whether Chu-spans form a double   category.  While this turns out not to be the case, we are led to   considering a generalization of the construction to paths of 1-cells in   the base, leading to two hierarchies of closed bicategories, one for   linear paths and one for loops.  The possibility of moving beyond   paths, respectively, loops of the same length is indicated.\nFinally, Chu-spans in rel are identified as bipartite state  transition   systems.  Even though their composition may fail here due to the   lack of pullbacks in rel, basic game-theoretic constructions can   be performed on cyclic Chu-spans.  These are available in all   symmetric monoidal closed categories with finite products.  If   pullbacks exist as well, the bicategory of cyclic Chu-spans inherits   a monoidal structure that on objects coincides with the categorical   product."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/5/17-05abs.html", "title": "Approximable Concepts, Chu spaces, and  information systems", "authors": "Guo-Qiang Zhang and Gongqin Shen", "keywords": ["Formal concept analysis", "domain theory", "Chu spaces"], "abstract": "This paper serves to bring three independent but important areas of computer science to a common meeting point: Formal Concept Analysis (FCA), Chu Spaces, and Domain Theory (DT). Each area is given a perspective or reformulation that is conducive to the flow of ideas and to the exploration of cross-disciplinary connections. Among other results, we show that the notion of state in Scott's information system corresponds precisely to that of formal concepts in FCA with respect to all finite Chu spaces, and the entailment relation corresponds to ``association rules\". We introduce, moreover, the notion of approximable concept and show that approximable concepts represent algebraic lattices which are identical to Scott domains except the inclusion of a top element. This notion serves as a stepping stone in recent work in which a new notion of morphism on formal contexts results in a category equivalent to (a) the category of complete algebraic lattices and Scott continuous functions, and (b) a category of information systems and approximable mappings."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/3/17-03abs.html", "title": "A Parigot-style linear $\\lambda$-calculus for full intuitionistic linear  logic", "authors": "Valeria de Paiva and Eike Ritter", "keywords": ["linear logic", "$\\lambda\\mu$-calculus", "Curry-Howard isomorphism"], "abstract": "This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL),  an intriguing variation of  multiplicative linear logic, due to Hyland and de Paiva.  The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequent-calculus formulation has intrinsic multiple conclusions. From the intrinsic multiple conclusions comes the  inspiration to modify Parigot's natural deduction systems for classical logic, to produce a natural deduction formulation and a term assignment system for FILL."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/2/17-02abs.html", "title": "Coherence of the Double Involution on *-Autonomous Categories", "authors": "J.R.B. Cockett, M. Hasegawa and R.A.G. Seely", "keywords": [], "abstract": "We show that any free *-autonomous category is equivalent (in a strict sense) to a free *-autonomous category in which the double-involution $(-)^{**}$ is the identity functor and the canonical isomorphism $A\\simeq A^{**}$ is an identity arrow for all $A$."},
{"url": "http://www.tac.mta.ca/tac/volumes/17/1/17-01abs.html", "title": "The Chu construction: history of an idea", "authors": "Michael Barr", "keywords": ["Chu categories", "*-autonomous categories", "history"], "abstract": "This paper describes the historical background and motivation involved in the discovery (or invention) of Chu categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/22/18-22abs.html", "title": "Familial 2-functors and parametric right adjoints", "authors": "Mark Weber", "keywords": ["nerves", "parametric right adjoints", "operads", "familial 2-functors", "fibrations", "2-categories"], "abstract": "We define and study familial 2-functors primarily with a view to the development of the 2-categorical approach to operads of [Weber, 2005].  Also included in this paper is a result in which the well-known characterisation of a category as a simplicial set via the Segal condition, is generalised to a result about nice monads on cocomplete categories. Instances of this general result can be found in [Leinster, 2004], [Berger, 2002] and [Moerdijk-Weiss, 2007b]. Aspects of  this general theory are then used to show that the composite 2-monads of [Weber, 2005]  that describe symmetric and braided analogues of the $\\omega$-operads of [Batanin, 1998], are cartesian 2-monads and their underlying  endo-2-functor is familial. Intricately linked to the notion of familial 2-functor is the theory of fibrations in a finitely complete 2-category [Street, 1974]  [Street, 1980], and those aspects of that theory that we require, that weren't discussed in [Weber, 2007], are reviewed here."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/21/18-21abs.html", "title": "Quotients of tannakian categories", "authors": "J.S. Milne", "keywords": ["gerbes", "motives", "tannakian categories"], "abstract": "We classify the \"quotients\" of a tannakian category in which the objects of a tannakian subcategory become trivial, and we examine the properties of such quotient categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/20/18-20abs.html", "title": "A universal property of the monoidal 2-category of cospans of ordinals and  surjections", "authors": "M. Menni, N. Sabadini and R. F. C. Walters", "keywords": ["2-categories", "monoidal categories", "structured objects"], "abstract": "We prove that the monoidal 2-category of cospans of ordinals and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/19/18-19abs.html", "title": "Collared cospans, cohomotopy and TQFT  (cospans in algebraic topology, II)", "authors": "Marco Grandis", "keywords": ["spans", "cospans", "bicategory", "weak double category", "homotopy pushout", "cohomotopy functors", "cobordism", "knots", "links", "tangles", "topological  quantum field theories"], "abstract": "Topological cospans and their concatenation, by pushout, appear in the theories of tangles, ribbons, cobordisms, etc. Various algebraic invariants have been introduced for their study, which it would be interesting to link with the standard tools of Algebraic Topology, (co)homotopy and (co)homology functors.\nHere we introduce collarable (and collared)  cospans between topological spaces. They generalise the cospans which appear in the previous theories, as a consequence of a classical theorem on manifolds with boundary. Their interest lies in the fact that their concatenation is realised by means of homotopy pushouts. Therefore,  cohomotopy functors induce `functors' from collarable cospans to  spans of sets, providing - by linearisation - topological quantum field theories (TQFT) on manifolds and their cobordisms. Similarly, (co)homology and homotopy functors take collarable cospans to relations of abelian groups or (co)spans of groups, yielding other `algebraic' invariants.\nThis is the second paper in a series devoted to the study of cospans in Algebraic Topology. It is practically independent from the first, which deals with higher cubical cospans in abstract categories. The third article will proceed from both, studying cubical topological cospans and their collared version."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/16/18-16abs.html", "title": "Nuclei of categories with tensor products", "authors": "Alexei Davydov", "keywords": ["monoidal categories", "monoidal functors", "quasi-bialgebras"], "abstract": "Following the analogy between algebras (monoids) and monoidal categories the construction of nucleus for non-associative algebras is simulated on the categorical level.\nNuclei of categories of modules are considered as an example."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/15/18-15abs.html", "title": "Enriched model categories and an application to additive endomorphism spectra", "authors": "Daniel Dugger and Brooke Shipley", "keywords": ["model categories", "symmetric spectra", "endomorphism ring"], "abstract": "We define the notion of an additive model category and prove that any stable, additive, combinatorial model category $\\cal M$ has a model enrichment over $Sp^\\Sigma(sAb)$ (symmetric spectra based on simplicial abelian groups).  So to any object $X$ in $\\cal M$ one can attach an endomorphism ring object, denoted $hEnd_ad(X)$, in the category $Sp^\\Sigma(sAb)$.  We establish some useful properties of these endomorphism rings.\nWe also develop a new notion in enriched category theory which we call `adjoint modules'.  This is used to compare enrichments over one symmetric monoidal model category with enrichments over a Quillen equivalent one.  In particular, it is used here to compare enrichments over $\\Sp^\\Sigma(s\\Ab)$ and chain complexes."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/14/18-14abs.html", "title": "Dense morphisms of monads", "authors": "Panagis Karazeris, Jiri Velebil", "keywords": ["Definable operation", "monad morphism", "locally finitely           presentable category"], "abstract": "Given an arbitrary locally finitely presentable category $K$ and finitary monads $T$ and $S$ on $K$, we characterize monad morphisms $\\alpha: S\\to T$ with the property that the induced functor $\\alpha_*: K^T \\to K^ S$ between the categories of Eilenberg-Moore algebras is fully faithful. We call such monad morphisms dense and give a characterization of them in the spirit of Beth's definability theorem: $\\alpha$ is a dense monad morphism if and only if every $T$-operation is explicitly defined using $S$-operations. We also give a characterization in terms of epimorphic property of $\\alpha$ and clarify the connection between various notions of epimorphisms between monads."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/13/18-13abs.html", "title": "Coherent unit actions on regular operads and Hopf algebras", "authors": "Kurusch Ebrahimi-Fard and Li Guo", "keywords": ["dendriform algebras", "coherent unit actions", "regular operads", "Hopf algebras"], "abstract": "J.-L. Loday introduced the concept of coherent unit actions on a regular operad and showed that such actions give Hopf algebra structures on the free algebras. Hopf algebras obtained this way include the Hopf algebras of shuffles, quasi-shuffles and planar rooted trees. We characterize coherent unit actions on binary quadratic regular operads in terms of linear equations of the generators of the operads. We then use these equations to classify operads with coherent unit actions. We further show that coherent unit actions are preserved under taking products and thus yield Hopf algebras on the free object of the product operads when the factor operads have coherent unit actions. On the other hand, coherent unit actions are never preserved under taking the dual in the operadic sense except for the operad of associative algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/12/18-12abs.html", "title": "Higher cospans and weak cubical categories (Cospans in algebraic topology, I)", "authors": "Marco Grandis", "keywords": ["weak cubical category", "multiple category", "double category", "cubical sets", "spans", "cospans"], "abstract": "We define a notion of  weak cubical category, abstracted from the structure of n-cubical cospans $x : \\wedge^n \\to X$ in a category $X$ where $\\wedge$ is the `formal cospan' category. These diagrams form a cubical set with compositions $x +_i y$ in all directions, which are computed using pushouts and behave `categorically' in a weak sense, up to suitable comparisons. Actually, we work with a `symmetric cubical structure', which includes the transposition symmetries, because this allows for a strong simplification of the coherence conditions. These notions will be used in subsequent papers to study topological cospans and their use in Algebraic Topology, from tangles to cobordisms of manifolds.\nWe also introduce the more general notion of a multiple category, where - to start with - arrows belong to different sorts, varying in a countable family, and symmetries must be dropped. The present examples seem to show that the symmetric cubical case is better suited for topological applications."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html", "title": "Towards an n-category of cobordisms", "authors": "Eugenia Cheng and Nick Gurski", "keywords": ["n-category", "operad", "topological quantum field theory", "cobordism"], "abstract": "We discuss an approach to constructing a weak n-category of cobordisms. First we present a generalisation of Trimble's definition of n-category which seems most appropriate for this construction; in this definition composition is parametrised by a contractible operad.  Then we show how to use this definition to define the n-category nCob, whose k-cells are k-cobordisms, possibly with corners.  We follow Baez and Langford in using ``manifolds embedded in cubes'' rather than general manifolds.  We make the construction for 1-manifolds embedded in 2- and 3-cubes.  For general dimensions k and n we indicate what the construction should be."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/9/18-09abs.html", "title": "More morphisms between bundle gerbes", "authors": "Konrad Waldorf", "keywords": ["2-category", "bundle gerbe", "holonomy"], "abstract": "Usually bundle gerbes are considered as objects of a 2-groupoid, whose 1-morphisms, called stable isomorphisms, are all invertible.  I introduce new 1-morphisms which include stable isomorphisms, trivializations and bundle gerbe modules. They fit into the structure of a 2-category of bundle gerbes, and lead to natural definitions of surface holonomy for closed surfaces, surfaces with boundary, and unoriented closed surfaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/8/18-08abs.html", "title": "Quasi locally connected toposes", "authors": "Marta Bunge and Jonathon Funk", "keywords": ["complete spreads", "distributions", "zero-dimensional locales", "comprehensive  factorization"], "abstract": "We have shown that complete spreads (with a locally connected domain) over a bounded topos E (relative to S) are `comprehensive' in the sense that they are precisely the second factor of a factorization associated with an instance of the comprehension scheme involving S-valued distributions on E. Lawvere has asked whether the `Michael coverings' (or complete spreads with a definable dominance domain) are comprehensive in a similar fashion. We give here a positive answer to this question. In order to deal effectively with the comprehension scheme in this context, we introduce a notion of an `extensive topos doctrine,' where the extensive quantities (or distributions) have values in a suitable subcategory of what we call `locally discrete' locales. In the process we define what we mean by a quasi locally connected topos, a notion that we feel may be of interest in its own right."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/11/18-11abs.html", "title": "The theory of core algebras: its completeness", "authors": "Peter Freyd", "keywords": ["core", "cored category", "abstract core algebra", "critical lemma", "no-lost-variables", "base cancellation"], "abstract": "The core of a category (first defined in ``Core algebra revisited'' Theoretical Computer Science, Vol 375, Issues 1-3, pp 193-200) has the structure of an abstract core algebra (first defined in the same place). A question was left open: is there more structure yet to be defined? The answer is no: it is shown that any operation on an object arising from the fact that the object is the core of its category can be defined using only the constant and two binary operations that appear in the definition of abstract core algebra. In the process a number of facts about abstract core algebras must be developed."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/7/18-07abs.html", "title": "Monad compositions I: general constructions and recursive distributive laws", "authors": "Ernie Manes and Philip Mulry", "keywords": ["distributive law", "linear equation"], "abstract": "New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursively-defined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a  monad."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/6/18-06abs.html", "title": "CCD lattices in presheaf categories", "authors": "G. S. H. Cruttwell, F. Marmolejo and R. J. Wood", "keywords": ["Constructive complete distributivity", "presheaves", "Beck-Chevalley condition", "Frobenius reciprocity", "change of base"], "abstract": "In this paper we give a characterization of constructively completely distributive (CCD) lattices in presheaves on C, for C a small category with pullbacks."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/5/18-05abs.html", "title": "Pullback and finite coproduct preserving functors between categories of permutation representations: Corrigendum and Addendum", "authors": "Elango Panchadcharam and Ross Street", "keywords": ["topos", "permutation representations", "limits and colimits", "adjunction", "Mackey functors"], "abstract": "Francisco Marmolejo pointed out a mistake in the statement of Proposition 4.4 in our TAC paper (Vol. 16, No. 28). The mistaken version is used later in that paper. Our purpose here is to correct the error by providing an explicit description of the finite coproduct completion of the dual of the category of connected G-sets. The description uses the distinguished morphisms of a factorization system on the category of G-sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/4/18-04abs.html", "title": "On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups", "authors": "Joao Faria Martins and Timothy Porter", "keywords": ["Categorical Groups", "Crossed Modules", "Cohomology of Crossed Modules", "State Sum Invariants of Manifolds", "Dijkgraaf-Witten Invariant", "Yetter's Invariant"], "abstract": "We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(\\cal G))$, where \\cal G is a crossed module and B(\\cal G) is its classifying space.  From this formulation, there follows that Yetter's invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module \\cal G. We use this interpretation to define a twisting of Yetter's Invariant by cohomology classes of crossed modules, defined as cohomology classes of their classifying spaces, in the form of a state sum invariant. In particular, we obtain an extension of the Dijkgraaf-Witten Invariant of manifolds to categorical groups. The straightforward extension to crossed complexes is also considered."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/1/18-01abs.html", "title": "Bicat is not triequivalent to Gray", "authors": "Stephen Lack", "keywords": ["bicategory", "tricategory", "Gray-category", "coherence"], "abstract": "Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications. Gray is the subtricategory of 2-categories, 2-functors, pseudonatural transformations, and modifications. We show that these two tricategories are not triequivalent."},
{"url": "http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html", "title": "Cartesian bicategories II", "authors": "A. Carboni, G.M. Kelly, R.F.C. Walters, and R.J. Wood", "keywords": ["bicategory", "finite products", "monoidal bicategory"], "abstract": "The notion of cartesian bicategory, introduced by  Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory."},
{"url": "http://www.tac.mta.ca/tac/volumes/19/5/19-05abs.html", "title": "Iterative algebras: How iterative are they?", "authors": "J. Adamek, R. Borger, S. Milius, and J. Velebil", "keywords": ["iterative algebra", "guarded equation", "strict solution", "extensive category"], "abstract": "Iterative algebras, defined by the property that every guarded system of recursive equations has a unique solution, are proved to have a much stronger property: every system of recursive equations has a unique strict solution. Those systems that have a unique solution in every iterative algebra are characterized."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/3/18-03abs.html", "title": "Functional analysis on normed spaces: the  Banach space comparison", "authors": "M. Sioen, S. Verwulgen", "keywords": ["Banach space", "Monad", "Totally convex module", "Duality"], "abstract": "It is the aim of this paper to compute the category of Eilenberg-Moore algebras for the monad arising from the dual unit-ball functor on the category of (semi)normed spaces.  We show that this gives rise to a stronger algebraic structure than the totally convex one obtained from the closed unit ball functor on the category of Banach spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/18/2/18-02abs.html", "title": "Polarized category theory, modules, and game semantics", "authors": "J.R.B. Cockett and R.A.G. Seely", "keywords": ["polarized categories", "polarized linear logic", "game semantics", "theory of  communication"], "abstract": "Motivated by an analysis of Abramsky-Jagadeesan games, the paper considers a categorical semantics for a polarized notion of two-player games, a semantics which has close connections with the logic of (finite cartesian) sums and products, as well as with the multiplicative structure of linear logic.  In each case, the structure is polarized, in the sense that it will be modelled by two categories, one for each of two polarities, with a module structure connecting them. These are studied in considerable detail, and a comparison is made with a different notion of polarization due to Olivier Laurent: there is an adjoint connection between the two notions."},
{"url": "http://www.tac.mta.ca/tac/volumes/19/4/19-04abs.html", "title": "Q-modules are Q-suplattices", "authors": "Isar Stubbe", "keywords": ["Quantaloid", "quantale", "locale", "ordered sheaf", "module", "centre", "KZ doctrine"], "abstract": "It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, we refine a result of F. Borceux and E. Vitale."},
{"url": "http://www.tac.mta.ca/tac/volumes/19/2/19-02abs.html", "title": "Components, complements and the reflection formula", "authors": "Claudio Pisani", "keywords": ["categories over a base", "discrete fibrations", "reflection", "components", "tensor", "complement", "strong dinaturality", "limits and colimits", "atoms", "idempotents", "graphs and evolutive sets"], "abstract": "We illustrate the formula $ (\\downarrow p)x = \\Gamma_!(x/p) $, which gives the reflection $\\downarrow p$ of a category $p : P \\to X$ over  $X$ in  discrete fibrations. One of its proofs is based on a ``complement operator\" which takes a  discrete fibration $A$ to the functor $\\neg A$, right adjoint to $\\Gamma_!(A\\times-):Cat/X \\to Set$ and  valued in discrete opfibrations.  Some consequences and applications are presented."},
{"url": "http://www.tac.mta.ca/tac/volumes/19/3/19-03abs.html", "title": "Axiomatic Cohesion", "authors": "F. William Lawvere", "keywords": ["Cohesion", "qualities", "graphs", "nature of space"], "abstract": "The nature of the spatial background for classical analysis and for modern theories of continuum physics requires more than the partial invariants of locales and cohomology rings for its description. As Maxwell emphasized, this description has various levels of precision depending on the needs of investigation. These levels correspond to different categories of space, all of which have intuitively the feature of cohesion. Our aim here is to continue the axiomatic study of such categories, which involves the following aspects: I.      Categories of space as cohesive backgrounds II.     Cohesion versus non-cohesion; quality types III.    Extensive quality; intensive quality in its rarefied and  condensed aspects; the canonical qualities form and substance IV.     Non-cohesion within cohesion via constancy on infinitesimals V.      The example of reflexive graphs and their atomic numbers VI.     Sufficient cohesion and the Grothendieck condition VII.    Weak generation of a subtopos by a quotient topos I look forward to further work on each of these aspects, as well as development of categories of dynamical laws, constitutive relations, and other mathematical structures that naturally live in cohesive categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html", "title": "Framed bicategories and monoidal fibrations", "authors": "Michael Shulman", "keywords": [], "abstract": "In some bicategories, the 1-cells are `morphisms' between the   0-cells, such as functors between categories, but in others they are   `objects' over the 0-cells, such as bimodules, spans, distributors,   or parametrized spectra.  Many bicategorical notions do not work   well in these cases, because the `morphisms between 0-cells', such   as ring homomorphisms, are missing.  We can include them by using a   pseudo double category, but usually these morphisms also induce base   change functors acting on the 1-cells.  We avoid complicated   coherence problems by describing base change `nonalgebraically',   using categorical fibrations.  The resulting `framed bicategories'   assemble into 2-categories, with attendant notions of equivalence,   adjunction, and so on which are more appropriate for our examples   than are the usual bicategorical ones.\nWe then describe two ways to construct framed bicategories.  One is   an analogue of rings and bimodules which starts from one framed   bicategory and builds another.  The other starts from a `monoidal   fibration', meaning a parametrized family of monoidal categories,   and produces an analogue of the framed bicategory of spans.   Combining the two, we obtain a construction which includes both   enriched and internal categories as special cases."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/16/20-16abs.html", "title": "Cohomology theory in 2-categories", "authors": "Hiroyuki Nakaoka", "keywords": ["symmetric categorical group", "2-category", "cohomology", "exact sequence"], "abstract": "Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups SCG are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example. As a main theorem, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify the proofs, by analogy with abelian categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/19/1/19-01abs.html", "title": "Exponentiability in homotopy slices of", "authors": "Susan Niefield", "keywords": ["pseudo-exponentiable", "Kleisli bicategory", "homotopy slice", "pseudo-slice"], "abstract": "We prove a general theorem relating pseudo-exponentiable objects of a  bicategory K to those of the Kleisli bicategory of a pseudo-monad on K.  This theorem is  applied to obtain pseudo-exponentiable objects of the homotopy slices Top//B of the category of  topological spaces and the pseudo-slices Cat//B of the category of small categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/17/20-17abs.html", "title": "Propriétés universelles et extensions de Kan  dérivées", "authors": "Denis-Charles Cisinski", "keywords": ["derivator", "homotopy theory", "derived Kan extension", "homotopy colimit", "homotopy limit"], "abstract": "We prove that, given any small category A$=, the derivator HOT_A, corresponding to the homotopy theory of presheaves of homotopy types on A, is characterized by a natural universal property. In particular, the theory of Kan extensions extends to the setting of Grothendieck derivators."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/15/20-15abs.html", "title": "Isbell duality", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["duality", "fixed categories", "sober objects"], "abstract": "We develop in some generality the dualities that often arise when one object lies in two different categories.  In our examples, one category is equational and the other consists of the topological objects in a (generally different) equational category."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/14/20-14abs.html", "title": "Core varieties, extensivity, and rig geometry", "authors": "F. William Lawvere", "keywords": ["topos", "Frobenius", "dimension rigs"], "abstract": "The role of the Frobenius operations in analyzing finite spaces, as well as the extended algebraic geometry over rigs, depend partly on varieties (Birkhoffian inclusions of algebraic categories) that have coreflections as well as reflections and whose dual category of affine spaces is extensive. Even within the category of those rigs where  1 + 1 = 1, not only distributive lattices but also the function algebras of tropical geometry (where x + 1 = 1) and the dimension rigs of separable prextensive categories (where x + x^2 = x^2) enjoy those features. (Talk given at CT08, Calais.)"},
{"url": "http://www.tac.mta.ca/tac/volumes/20/13/20-13abs.html", "title": "Quotients of unital $A_\\infty$-categories", "authors": "Volodymyr Lyubashenko and Oleksandr Manzyuk", "keywords": ["$A_\\infty$-categories", "$A_\\infty$-categories", "$A_\\infty$-functors", "$A_\\infty$-transformations", "2-categories", "2-functors"], "abstract": "Assuming that $B$ is a full $A_\\infty$-subcategory of a unital $A_\\infty$-category $\\cc$ we construct the quotient unital  $A_\\infty$-category $\\cd=$`$\\cc/\\cb$'. It represents the $A_\\infty^u$-2-functor $A \\mapsto A_\\infty^u(C,A)_{mod B}$, which associates with a given unital $A_\\infty$-category $A$ the $A_\\infty$-category of unital $A_\\infty$-functors $C \\to A$, whose restriction to $B$ is  contractible. Namely, there is a unital $A_\\infty$-functor $e: C \\to D$ such that the composition $B \\hookrightarrow C \\to^e D$ is contractible, and for an arbitrary unital $A_\\infty$-category $A$ the restriction $A_\\infty$-functor  $(e\\boxtimes 1)M : A_\\infty^u(D,A)\\to A_\\infty^u(C,A)_{mod B}$ is an equivalence.\nLet $C_k$ be the differential graded category of differential graded $k$-modules. We prove that the Yoneda $A_\\infty$-functor $Y: A \\to A_\\infty^u(A^{op}, C_k)$ is a full embedding for an arbitrary unital $A_\\infty$-category $A$. In particular, such $A$ is $A_\\infty$-equivalent to a differential graded category with the same set  of objects."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/12/20-12abs.html", "title": "Colimits of representable algebra-valued functors", "authors": "George M. Bergman", "keywords": ["representable functor among varieties of algebras", "initial representable functor", "colimit of representable functors", "final coalgebra", "limit of coalgebras", "binar (set with one binary operation)", "semigroup", "monoid", "group", "ring", "Boolean ring", "Stone topological algebra"], "abstract": "If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C --> D we understand a functor which, when composed with the forgetful functor D --> Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined by D-coalgebra objects of C. Let Rep(C, D) denote the category of all such functors, a full subcategory of Cat(C, D, opposite to the category of D-coalgebras in C.\nIt is proved that Rep(C, D) has small colimits, and in certain situations, explicit constructions for the representing coalgebras are obtained.\nIn particular, Rep(C, D) always has an initial object. This is shown to be ``trivial'' unless C and D either both have no zeroary operations, or both have more than one derived zeroary operation. In those two cases, the functors in question may have surprisingly opulent structures.  It is also shown that every set-valued representable functor on C admits a universal morphism to a D-valued representable functor.  Several examples are worked out in detail, and areas for further investigation are noted."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/11/20-11abs.html", "title": "Star-autonomous functor categories", "authors": "Jeff Egger", "keywords": ["Linear distributive categories", "star-autonomous categories", "functor  categories"], "abstract": "We construct a star-autonomous structure on the functor category $K^J$, where $J$ is small, $K$ is small-complete, and both are star-autonomous. A weaker result, that $K^J$ admits a linear distributive structure, is also shown under weaker hypotheses. The latter leads to a deeper understanding of the notion of linear functor."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/6/20-06abs.html", "title": "Balanced category theory", "authors": "Claudio Pisani", "keywords": ["factorization systems", "final maps and discrete fibrations", "initial maps  and discrete opfibrations", "reflections", "internal sets and components", "slices and coslices", "colimiting  cones", "adjunctible maps", "dense maps", "cylinders and homotopy", "arrow intervals", "enrichment", "complements"], "abstract": "Some aspects of basic category theory are developed in a finitely complete category $\\cal C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this axiomatization of final and initial functors and discrete (op)fibrations, concepts such as components, slices and coslices, colimits and limits, left and right adjunctible maps, dense maps and arrow intervals, can be naturally defined in $\\cal C$, and several classical properties concerning them can be effectively proved.  For any object $X$ of $\\cal C$, by restricting $\\cal C/X$ to the slices or to the coslices of $X$, two dual ``underlying categories\" are obtained. These can be enriched over internal sets (discrete objects) of $\\cal C$: internal hom-sets are given by the components of the pullback of the corresponding slice and coslice of $X$. The construction extends to give functors $\\cal C \\to Cat$, which preserve (or reverse) slices and adjunctible maps and which can be enriched over internal sets too."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html", "title": "Algebraic real analysis", "authors": "Peter Freyd", "keywords": ["algebraic real analysis", "closed interval", "closed midpoint   algebra", "chromatic scale", "coalgebraic real analysis", "complete scale", "finitely presented scale", "free scale", "harmonic scale", "injective   scale", "lattice-ordered abelian group", "linear logic", "Lipschitz   extension", "Lukasiewicz logic", "midpoint algebra", "minor scale", "modal  logic", "ordered wedge", "scale", "semi-simple scale", "simple scale", "zoom   operator"], "abstract": "An effort to initiate the subject of the title: the basic tool is the study of the abstract closed interval equipped with certain equational structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/7/20-07abs.html", "title": "Category-theoretic models of linear Abadi and Plotkin logic", "authors": "Lars Birkedal and Rasmus E. Mogelberg and Rasmus L. Petersen", "keywords": ["Parametric polymorphism", "categorical semantics", "axiomatic domain theory", "recursive types", "fibrations"], "abstract": "This paper presents a sound and complete category-theoretic notion   of models for Linear Abadi and Plotkin Logic, a logic suitable for reasoning about   parametricity in combination with recursion. A subclass of these   called parametric LAPL structures can be seen as an   axiomatization of domain theoretic models of parametric   polymorphism, and we show how to solve general (nested) recursive   domain equations in these.   Parametric LAPL structures constitute a general notion of model of   parametricity in a setting with recursion.   In future papers we will demonstrate this by showing how many   different models of parametricity and recursion give rise to   parametric LAPL structures, including Simpson and Rosolini's set   theoretic models, a syntactic model based   on Lily and a model based  on admissible pers over a reflexive domain."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/9/20-09abs.html", "title": "Fundamental pushout toposes", "authors": "Marta Bunge", "keywords": ["topos", "fundamental progroupoid", "spreads", "zero dimensional locales", "covering projections"], "abstract": "The author introduced and employed certain `fundamental pushout toposes' in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. We replace connected components by constructively complemented, or definable, monomorphisms. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of Janelidze. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional locales.  In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/8/20-08abs.html", "title": "Kan extensions in double categories", "authors": "Marco Grandis and Robert Paré", "keywords": ["weak double category", "weak double functor", "Kan extension", "adjoint functor", "comma object"], "abstract": "This paper deals with Kan extensions in a weak double category.  Absolute Kan extensions are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/5/20-05abs.html", "title": "Coherence for pseudodistributive laws revisited", "authors": "F. Marmolejo and R. J. Wood", "keywords": ["pseudomonads", "pseudodistributive laws", "coherence"], "abstract": "In this paper we show that eight coherence conditions suffice for the definition of a pseudodistributive law between pseudomonads."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/4/20-04abs.html", "title": "Abelian groupoids and non-pointed additive categories", "authors": "Dominique Bourn", "keywords": ["Mal'tsev", "protomodular", "naturally Mal'tsev categories", "internal group", "Baer sum", "long cohomology sequence"], "abstract": "We show that, in any Mal'tsev (and a fortiori protomodular) category E, not only the fibre Grd_X E of internal groupoids above the object X is a naturally Mal'tsev category, but moreover it shares with the category Ab of abelian groups the property following which the domain of any split epimorphism is isomorphic with the direct sum of its codomain with its kernel. This allows us to point at a new class of ``non-pointed additive'' categories which is necessarily protomodular.  Actually this even gives rise to a larger classification table of non-pointed additive categories which gradually take place between the class of naturally Mal'tsev categories and the one of essentially affine categories. As an application, when furthermore the ground category E is efficiently regular, we get a new way to produce Baer sums in the fibres Grd_X E and, more generally, in the fibres n-Grd_X E."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/3/20-03abs.html", "title": "Frobenius objects in cartesian bicategories", "authors": "R.F.C Walters and R.J. Wood", "keywords": ["cartesian bicategory", "Frobenius object", "dual object", "groupoid"], "abstract": "Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, map(B)(T,A) is a groupoid."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/10/21-10abs.html", "title": "Descent for compact 0-dimensional spaces", "authors": "George Janelidze and Manuela Sobral", "keywords": ["comma categories", "effective descent", "effective F-descent"], "abstract": "Using the reflection of the category C of compact 0-dimensional topological spaces into the category of Stone spaces we introduce a concept of a fibration in C. We show that: (i) effective descent morphisms in C are the same as the surjective fibrations; (ii) effective descent morphisms in C with respect to the fibrations are all surjections."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/2/20-02abs.html", "title": "Generalized Brown representability in homotopy categories: Erratum", "authors": "Jiri Rosicky", "keywords": ["Quillen model category", "Brown representability", "triangulated category", "accessible category"], "abstract": "Propositions 4.2 and 4.3 of the author's article (Theory Appl. Categ. 14 (2005), 451-479) are not correct. We show that their use can be avoided and all remaining results remain correct.\nSee note on p. 24."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/8/21-08abs.html", "title": "Approximate Mal'tsev operations", "authors": "Dominique Bourn and Zurab Janelidze", "keywords": ["Mal'tsev category", "Mal'tsev operation", "unital category", "strongly unital  category", "subtractive category"], "abstract": "Let $X$ and $A$ be sets and $\\alpha:X\\to A$ a map between them. We call a map $\\mu:X\\times X\\times X\\to A$ an approximate Mal'tsev operation with approximation $\\alpha$, if it satisfies $\\mu(x,y,y) = \\alpha(x) = \\mu(y,y,x)$ for all $x,y\\in X$. Note that if $A = X$ and the approximation $\\alpha$ is an identity map, then $\\mu$ becomes an ordinary Mal'tsev operation. We prove the following two characterization theorems: a category $\\mathbb{X}$ is a Mal'tsev category if and only if in the functor category $\\mathbf{Set}^{\\mathbb{X}^\\mathrm{op}\\times\\mathbb{X}}$ there exists an internal approximate Mal'tsev operation $\\mathrm{hom}_{\\mathbb{X}}\\times \\mathrm{hom}_{\\mathbb{X}}\\times \\mathrm{hom}_{\\mathbb{X}}\\rightarrow A$ whose approximation $\\alpha$ satisfies a suitable condition; a regular category $\\mathbb{X}$ with finite coproducts is a Mal'tsev category, if and only if in the functor category $\\mathbb{X}^\\mathbb{X}$ there exists an internal approximate Mal'tsev co-operation $A\\rightarrow 1_\\mathbb{X}+1_\\mathbb{X}+1_\\mathbb{X}$ whose approximation $\\alpha$ is a natural transformation with every component a regular epimorphism in $\\mathbb{X}$. Note that in both of these characterization theorems, if require further the approximation $\\alpha$ to be an identity morphism, then the conditions there involving $\\alpha$ become equivalent to $\\mathbb{X}$ being a naturally Mal'tsev category."},
{"url": "http://www.tac.mta.ca/tac/volumes/20/1/20-01abs.html", "title": "Locally cartesian closed categories without chosen constructions", "authors": "Erik Palmgren", "keywords": ["anafunctor", "axiom of choice", "adjoint"], "abstract": "We show how to formulate the notion of locally cartesian closed category without chosen pullbacks, by the use of Makkai's theory of anafunctors."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/12/21-12abs.html", "title": "Relative injectivity as cocompleteness for a class of distributors", "authors": "Maria Manuel Clementino and Dirk Hofmann", "keywords": ["Quantale", "V-category", "monad", "topological theory", "distributor", "Yoneda lemma", "weighted colimit"], "abstract": "Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over SET. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show in particular that the description of several semantic domains presented in [M. Escardo and B. Flagg, Semantic domains, injective spaces and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated into the V-enriched setting."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/11/21-11abs.html", "title": "Analytic functors and weak pullbacks", "authors": "J. Adamek and J. Velebil", "keywords": ["analytic functor", "weak limit", "weak pullback"], "abstract": "For accessible set-valued functors it is well known that weak preservation of limits is equivalent to representability, and weak preservation of connected limits to familial representability. In contrast, preservation of weak wide pullbacks is equivalent to being a coproduct of quotients of $\\hom$-functors modulo groups of automorphisms. For finitary functors this was proved by Andr\\'e Joyal who called these functors analytic. We introduce a generalization of Joyal's concept from endofunctors of Set to endofunctors of a symmetric monoidal category."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/9/21-09abs.html", "title": "Notes on effective descent and projectivity in quasivarieties of universal  algebras", "authors": "Ana Helena Roque", "keywords": ["effective descent", "projectivity", "quasivariety"], "abstract": "We present sufficient conditions under which effective descent morphisms in a quasivariety of universal algebras are the same as regular epimorphisms and examples for which they are the same as regular epimorphisms satisfying projectivity."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/7/21-07abs.html", "title": "Extensions in the theory of lax algebras", "authors": "Christoph Schubert and Gavin J. Seal", "keywords": ["lax algebra", "Kleisli extension", "initial extension", "strata extension", "tower  extension"], "abstract": "Recent investigations of lax algebras - in generalization of Barr's relational algebras -make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and V-relations (where V is a unital quantale). For a given monad there may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, and present a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html", "title": "A convenient category for directed homotopy", "authors": "L. Fajstrup and J. Rosicky", "keywords": ["locally presentable category", "simplex-generated spaces", "directed homotopy", "dicovering"], "abstract": "We propose a convenient category for directed homotopy consisting of \"directed'' topological spaces generated by \"directed'' cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/6/21-06abs.html", "title": "Weakly Mal'cev categories", "authors": "N. Martins-Ferreira", "keywords": ["Admissible reflexive graph", "multiplicative graph", "internal category", "internal groupoid", "weakly Mal'cev category", "naturally weakly Mal'cev category", "Mal'cev variety of universal algebras"], "abstract": "We introduce a notion of weakly Mal'cev category, and show that:   (a) every internal reflexive graph in a weakly Mal'tsev category admits at most one multiplicative graph structure in the sense of Janelidze, and  such a structure always makes it an internal category; (b) (unlike the special case of Mal'tsev categories) there are weakly Mal'tsev categories in which not every internal category is an internal groupoid.  We also give a simplified characterization of internal groupoids among internal categories in this context."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/5/21-05abs.html", "title": "Completions in biaffine sets", "authors": "Elisabetta Felaco and Eraldo Giuli", "keywords": ["2affine set", "Zariski closure", "separated object", "complete object", "injective object", "bitopological space}"], "abstract": "The theory of completion of $T_0$ objects in categories of affine objects over a given complete category developed by the second author  is extended to the case of $T_0$ objects in categories of 2affine objects. In the paper the case of the category $Set$ and target object the two-point set is studied in detail and an internal characterization of 2affine sets is provided."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/4/21-04abs.html", "title": "Doubles for monoidal categories", "authors": "Craig Pastro and Ross Street", "keywords": ["monoidal centre", "Drinfeld double", "monoidal category", "Day convolution"], "abstract": "In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal V-category A.   When A is autonomous (= rigid = compact), he showed that the V-category (that we call Tamb(A)) of so-equipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z[A,V] of the convolution monoidal V-category [A, V]. Our paper extends these ideas somewhat. For general A, we construct a promonoidal V-category DA (which we suggest should be called the double of A) with an equivalence of [DA, V] with Tamb(A). When A is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a V-category Tamb_s(A) (respectively, Tamb_{ls}(A)) which is equivalent to the centre (respectively, lax centre) of [A, V]. We construct localizations D_sA and D_{ls}A of DA such that there are equivalences of Tamb_s(A) with [D_sA, V] and of Tamb_{ls}(A) with [D_{ls}A, V]. When A is autonomous, every Tambara module is strong; this implies an equivalence of Z[A, V] with [DA,V]."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/2/21-02abs.html", "title": "Limit preserving full embeddings", "authors": "V. Trnkova  and J. Sichler", "keywords": ["universal algebra", "unary algebra", "limit", "full embedding", "limit preserving functor"], "abstract": "We prove that every small strongly connected category k has a full embedding preserving all limits existing in k into a category of unary universal algebras.  The number of unary operations can be restricted to |mor k| in case when k has a terminal object and only preservation of limits over finitely many objects is desired.  And all limits existing in such a category k are preserved by a full embedding of k into the category of all algebraic systems with |mor k| unary operation and one unary relation."},
{"url": "http://www.tac.mta.ca/tac/volumes/21/3/21-03abs.html", "title": "Protolocalisations of exact Mal'cev categories", "authors": "Francis Borceux, Marino Gran, Sandra Mantovani", "keywords": ["protolocalisation", "radical theory", "homological category", "semi-abelian  category"], "abstract": "A protolocalisation of a regular category is a full reflective regular subcategory, whose reflection preserves pullbacks of regular epimorphisms along arbitrary morphisms. We devote special attention to the epireflective protolocalisations of an exact Mal'cev category; we characterise them in terms of a corresponding closure operator on equivalence relations. We give some examples in algebra and in topos theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/23/22-23abs.html", "title": "Cartesian differential categories", "authors": "R.F. Blute, J.R.B. Cockett and R.A.G. Seely", "keywords": ["monoidal categories", "differential categories", "Kleisli categories", "differential operators"], "abstract": "This paper revisits the authors' notion of a differential category from a different perspective.  A differential category is an additive symmetric monoidal category with a comonad (a \"coalgebra modality\") and a differential combinator. The morphisms of a differential category should be thought of as the linear maps; the differentiable or smooth maps would then be morphisms of the coKleisli category.  The purpose of the present paper is to directly axiomatize differentiable maps and thus to move the emphasis from the linear notion to structures resembling the coKleisli category. The result is a setting with a more evident and intuitive relationship to the familiar notion of calculus on smooth maps. Indeed a primary example is the category whose objects are Euclidean spaces and whose morphisms are smooth maps.\nA Cartesian differential category is a Cartesian left additive category which possesses a Cartesian differential operator. The differential operator itself must satisfy a number of equations, which guarantee, in particular, that the differential of any map is `\"linear\" in a suitable sense.\nWe present an analysis of the basic properties of Cartesian differential categories.  We show that under modest and natural assumptions, the coKleisli category of a differential category is Cartesian differential.  Finally we present a (sound and complete) term calculus for these categories which allows their structure to be analysed using essentially the same language one might use for traditional multi-variable calculus."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/20/22-20abs.html", "title": "Algebraic categories whose projectives are explicitly free", "authors": "Matías Menni", "keywords": ["monads", "combinatorics", "projective objects", "free objects"], "abstract": "Let M = (M, m, u) be a monad and let (MX, m) be the free M-algebra on the object X. Consider an M-algebra (A, a), a retraction r : (MX, m) --> (A, a) and a section t : (A, a) --> (MX, m) of r. The retract  (A, a) is not free in general. We observe that for many monads with a `combinatorial flavor' such a retract is not only a free algebra  (MA_0, m), but it is also the case that the object A_0 of generators is determined in a canonical way by the section t. We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html", "title": "Homotopical interpretation of globular complex by multipointed d-space", "authors": "Philippe Gaucher", "keywords": ["homotopy", "directed homotopy", "combinatorial model category", "simplicial category", "topological category", "delta-generated   space", "d-space", "globular complex", "time flow"], "abstract": "Globular complexes were introduced by E. Goubault and the author to   model higher dimensional automata.  Globular complexes are   topological spaces equipped with a globular decomposition which is   the directed analogue of the cellular decomposition of a CW-complex.   We prove that there exists a combinatorial model category such that   the cellular objects are exactly the globular complexes and such   that the homotopy category is equivalent to the homotopy category of   flows.  The underlying category of this model category is a variant   of M. Grandis' notion of d-space over a topological space colimit   generated by simplices. This result enables us to understand the   relationship between the framework of flows and other works in   directed algebraic topology using d-spaces. It also enables us to   prove that the underlying homotopy type functor of flows can be   interpreted up to equivalences of categories as the total left   derived functor of a left Quillen adjoint."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/19/22-19abs.html", "title": "A brief review of abelian categorifications", "authors": "Mikhail Khovanov, Volodymyr Mazorchuk and Catharina Stroppel", "keywords": ["category", "functor", "abelian categorification", "braid group", "Hecke algebra", "Weyl algebra"], "abstract": "This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra $sl_n$. The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/18/22-18abs.html", "title": "Higher-dimensional categories with finite derivation type", "authors": "Yves Guiraud and Philippe Malbos", "keywords": ["n-category", "rewriting", "polygraph", "finite derivation type", "low-dimensional topology"], "abstract": "We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalising the one introduced by Squier for word rewriting systems. We characterise this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/21/22-21abs.html", "title": "Vector fields and flows on differentiable stacks", "authors": "Richard Hepworth", "keywords": ["Stacks", "differentiable stacks", "orbifolds", "vector fields", "flows"], "abstract": "This paper introduces the notions of vector field and flow on a general differentiable stack.  Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell.  This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds.  It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie algebra of vector fields."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/15/22-15abs.html", "title": "", "authors": "Jacques Penon", "keywords": ["category", "internal category", "monad", "lax algebra", "T-category"], "abstract": "With the representable T-categories, we form a connection  between two concepts, both owed to A. Burroni : on the one hand, the one of T-category, and, on the other hand, the one of T-lax algebra. Both of them generalise the concept of algebra on a monad  T."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/16/22-16abs.html", "title": "Epimorphic covers make $R^+_G$ a site, for profinite $G$", "authors": "Daniel G. Davis", "keywords": ["site", "profinite group", "finite discrete $G$-sets", "presheaves of spectra", "Lubin-Tate spectrum", "continuous $G$-spectrum"], "abstract": "Let $G$ be a non-finite profinite group and let $G-Sets_{df}$ be the canonical site of finite discrete $G$-sets. Then the category $R^+_G$, defined by Devinatz and Hopkins, is the category obtained by considering $G-Sets_{df}$ together with the profinite $G$-space $G$ itself, with morphisms being continuous $G$-equivariant maps. We show that $R^+_G$ is a site when equipped with the pretopology of epimorphic covers. We point out that presheaves of spectra on $R^+_G$ are an efficient way of organizing the data that is obtained by taking the homotopy fixed points of a continuous $G$-spectrum with respect to the open subgroups of $G$. Additionally, utilizing the result that $R^+_G$ is a site, we describe various model category structures on the category of presheaves of spectra on $R^+_G$ and make some observations about them."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/17/22-17abs.html", "title": "Isbell duality for modules", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["duality", "weak torsion", "sober objects"], "abstract": "The purpose of this paper is to extend the results of  TAC Vol. 20, No. 15 from the case of abelian groups  (Z-modules)  to that of modules over a large class of not necessarily commutative rings."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/14/22-14abs.html", "title": "Eventually cyclic spectra of parameterized flows", "authors": "John F. Kennison", "keywords": ["Parameterized Boolean flow", "dynamical systems", "spectrum", "sheaf"], "abstract": "This paper continues the work of our previous papers, The cyclic spectrum of a Boolean flow TAC 10, 392-419 and Spectra of finitely generated Boolean flows TAC 16, 434-459.  We define eventually cyclic Boolean flows and the eventually cyclic spectrum of a Boolean flow. We show that this spectrum, as well as the spectra defined in our earlier papers, extend to parametrized flows on Stone spaces and on compact Hausdorff space when symbolic dynamics is used. An example shows that the cyclic spectrum for a parameterized flow is sometimes over a non-spatial locale."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/13/22-13abs.html", "title": "An embedding theorem for Hilbert categories", "authors": "Chris Heunen", "keywords": ["(pre-)Hilbert category", "embedding", "monoidal Abelian category"], "abstract": "We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all finite (co)limits. An intermediate result that is important in its own right is that the scalars in such a category necessarily form an involutive field. In case of a Hilbert category, the embedding extends to the category of Hilbert spaces and continuous linear maps. The axioms for (pre-)Hilbert categories are weaker than the axioms found in other approaches to axiomatizing 2-Hilbert spaces. Neither enrichment nor a complex base field is presupposed. A comparison to other approaches will be made in the introduction."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html", "title": "Weak distributive laws", "authors": "Ross Street", "keywords": ["monad", "triple", "distributive law", "weak bialgebra"], "abstract": "Distributive laws between monads (triples) were defined by Jon Beck in the 1960s. They were generalized to monads in 2-categories and noticed to be monads in a 2-category of monads. Mixed distributive laws are comonads in the 2-category of monads; if the comonad has a right adjoint monad, the mate of a mixed distributive law is an ordinary distributive law. Particular cases are the entwining operators between algebras and coalgebras. Motivated by work on weak entwining operators, we define and study a weak notion of distributive law for monads. In particular, each weak distributive law determines a wreath product monad (in the terminology of Lack and Street); this gives an advantage over the mixed  case."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/10/22-10abs.html", "title": "On the categorical semantics of elementary linear logic", "authors": "Olivier Laurent", "keywords": ["monoidal categories", "elementary linear logic", "categorical   logic", "denotational semantics", "coherent spaces"], "abstract": "We introduce the notion of elementary Seely category as a   notion of categorical model of Elementary Linear Logic (ELL)   inspired from Seely's definition of models of Linear Logic (LL).   In order to deal with additive connectives in ELL, we use the   approach of Danos and Joinet. From the categorical   point of view, this requires us to go outside the usual   interpretation of connectives by functors. The $!$ connective is   decomposed into a pre-connective $\\sharp$ which is interpreted by a   whole family of functors (generated by $\\id$, $\\tens$ and $\\with$).    As an application, we prove the stratified coherent model and the   obsessional coherent model to be elementary Seely categories and   thus models of ELL."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/11/22-11abs.html", "title": "Enriched Orthogonality and Equivalences", "authors": "M. Golasinski and L. Stramaccia", "keywords": ["ANR-space", "enriched orthogonality", "groupoid", "homotopy structure", "prospace", "shape equivalence", "simplicial set", "${\\mathcal V}$-orthogonal object", "strongly fibered space"], "abstract": "In this paper, we consider an enriched orthogonality for classes of spaces, with respect to groupoids, simplicial sets and spaces themselves. This point of view allows one to characterize homotopy equivalences, shape and strong shape equivalences.  We show that there exists a class of spaces, properly containing ANR-spaces, for which shape and strong shape equivalences coincide. For such a class of spaces homotopy orthogonality implies enriched orthogonality."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/8/22-08abs.html", "title": "Algebraic colimit calculations in homotopy theory using fibred and cofibred categories", "authors": "Ronald Brown and Rafael Sivera", "keywords": ["higher homotopy van Kampen theorems", "homotopical excision", "colimits", "fibred and cofibred categories", "groupoids", "modules", "crossed modules"], "abstract": "Higher Homotopy van Kampen Theorems allow some colimit calculations of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases.  A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical applications are to pairs of spaces with several base points; we also describe briefly applications to triads."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/9/22-09abs.html", "title": "Covering space theory for directed topology", "authors": "Eric Goubault, Emmanuel Haucourt, Sanjeevi Krishnan", "keywords": ["pospace", "covering space", "directed topology"], "abstract": "The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a ``local preorder'' encoding control flow. In the case where time does not loop, the ``locally preordered'' state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a ``locally monotone'' covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/5/22-05abs.html", "title": "Categories with slicing", "authors": "Thorsten Palm", "keywords": ["slice categories", "presheaf category", "polytopic sets"], "abstract": "Prior work towards the subject of higher-dimensional categories gives rise to several examples of a category over $Cat$ to which the slice-category construction can be lifted universally. The present paper starts by supplying this last clause with a precise meaning. It goes on to establish for any such category a certain embedding in a presheaf category, to describe the image, and hence to derive conditions collectively sufficient for that functor to be an equivalence. These conditions are met in the foremost of the examples: the category of dendrotopic sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/3/22-03abs.html", "title": "Searching for more absolute CR-epic spaces", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["absolute CR-epics", "countable neighbourhood property", "Alster's condition  Dieudonne plank"], "abstract": "We continue our examination of absolute CR-epic spaces, or spaces with the property that any embedding induces an epimorphism, in the category of commutative rings, between their rings of continuous functions.  We examine more closely the deleted plank construction, which generalizes the Dieudonne construction, and yields absolute CR-epic spaces which are not Lindelof.  For the Lindelof case, an earlier paper has shown the usefulness of the countable neighbourhood property, CNP, and the Alster condition (where CNP means that the space is a P-space in any compactification and the Alster condition says that any cover of the space by $G_\\delta$ sets has a countable subcover, provided each compact subset can be covered by a finite subset.)  In this paper, we find further properties of Lindelof CNP spaces and of Alster spaces, including constructions that preserve these properties and conditions equivalent to these properties.  We explore the outgrowths of such spaces and find several examples that answer open questions in our previous work."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/6/22-06abs.html", "title": "Tensor-triangulated categories and dualities", "authors": "Baptiste Calmès and Jens Hornbostel", "keywords": ["closed monoidal category", "commutative diagram", "duality", "Witt group"], "abstract": "In a triangulated closed symmetric monoidal category, there are natural dualities induced by the internal Hom. Given a monoidal exact functor $f^*$ between two such categories and adjoint couples $(f^*,f_*)$, $(f_*,f^!)$, we establish the commutative diagrams necessary for $f^*$ and $f_*$ to respect certain dualities, for a projection formula to hold between them (as duality preserving exact functors) and for classical base change and composition formulas to hold when such duality preserving functors are composed. This framework allows us to define push-forwards for Witt groups, for example."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/4/22-04abs.html", "title": "On endomorphism algebras of separable monoidal functors", "authors": "Brian Day and Craig Pastro", "keywords": ["separable fibre functor", "Tannaka reconstruction", "bialgebra", "von Neumann core"], "abstract": "We show that the (co)endomorphism algebra of a sufficiently separable ``fibre'' functor into $Vect_k$, for $k$ a field of characteristic 0, has the structure of what we call a ``unital'' von Neumann core in $Vect_k$. For $Vect_k$, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in $Set$ is again that of a group."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/2/22-02abs.html", "title": "On deformations of pasting diagrams", "authors": "D. N. Yetter", "keywords": ["pasting diagrams", "pasting schemes", "deformation theory"], "abstract": "We adapt the work of Power to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts.  We provide a deformation theory for pasting diagrams valued in the 2-category of k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack, proving the standard results.  Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/13/23-13abs.html", "title": "What are sifted colimits?", "authors": "J. Adamek, J. Rosicky, E. M. Vitale", "keywords": ["sifted colimit", "reflexive coequalizer", "filtered colimit"], "abstract": "Sifted colimits, important for algebraic theories, are \"almost\" just the combination of filtered colimits and reflexive coequalizers.  For example, given a finitely cocomplete category $\\cal A$, then a functor with domain $\\cal A$ preserves sifted colimits iff it preserves filtered colimits and reflexive coequalizers. But for general categories $\\cal A$ that statement is not true:  we provide a counter-example."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/11/23-11abs.html", "title": "The pointed subobject functor, 3 x 3 lemmas, and subtractivity of spans", "authors": "Zurab Janelidze", "keywords": ["subtractive category", "normal category", "homological category", "homological diagram lemmas", "diagram chasing"], "abstract": "The notion of a subtractive category recently introduced by the author, is a pointed categorical counterpart of the notion of a subtractive variety of universal algebras in the sense of A.~Ursini (recall that a variety is subtractive if its theory contains a constant 0 and a binary term s satisfying s(x,x)=0 and s(x,0)=x). Let us call a pointed regular category $\\mathbb{C}$ normal if every regular epimorphism in $\\mathbb{C}$ is a normal epimorphism. It is well known that any homological category in the sense of F. Borceux and D. Bourn is both normal and subtractive. We prove that in any subtractive normal category, the upper and lower $3\\times 3$ lemmas hold true, which generalizes a similar result for homological categories due to D. Bourn (note that the middle $3\\times 3$ lemma holds true if and only if the category is homological). The technique of proof is new: the pointed subobject functor $\\mathcal{S}=\\mathrm{Sub}(-):\\mathbb{C}\\rightarrow\\mathbf{Set}_*$ turns out to have suitable preservation/reflection properties which allow us to reduce the proofs of these two diagram lemmas to the standard diagram-chasing arguments in $\\mathbf{Set}_*$ (alternatively, we could use the more advanced embedding theorem for regular categories due to M.~Barr). The key property of $\\mathcal{S}$, which allows to obtain these diagram lemmas, is the preservation of subtractive spans. Subtractivity of a span provides a weaker version of the rule of subtraction --- one of the elementary rules for chasing  diagrams in abelian categories, in the sense of S. Mac Lane. A pointed regular category is subtractive if and only if every span in it is subtractive, and moreover, the functor $\\mathcal{S}$ not only preserves but also reflects subtractive spans. Thus, subtractivity seems to be exactly what we need in order to prove the upper/lower $3\\times 3$ lemmas in a normal category. Indeed, we show that a normal category is subtractive if and only if these $3\\times 3$ lemmas hold true in it. Moreover, we show that for any pointed regular category $\\mathbb{C}$ (not necessarily a normal one), we have: $\\mathbb{C}$ is subtractive if and only if the lower $3\\times 3$ lemma holds true in $\\mathbb{C}$."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/10/23-10abs.html", "title": "A metric tangential calculus", "authors": "Elisabeth Burroni and  Jacques Penon", "keywords": ["differential calculus", "jets", "metric spaces", "categories"], "abstract": "The metric jets, introduced here, generalize the jets (at order one) of Charles Ehresmann. In short, for a ``good'' map f (said to be ``tangentiable'' at a) between metric spaces, we define its metric jet tangent at a (composed of all the maps which are locally lipschitzian at     a and tangent to f at a) called the ``tangential'' of f at a, and denoted Tf_a. So, in this metric context, we define a ``new differentiability'' (called ``tangentiability'') which extends the classical differentiability (while preserving most of its properties) to new maps, traditionally pathologic."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/9/23-09abs.html", "title": "Star-multiplicative graphs in pointed protomodular categories", "authors": "N. Martins-Ferreira", "keywords": ["Internal category", "internal groupoid", "reflexive graph", "multiplicative graph", "star-multiplicative graph", "jointly epic pair", "admissible pair", "jointly epic split extension", "split short five lemma", "pointed protomodular"], "abstract": "Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in the semiabelian context.\nWhen the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of Janelidze."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/8/23-08abs.html", "title": "The third cohomology group classifies double central extensions", "authors": "Diana Rodelo and Tim Van der Linden", "keywords": ["cohomology", "categorical Galois theory", "semi-abelian category", "higher central extension", "Baer sum"], "abstract": "We characterise the double central extensions in a semi-abelian category in terms of commutator conditions. We prove that the third cohomology group H^3(Z,A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/7/23-07abs.html", "title": "Strongly separable morphisms in general categories", "authors": "G. Janelidze and W. Tholen", "keywords": ["separable morphism", "strongly separable morphism", "separated morphism", "compact morphism", "covering morphism", "factorization system", "effective descent morphism", "Galois theory", "lextensive category"], "abstract": "We clarify the relationship between separable and covering morphisms in general categories by introducing and studying an intermediate class of morphisms that we call strongly separable."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/4/23-04abs.html", "title": "Snake Lemma in incomplete relative homological categories", "authors": "Tamar Janelidze", "keywords": ["incomplete relative homological category", "homological category", "normal epimorphism", "snake lemma"], "abstract": "The purpose of this paper is to prove a new, incomplete-relative, version of Non-abelian Snake Lemma, where \"relative\" refers to a distinguished class of normal epimorphisms in the ground category, and \"incomplete\" refers to omitting all completeness/cocompleteness assumptions not involving that class."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/6/23-06abs.html", "title": "Internal crossed modules and  Peiffer Condition", "authors": "Sandra Mantovani, Giuseppe Metere", "keywords": ["internal crossed module", "reflexive graph", "internal action", "semiabelian category"], "abstract": "In this paper we show that in a homological category in the sense of F. Borceux and D. Bourn,  the notion of an internal precrossed module corresponding to a star-multiplicative graph, in the sense of G. Janelidze, can be obtained by directly internalizing  the usual axioms of a crossed module, via equivariance. We then exhibit some sufficient conditions on a homological category under which this notion coincides with the notion of an internal crossed module due to G. Janelidze. We show that this is the case for any category of distributive  $\\Omega_2$-groups, in particular for the categories of groups with operations in the sense of G. Orzech."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/5/23-05abs.html", "title": "Monadic approach to Galois descent and cohomology", "authors": "Francis Borceux, Stefaan Caenepeel and George Janelidze", "keywords": ["Descent theory", "Galois theory", "monadic functor", "group cohomology"], "abstract": "We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads directly, and our monadic approach to Galois descent makes no reference to Grothendieck descent theory at all. In order to make Galois descent constructions perfectly clear, we also describe their connections with some other related constructions of categorical algebra, and make various explicit calculations, especially with 1-cocycles and 1-dimensional non-abelian cohomology, usually omitted in the literature."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html", "title": "The comprehensive factorization and torsors", "authors": "Ross Street and Dominic Verity", "keywords": ["torsor", "internal category", "exponentiable morphism", "discrete fibration", "final functor", "comprehensive factorization", "locally isomorphic"], "abstract": "This is an expanded, revised and corrected version of the first author's 1981 preprint. The discussion of one-dimensional cohomology $H^{1}$ in a fairly general category E involves passing to the 2-category Cat(E) of categories E.  In particular, the coefficient object is a category B in E and the torsors that $H^{1}$  classifies are particular functors in E. We only impose conditions on E that are satisfied also by Cat(E)  and argue that $H^{1}$ for Cat(E) is a kind of $H^{2}$ for E, and so on recursively. For us, it is too much to ask E to be a topos (or even internally complete) since, even if E is, Cat(E) is not. With this motivation, we are led to examine morphisms in E which act as internal families and to internalize the comprehensive factorization of functors into a final functor followed by a discrete fibration. We define B-torsors for a category B in E and prove clutching and classification theorems. The former theorem clutches Cech cocycles to construct torsors while the latter constructs a coefficient category to classify structures locally isomorphic to members of a given internal family of structures. We conclude with applications to examples."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/2/23-02abs.html", "title": "Homology of n-fold groupoids", "authors": "Tomas Everaert and Marino Gran", "keywords": ["Protoadditive functor", "categorical Galois theory", "internal groupoid", "semi-abelian category", "homology", "Hopf formula"], "abstract": "Any semi-abelian category A appears, via the discrete functor, as a full replete reflective subcategory of the semi-abelian category of internal groupoids in A. This allows one to study the homology of $n$-fold internal groupoids with coefficients in a semi-abelian category A, and to compute explicit higher Hopf formulae. The crucial concept making such computations possible is the notion of protoadditive functor, which can be seen as a natural generalisation of the notion of additive functor."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/12/23-12abs.html", "title": "A categorical approach to integration", "authors": "Reinhard Börger", "keywords": ["internal Boolean algebra", "universal measure", "multiplicative measure", "product measure", "Boolean algebra of idempotents", "symmetric monoidal closed category", "cartesian closed category"], "abstract": "We present a general treatment of measures and integrals in certain (monoidal closed) categories. Under appropriate conditions the integral can be defined by a universal property, and the universal measure is at the same time a universal multiplicative measure. In the multiplicative case this assignment is right adjoint to the formation of the Boolean algebra of idempotents. Now coproduct preservation yields an approach to product measures."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/1/22-01abs.html", "title": "Duality for CCD lattices", "authors": "Francisco Marmolejo, Robert Rosebrugh, and R.J. Wood", "keywords": ["adjunction", "completely distributive", "idempotent", "monadic", "proarrow equipment", "cauchy complete"], "abstract": "The 2-category of constructively completely distributive lattices is shown to be bidual to a 2-category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2-category of ordered sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/20/24-20abs.html", "title": "Strictification of categories weakly enriched in symmetric monoidal categories", "authors": "Bertrand J. Guillou", "keywords": ["Bimonoidal category", "coherence"], "abstract": "We show that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a \"many 0-cells\" version of the strictification of bimonoidal categories to strict ones."},
{"url": "http://www.tac.mta.ca/tac/volumes/23/1/23-01abs.html", "title": "Action accessibility for categories of interest", "authors": "Andrea Montoli", "keywords": ["protomodular categories", "action accessible categories", "categories of interest", "centralizers"], "abstract": "We prove that every category of interest (in the sense of G. Orzech) is action accessible in the sense of Bourn and Janelidze. This fact allows us to give an intrinsic description of centers and centralizers in this class of categories. We give also some new examples of categories of interest, mainly arising from Loday's papers."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/21/24-21abs.html", "title": "A unified framework for generalized multicategories", "authors": "G.S.H. Cruttwell and Michael A. Shulman", "keywords": ["Enriched categories", "change of base", "monoidal categories", "double categories", "multicategories", "operads", "monads"], "abstract": "Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces.  In each case, generalized multicategories are defined as the ``lax algebras'' or ``Kleisli monoids'' relative to a ``monad'' on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered.  We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/18/24-18abs.html", "title": "Higher Dimensional Algebra VII: Groupoidification", "authors": "John C. Baez, Alexander E. Hoffnung, and Christopher D. Walker", "keywords": ["categorification", "groupoid", "Hecke algebra", "Hall algebra", "quantum theory"], "abstract": "Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids.  We introduce this idea with a detailed exposition of `degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators.  Then we present three applications of groupoidification.  The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams.  The second application is to Hecke algebras.  We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter $q$ is a prime power.  We illustrate this with the simplest nontrivial example, coming from the $A_2$ Dynkin diagram.  In this example we show that the solution of the Yang--Baxter equation built into the $A_2$ Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field $\\mathbb{F}_q$.  The third application is to Hall algebras.  We explain how the standard construction of the Hall algebra from the category of $\\mathbb{F}_q$ representations of a simply-laced quiver can be seen as an example of degroupoidification.  This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/17/24-17abs.html", "title": "Internal profunctors and commutator theory;", "authors": "Dominique Bourn", "keywords": ["Mal'cev categories", "centralizers", "profunctor", "Schreier-Mac Lane extension  theorem", "internal groupoid", "Galois groupoid"], "abstract": "We clarify the relationship between internal profunctors and connectors on pairs (R,S) of equivalence relations which originally appeared in our new profunctorial approach of the Schreier-Mac Lane extension theorem.  This clarification allows us to extend this Schreier-Mac Lane theorem to any exact Mal'cev category with centralizers.  On the other hand, still in the Mal'cev context and in respect to the categorical Galois theory associated with a reflection I, it allows us to produce the faithful action of a certain abelian group on the set of classes (up to isomorphism) of I-normal extensions having a given Galois groupoid."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/16/24-16abs.html", "title": "On the duality between trees and disks", "authors": "David Oury", "keywords": ["delta", "disk", "duality", "globular set", "omega-category", "theta-category", "tree"], "abstract": "A combinatorial category Disk was introduced by André Joyal to play a role in his definition of weak $\\omega$-category. He defined the category $\\Theta$ to be dual to Disk. In the ensuing literature, a more concrete description of $\\Theta$ was provided. In this paper we provide another proof of the dual equivalence and introduce various categories equivalent to Disk or $\\Theta$, each providing a helpful viewpoint."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html", "title": "The span construction", "authors": "Robert Dawson, Robert Paré, Dorette Pronk", "keywords": ["Double categories", "Span construction", "Localizations", "Companions", "Conjoints", "Adjoints"], "abstract": "We present two generalizations of the Span construction. The first generalization gives Span of a category with all pullbacks as a (weak) double category. This double category Span A can be viewed as the free double category on the vertical category A where every vertical arrow has both a companion and a conjoint (and these companions and conjoints are adjoint to each other). Thus defined, Span : Cat --> Doub becomes a 2-functor, which is a partial left bi-adjoint to the forgetful functor Vrt : Doub --> Cat, which sends a double category to its category of vertical arrows.\nThe second generalization gives Span of an arbitrary category as an oplax normal double category. The universal property can again be given in terms of companions and conjoints and the presence of their composites. Moreover, Span A is universal with this property in the sense that Span : Cat --> OplaxNDoub is left bi-adjoint to the forgetful functor which sends an oplax double category to its vertical arrow category."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/15/24-15abs.html", "title": "A logic for categories", "authors": "Claudio Pisani", "keywords": ["Temporal doctrine", "internally enriched hom", "tensor and (co)limits", "Frobenius  law", "adjunction-like laws", "quantification formulas"], "abstract": "We present a doctrinal approach to category theory, obtained by abstracting from the indexed inclusion (via discrete fibrations and opfibrations) of left and of right actions of X in Cat in categories over X. Namely, a ``weak temporal doctrine'' consists essentially of two indexed functors with the same codomain such that the induced functors have both left and right adjoints satisfying some exactness conditions, in the spirit of categorical logic.\nThe derived logical rules include some adjunction-like laws involving the truth-values-enriched hom and tensor functors, which condense several basic categorical properties and display a nice symmetry. The symmetry becomes more apparent in the slightly stronger context of ``temporal doctrines'', which we initially treat and which include as an instance the inclusion of lower and upper sets in the parts of a poset, as well as the inclusion of left and right actions of a graph in the graphs over it."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/14/24-14abs.html", "title": "On *-autonomous categories of topological modules", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["*-autonomous", "chu category", "high and wide subcategory"], "abstract": "Let R be a commutative ring whose complete ring of quotients is R-injective.  We show that the category of topological R-modules contains a full subcategory that is *-autonomous using R itself as dualizing object.  In order to do this, we develop a new variation on the category chu(D,R), where D is the category of discrete R-modules: the high wide subcategory, which we show equivalent to the category of reflexive topological modules."},
{"url": "http://www.tac.mta.ca/tac/volumes/22/7/22-07abs.html", "title": "Lois distributives. Applications aux automates stochastiques", "authors": "Elisabeth Burroni", "keywords": ["category", "monad", "distributive law", "probability. stochastic automata"], "abstract": "Deterministic automata are algebras of the monad $T_M$ associated to a free monoid $M$. To extend to nondeterministic and stochastic automata such a monadic formalism, it is suitable to resort to a notion richer than the one of monad, but equally basic: the notion of distributive law between two monads. The notion of algebra on a monad is then generalized by the one of algebra for a distributive law. The nondeterministic and stochastic automata are precisely algebras for distributive laws whose first monad is $T_M$. If the nondeterministic case involves a distributive law between $T_M$ and the well-known power set monad, the stochastic case involves a distributive law between $T_M$ (where, here, $M$ is a measurable monoid) and the probability monad. This allows presentation of the stochastic automata as algebras for this distributive law. This paper taking place at the confluence of category, automata and probability theories, we have, for the convenience of the reader not aware of each area, made useful reviews about these subjects (in several appendices). We also recall the detailed construction of the probability monad; and  we construct precisely the distributive law which links it to the monad $T_M$."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/11/24-11abs.html", "title": "Tensor products of sup-lattices and generalized sup-arrows", "authors": "T. Kenney and R.J. Wood", "keywords": ["adjunction", "tensor product", "totally below", "CCD", "idempotent"], "abstract": "An alternative description of the tensor product of sup-lattices is given with yet another description provided for the tensor product in the special case of CCD sup-lattices. In the course of developing the latter, properties of sup-preserving  functions and the totally below relation are generalized to  not-necessarily-complete ordered sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/10/24-10abs.html", "title": "The Hopf algebra of Möbius intervals", "authors": "F. W. Lawvere and M. Menni", "keywords": ["Möbius category", "incidence algebra"], "abstract": "An unpublished result by the first author states that there exists a Hopf algebra $H$ such that for any Möbius category $\\cal C$ (in the sense of Leroux) there exists a canonical algebra morphism from the dual $H^*$ of $H$ to the incidence algebra of $\\cal C$. Moreover, the Möbius inversion principle in incidence algebras follows from a `master' inversion result in $H^*$. The underlying module of $H$ was originally defined as the free module on the set of iso classes of Möbius intervals, i.e. Möbius categories with initial and terminal objects. Here we consider a category of Möbius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Möbius intervals leads also to two new characterizations of Möbius categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/9/24-09abs.html", "title": "Finitely presentable morphisms in exact sequences", "authors": "Michel Hébert", "keywords": ["finitely presentable morphism", "abelian category", "Grothendieck category"], "abstract": "Let $\\cal K$ be a locally finitely presentable category. If $\\cal K$ is  abelian  and the sequence  $$ 0 \\to K \\to^k X \\to^c C \\to 0$$  is short exact, we show that 1) $K$ is finitely generated iff $c$ is finitely  presentable; 2) $k$ is finitely presentable iff $C$ is finitely presentable. The ``if\" directions fail for semi-abelian varieties. We show that all but (possibly) 2)(if) follow from  analogous properties which hold in all locally finitely presentable categories. As for 2)(if), it holds as soon as $\\cal K$ is also co-homological,  and all its strong epimorphisms are regular. Finally, locally finitely  coherent (resp. noetherian) abelian categories are characterized as those for which  all finitely presentable morphisms have finitely generated (resp. presentable)  kernel objects."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/8/24-08abs.html", "title": "On modified Reedy and modified projective model structures", "authors": "Mark W. Johnson", "keywords": ["Diagram category", "Quillen model category", "Reedy model category", "algebraic K-theory"], "abstract": "Variations on the notions of Reedy model structures and projective model structures on categories of diagrams in a model category are introduced. These allow one to choose only a subset of the entries when defining weak equivalences, or to use different model categories at different entries of the diagrams.  As a result, a bisimplicial model category that can be used to recover the algebraic K-theory for any Waldhausen subcategory of a model category is produced."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/7/24-07abs.html", "title": "Transversal homotopy theory", "authors": "Jonathan Woolf", "keywords": ["Stratified space", "homotopy theory"], "abstract": "Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy monoids, one for each natural number. The assignment is functorial for a natural class of maps which we call stratified normal submersions. When the stratification is trivial the transversal homotopy monoids are isomorphic to the usual homotopy groups. We compute some simple examples and explore the elementary properties of these invariants.  We also assign `higher invariants', the transversal homotopy categories, to each Whitney stratified manifold. These have a rich structure; they are rigid monoidal categories for n > 1 and ribbon categories for n > 2. As an example we show that the transversal homotopy categories of a sphere, stratified by a point and its complement, are equivalent to categories of framed tangles."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/6/24-06abs.html", "title": "Topos theoretic aspects of semigroup actions", "authors": "Jonathon Funk and Pieter Hofstra", "keywords": ["inverse semigroup", "semigroup action", "torsor", "classifying topos"], "abstract": "We define the notion of a torsor for an inverse semigroup, which is based on semigroup actions, and prove that this is precisely the structure classified by the topos associated with an inverse semigroup. Unlike in the group case, not all set-theoretic torsors are isomorphic: we shall give a complete description of the category of torsors. We explain how a semigroup prehomomorphism gives rise to an adjunction between a restrictions-of-scalars functor and a tensor product functor, which we relate to the theory of covering spaces and E-unitary semigroups. We also interpret for semigroups the Lawvere-product of a sheaf and distributio$ and finally, we indicate how the theory might be extended to general semigroups, by defining a notion of torsor and a classifying topos for those."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/5/24-05abs.html", "title": "On a Conjecture by J.H.Smith", "authors": "George Raptis", "keywords": ["combinatorial model category", "accessible category", "simplicial sets"], "abstract": "We show that the class of weak equivalences of a combinatorial model  category can be detected by an accessible functor into simplicial sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html", "title": "Monads as Extension Systems - No Iteration is Necessary", "authors": "F.  Marmolejo and R. J. Wood", "keywords": ["extension systems", "monads", "distributive laws", "wreaths", "profunctors"], "abstract": "We introduce a description of the algebras for a monad in terms of extension systems, similar to the one for monads given by  Manes. We rewrite distributive laws for monads and wreaths in terms of this description, avoiding the iteration of the functors involved. We give a profunctorial explanation of why Manes' description of monads in terms of extension systems works."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/19/24-19abs.html", "title": "Hopf monoidal comonads", "authors": "Dimitri Chikhladze, Stephen Lack, and Ross Street", "keywords": ["quantum category", "monoidal comonad", "monoidal bicategory", "quantum groupoid", "bialgebroid", "Hopf monoid", "closed category", "pseudomonoid"], "abstract": "We generalize to the context internal to an autonomous monoidal bicategory the work of Bruguieres, Virelizier, and the second-named author on lifting closed structure on a monoidal category to the category of Eilenberg-Moore algebras for an opmonoidal monad. The result then applies to quantum categories and bialgebroids."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/2/24-02abs.html", "title": "The Frobenius relations meet linear distributivity", "authors": "J.M. Egger", "keywords": ["Frobenius algebras", "linear distributive categories"], "abstract": "The notion of Frobenius algebra originally arose in ring theory, but it is a fairly easy observation that this notion can be extended to arbitrary monoidal categories. But, is this really the correct level of generalisation?  For example, when studying Frobenius algebras in the *-autonomous category $\\Sup$, the standard concept using only the usual tensor product is less interesting than a similar one in which both the usual tensor product and its de Morgan dual (par) are used.  Thus we maintain that the notion of linear-distributive category (which has both a tensor and a par, but is nevertheless more general than the notion of monoidal category) provides the correct framework in which to interpret the concept of Frobenius algebra."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/3/24-03abs.html", "title": "Joyal's arithmetic universe as list-arithmetic pretopos", "authors": "Maria Emilia Maietti", "keywords": ["Pretopoi", "dependent type theory", "categorical logic"], "abstract": "We explain in detail why the notion of list-arithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel's incompleteness results."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/1/24-01abs.html", "title": "Bicategories of spans as cartesian bicategories", "authors": "Stephen Lack, R.F.C. Walters, and R.J. Wood", "keywords": ["bicategory", "finite products", "discrete", "comonad", "Eilenberg-Moore object"], "abstract": "Bicategories of spans are characterized as cartesian bicategories in which every comonad has an Eilenberg-Moore object and every left adjoint arrow is comonadic."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/22/25-22abs.html", "title": "Symbolic dynamics and the category of graphs", "authors": "Terrence Bisson and Aristide Tsemo", "keywords": ["category of graphs", "Quillen model structure", "walks", "symbolic  dynamics", "coverings"], "abstract": "Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, $C^*$-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a \"finite-level\" homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism.   We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if $B$ is basal and separated then the walk spaces for $X$ and $B$ are topologically conjugate if and only if $X$ and $B$ are homotopically equivalent for our model structure."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/21/25-21abs.html", "title": "Differential restriction categories", "authors": "J.R.B. Cockett, G.S.H. Cruttwell,  and J. D. Gallagher", "keywords": ["Differential restriction categories", "Rational functions and the  Rational monad", "Join completion", "Classical completion"], "abstract": "We combine two recent ideas: cartesian differential categories, and restriction categories.  The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $R^n$ in a way that is completely algebraic.  We also give other models   for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/19/25-19abs.html", "title": "Countable meets in coherent spaces with applications to the cyclic spectrum", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["countable localic meets of subspaces", "Boolean flows", "cyclic spectrum"], "abstract": "This paper reviews the basic properties of coherent spaces, characterizes them, and proves a theorem about countable meets of open sets.  A number of examples of coherent spaces are given, including the set of all congruences (equipped with the Zariski topology) of a model of a theory based on a set of partial operations. We also give two alternate proofs of the main theorem, one using a theorem of Isbell's and a second using an unpublished theorem of Makkai's.  Finally, we apply these results to the Boolean cyclic spectrum and give some relevant examples."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/20/25-20abs.html", "title": "On reflective-coreflective equivalence and associated pairs", "authors": "Erik Bédos, S. Kaliszewski, and John Quigg", "keywords": ["reflective and coreflective subcategories", "equivalent categories", "associated pairs of subcategories"], "abstract": "We show that a reflective/coreflective pair of full subcategories satisfies a ``maximal-normal''-type equivalence if and only if it is an associated pair in the sense of Kelly and Lawvere."},
{"url": "http://www.tac.mta.ca/tac/volumes/24/12/24-12abs.html", "title": "Lax Presheaves and Exponentiability", "authors": "Susan Niefield", "keywords": ["span", "relation", "partial map", "topos", "cartesian closed", "exponentiable", "presheaf"], "abstract": "The category of Set-valued presheaves on a small category B is a topos. Replacing Set by a bicategory S whose objects are sets and morphisms are spans, relations, or partial maps, we consider a category Lax(B, S) of S-valued lax functors on B.  When S = Span, the resulting category is equivalent to Cat/B, and hence, is rarely even cartesian closed. Restricting this equivalence gives rise to exponentiability characterizations for Lax(B, Rel) by Niefield and for Lax(B, Par) in this paper.  Along the way, we obtain a characterization of those B for which the category UFL/B is a coreflective subcategory of Cat/B, and hence, a topos."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/18/25-18abs.html", "title": "Flows: cocyclic and almost cocyclic", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["flow on compact spaces", "periodic and cocyclic flows", "almost cocyclic flows"], "abstract": "A flow on a compact Hausdorff space is an automorphism.  Using the closed structure on the category of uniform spaces, a flow gives rise, by iteration, to an action of the integers on the topological group of automorphisms of the object.  We study special classes of flows:   periodic, cocyclic, and almost cocyclic, mainly in term of the possibility of extending this action continuously to various compactifications of the integers."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/16/25-16abs.html", "title": "Semidirect products and crossed modules in varieties of right $\\Omega$-loops", "authors": "Edward B. Inyangala", "keywords": ["semidirect products", "variety of right loops", "crossed module", "precrossed module"], "abstract": "We present a new explicit construction of categorical semidirect products in an arbitrary variety V of right $\\Omega$-loops and use it to obtain simplified descriptions of internal precrossed and crossed modules in V."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/17/25-17abs.html", "title": "Yoneda theory for double categories", "authors": "Robert Paré", "keywords": ["Double category", "lax functor", "module", "modulation", "representable", "Yoneda lemma"], "abstract": "Representables for double categories are defined to be lax morphisms into a certain double category of sets. We show that horizontal transformations from representables into lax morphisms correspond to elements of that lax morphism. Vertical arrows give rise to modules between representables. We establish that the Yoneda embedding is a strong morphism of lax double categories which is horizontally full and faithful and dense."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/15/25-15abs.html", "title": "The Faà di Bruno construction", "authors": "J.R.B. Cockett and R.A.G. Seely", "keywords": ["Higher-order chain rule", "Cartesian differential categories", "bundle fibration", "coalgebras"], "abstract": "In the context of Cartesian differential categories, the structure of the first-order chain rule gives rise to a fibration, the ``bundle category''.  In the present paper we generalise this to the higher-order chain rule (originally developed in the traditional setting by Faà di Bruno in the nineteenth century); given any Cartesian differential category X, there is a ``higher-order chain rule fibration'' Faa(X) --> X over it. In fact, Faa is a comonad (over the category of Cartesian left (semi-)additive categories). Our main theorem is that the coalgebras for this comonad are precisely the Cartesian differential categories.  In a sense, this result affirms the ``correctness'' of the notion of Cartesian differential categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/14/25-14abs.html", "title": "On involutive monoidal categories", "authors": "J.M. Egger", "keywords": ["involutive monoidal categories", "dagger pivotal categories", "braidings", "balances", "coherence theorems"], "abstract": "In this paper, we consider a non-posetal analogue of the notion of involutive quantale; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets to categories; we also link our subject with other notions already in the literature, such as balanced monoidal categories  and dagger pivotal categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/13/25-13abs.html", "title": "Covariant presheaves and subalgebras", "authors": "Ulrich Höhle", "keywords": ["Involutive quantale", "involutive quantaloid", "symmetric ${\\cal Q}$-category", "covariant presheaf", "monad of weak singletons", "classifiable subalgebra", "closed left ideal"], "abstract": "For small involutive and integral quantaloids ${\\cal Q}$ it is shown that covariant presheaves on symmetric ${\\cal Q}$-categories are equivalent to certain subalgebras of a specific monad on the category of symmetric ${\\cal Q}$-categories. This construction is related to a weakening of the subobject classifier axiom which does not require the classification of all subalgebras, but only guarantees that classifiable subalgebras are uniquely classifiable. As an application the identification of closed left ideals of non-commutative $C^*$-algebras with certain  \"open\", subalgebras of freely generated algebras is given."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/12/25-12abs.html", "title": "Towards a homotopy theory of higher dimensional transition systems", "authors": "Philippe Gaucher", "keywords": ["higher dimensional transition system", "locally presentable   category", "topological category", "combinatorial model category", "left determined model category", "Bousfield localization", "bisimulation"], "abstract": "We proved in a previous work that Cattani-Sassone's higher   dimensional transition systems can be interpreted as a   small-orthogonality class of a topological locally finitely   presentable category of weak higher dimensional transition   systems. In this paper, we turn our attention to the full   subcategory of weak higher dimensional transition systems which are   unions of cubes. It is proved that there exists a left proper   combinatorial model structure such that two objects are weakly   equivalent if and only if they have the same cubes after   simplification of the labelling. This model structure is obtained by   Bousfield localizing a model structure which is left determined with   respect to a class of maps which is not the class of monomorphisms.   We prove that the higher dimensional transition systems   corresponding to two process algebras are weakly equivalent if and   only if they are isomorphic. We also construct a second Bousfield   localization in which two bisimilar cubical transition systems are   weakly equivalent.  The appendix contains a technical lemma about   smallness of weak factorization systems in coreflective   subcategories which can be of independent interest.  This paper is a first step towards a homotopical interpretation of bisimulation for   higher dimensional transition systems."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/11/25-11abs.html", "title": "Symmetry and Cauchy completion of quantaloid-enriched categories", "authors": "Hans Heymans and Isar Stubbe", "keywords": ["Quantaloid", "enriched category", "symmetry", "Cauchy completion"], "abstract": "We formulate an elementary condition on an involutive quantaloid $Q$  under which there is a distributive law from the Cauchy completion monad  over the symmetrisation comonad on the category of $Q$-enriched  categories. For such quantaloids, which we call Cauchy-bilateral  quantaloids, it follows that the Cauchy completion of any symmetric  $Q$-enriched category is again symmetric. Examples include Lawvere's  quantale of non-negative real numbers and Walters' small quantaloids of  closed cribles."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/9/25-09abs.html", "title": "A small observation on co-categories", "authors": "Peter LeFanu Lumsdaine", "keywords": ["Co-categories", "co-groupoids", "coherent categories", "coherent logic"], "abstract": "Various concerns suggest looking for internal co-categories in categories with strong logical structure.  It turns out that in any coherent category E, all co-categories are co-equivalence relations."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/8/25-08abs.html", "title": "Model-categories of coalgebras over operads", "authors": "Justin R. Smith", "keywords": ["operads", "cofree coalgebras"], "abstract": "This paper constructs model structures on the categories of coalgebras and pointed irreducible coalgebras over an operad whose components are projective, finitely generated in each dimension, and satisfy a condition that allows one to take tensor products with a unit interval. The underlying chain-complex is assumed to be unbounded and the results for bounded coalgebras over an operad are derived from the unbounded case."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/10/25-10abs.html", "title": "Higher categorified algebras versus bounded homotopy algebras", "authors": "David Khudaverdyan, Ashis Mandal, and Norbert Poncin", "keywords": ["Higher category", "homotopy algebra", "monoidal category", "Eilenberg-Zilber map"], "abstract": "We define Lie 3-algebras and prove that these are in 1-to-1 correspondence with the 3-term Lie infinity algebras whose bilinear and trilinear maps vanish in degree (1,1) and in total degree 1, respectively. Further, we give an answer to a question of Roytenberg pertaining to the use of the nerve and normalization functors in the study of the relationship between categorified algebras and truncated sh algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/7/25-07abs.html", "title": "An embedding theorem for adhesive categories", "authors": "Stephen Lack", "keywords": ["adhesive category", "topos", "embedding theorem"], "abstract": "Adhesive categories are categories which have pushouts with one leg a monomorphism, all pullbacks, and certain exactness conditions relating these pushouts and pullbacks. We give a new proof of the fact that every topos is adhesive. We also prove a converse:  every small adhesive category has a fully faithful functor in a topos, with the functor preserving the all the structure. Combining these two results, we see that the exactness conditions in the definition of adhesive category are exactly the relationship between pushouts along monomorphisms and pullbacks which hold in any topos."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/6/25-06abs.html", "title": "Reflective-coreflective equivalence", "authors": "Erik Bédos, S. Kaliszewski, and John Quigg", "keywords": ["adjoint functors", "reflective and coreflective subcategories", "equivalent  categories", "$C^*$-algebras", "coactions", "quantum groups"], "abstract": "We explore a curious type of equivalence between certain pairs of  reflective and coreflective subcategories. We illustrate with examples involving  noncommutative duality for $C^*$-dynamical systems and compact quantum groups, as well as  examples where the subcategories are actually isomorphic."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/5/25-05abs.html", "title": "Monoidal functor categories and graphic Fourier transforms", "authors": "Brian J. Day", "keywords": ["monoidal category", "promonoidal category", "convolution", "Fourier transform"], "abstract": "This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures. There is a close resemblance to convolution products and the Wiener algebra (of transforms) in functional analysis. The analysis term ``kernel'' (of a distribution) has also been adapted below in connection with certain special types of ``distributors'' (in the terminology of J. Benabou) or ``modules'' (in the terminology of R. Street) in category theory. In using the term ``graphic'', in a very broad sense, we are clearly distinguishing the categorical methods employed in this article from standard Fourier and wavelet mathematics. The term ``graphic'' also applies to promultiplicative graphs, and related concepts, which can feature prominently in the theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html", "title": "Comparative smootheology", "authors": "Andrew Stacey", "keywords": ["generalised smooth spaces"], "abstract": "We compare various different definitions of \"the category of smooth objects\". The definitions compared are due to Chen, Frolicher, Sikorski, Smith, and Souriau.  The method of comparison is to construct functors between the categories that enable us to see how the categories relate to each other.  Our method of study involves finding a general context into which these categories can be placed. This involves considering categories wherein objects are considered in relation to a certain collection of standard test objects. This therefore applies beyond the question of categories of smooth spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/3/25-03abs.html", "title": "Remarks on punctual local connectedness", "authors": "Peter Johnstone", "keywords": ["axiomatic cohesion", "locally conected topos"], "abstract": "We study the condition, on a connected and locally connected geometric morphism $p : \\cal E \\to \\cal S$, that the canonical natural transformation $p_*\\to p_!$ should be (pointwise) epimorphic - a condition which F.W. Lawvere called the `Nullstellensatz', but which we prefer to call `punctual local connectedness'. We show that this condition implies that $p_!$ preserves finite products, and that, for bounded morphisms between toposes with natural number objects, it is equivalent to being both local and hyperconnected."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/27/26-27abs.html", "title": "Tangled circuits", "authors": "R. Rosebrugh, N. Sabadini and R. F. C. Walters", "keywords": ["circuit diagram", "braided monoidal category", "tangle algebra"], "abstract": "We consider commutative Frobenius algebras in braided strict monoidal categories in the study of the circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras. For example, we show how Armstrong's description of knot colourings and knot groups fit into this context."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html", "title": "Internal categories, anafunctors and localisations", "authors": "David Michael Roberts", "keywords": ["internal categories", "anafunctors", "localization", "bicategory of fractions"], "abstract": "In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site $S$, one can form a bicategorical localisation of various 2-categories of internal categories or groupoids at weak equivalences using anafunctors as 1-arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on $S$."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/2/25-02abs.html", "title": "A remark about the Connes fusion tensor product", "authors": "Andreas Thom", "keywords": ["Connes fusion tensor product", "von Neumann algebras"], "abstract": "We analyze the algebraic structure of the Connes fusion tensor product (CFTP) in the case of bi-finite Hilbert modules over a von Neumann algebra M. It turns out that all complications in its definition disappear if one uses the closely related bi-modules of bounded vectors. We construct an equivalence of monoidal categories with duality between a category of Hilbert bi-modules over M with CFTP and some natural category of bi-modules over M with the usual relative algebraic tensor product."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/28/26-28abs.html", "title": "Décalage and Kan's simplicial loop group functor", "authors": "Danny Stevenson", "keywords": ["simplicial loop group", "d\\'{e}calage", "Artin-Mazur total simplicial set"], "abstract": "Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur.  There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy equivalence for any bisimplicial set.  In this paper we will give a new, elementary proof of this result.  As an application, we will revisit Kan's simplicial loop group functor $G$.  We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane's classifying complex functor $\\overline{W}$. We will give a new, short, proof of Kan's result that the unit map for the adjunction $G\\dashv \\overline{W}$ is a weak homotopy equivalence for reduced simplicial sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/25/1/25-01abs.html", "title": "A category of quantum categories", "authors": "Dimitri Chikhladze", "keywords": ["quantum category", "monoidal category", "comonad"], "abstract": "Quantum categories were introduced by Day and Street as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/25/26-25abs.html", "title": "Site characterizations for geometric invariants of toposes", "authors": "Olivia Caramello", "keywords": ["Grothendieck topos", "site characterizations", "geometric logic"], "abstract": "We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given `geometric' invariant as a property of its sites of definition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to specific invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos), obtaining explicit site characterizations for them."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/26/26-26abs.html", "title": "Span, cospan, and other double categories", "authors": "Susan Niefield", "keywords": ["double category", "lax functor", "(co)span", "(co)tabulator", "companion", "conjoint", "symmetric algebra"], "abstract": "Given a double category $\\mathbb D$ such that $\\mathbb D_0$ has pushouts, we characterize oplax/lax adjunctions between $\\mathbb D$ and $Cospan(\\mathbb D_0)$ for which the right adjoint is normal and restricts to the identity on $\\mathbb D_0$, where $Cospan(\\mathbb D_0)$ is the double category on $\\mathbb D_0$ whose vertical morphisms are cospans.  We show that such a pair exists if and only if $\\mathbb D$ has companions, conjoints, and 1-cotabulators. The right adjoints are induced by the companions and conjoints, and the left adjoints by the 1-cotabulators.  The notion of a 1-cotabulator is a common generalization of the symmetric algebra of a module and Artin-Wraith glueing of toposes, locales, and topological spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/24/26-24abs.html", "title": "Isotropy and crossed toposes", "authors": "Jonathon Funk, Pieter Hofstra and Benjamin Steinberg", "keywords": ["Topos theory", "inverse semigroups", "\\'etale groupoids", "isotropy groups", "crossed modules"], "abstract": "Motivated by constructions in the theory of inverse semigroups and etale groupoids, we define and investigate the concept of isotropy from a topos-theoretic perspective. Our main conceptual tool is a monad on the category of grouped toposes. Its algebras correspond to a generalized notion of crossed module, which we call a crossed topos. As an application, we present a topos-theoretic characterization and generalization of the `Clifford, fundamental' sequence associated with an inverse semigroup."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/23/26-23abs.html", "title": "Duality and traces for indexed monoidal categories", "authors": "Kate Ponto and Michael Shulman", "keywords": ["duality", "trace", "monoidal category", "indexed category", "fiberwise duality"], "abstract": "By the Lefschetz fixed point theorem, if an endomorphism of a   topological space is fixed-point-free, then its Lefschetz number   vanishes.  This necessary condition is not usually sufficient,   however; for that we need a refinement of the Lefschetz number called   the Reidemeister trace.  Abstractly, the Lefschetz number is a trace   in a symmetric monoidal category, while the Reidemeister trace is a   trace in a bicategory; in this paper we relate these contexts using   indexed symmetric monoidal categories.\nIn particular, we will show that for any symmetric monoidal category   with an associated indexed symmetric monoidal category, there is an   associated bicategory which produces refinements of trace analogous to   the Reidemeister trace.  This bicategory also produces a new notion of   trace for parametrized spaces with dualizable fibers, which refines   the obvious ``fiberwise'' traces by incorporating the action of the   fundamental group of the base space.  We also advance the basic theory   of indexed monoidal categories, including introducing a string diagram   calculus which makes calculations much more tractable.  This abstract   framework lays the foundation for generalizations of these ideas to   other contexts."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/22/26-22abs.html", "title": "Bimonadicity and the explicit basis property", "authors": "Matias Menni", "keywords": ["(co)monads", "projective objects", "descent", "modular categories", "Peano algebras"], "abstract": "Let ${L\\dashv R:\\cal X \\rightarrow\\cal Y}$ be an adjunction with $R$ monadic and $L$ comonadic. Denote the induced monad on $\\cal Y$ by $M$ and the induced comonad on $\\calX$ by $C$. We characterize those $C$ such that $M$ satisfies the Explicit Basis property. We also discuss some new examples and results motivated by this characterization."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/21/26-21abs.html", "title": "Yoneda representations of flat functors and classifying toposes", "authors": "Olivia Caramello", "keywords": ["Classifying topos", "Yoneda lemma", "flat functor", "theory of presheaf type"], "abstract": "We obtain semantic characterizations, holding for any Grothendieck site $(C, J)$, for the models of a theory classified by a topos of the form $Sh(C,J)$ in terms of the models of a theory classified by a topos $[C^{op}, Set]$. These characterizations arise from an appropriate representation of flat functors into Grothendieck toposes based on an application of the Yoneda Lemma in conjunction with ideas from indexed category theory, and turn out to be relevant also in different contexts, in particular for addressing questions in classical Model Theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/20/26-20abs.html", "title": "Modeling stable one-types", "authors": "Niles Johnson, Angélica M. Osorno", "keywords": ["stable homotopy one-type", "Picard groupoid"], "abstract": "Classification of homotopy $n$-types has focused on developing algebraic categories which are equivalent to categories of $n$-types.  We expand this theory by providing algebraic models of homotopy-theoretic constructions for stable one-types.  These include a model for the Postnikov one-truncation of the sphere spectrum, and for its action on the model of a stable one-type.   We show that a bicategorical cokernel introduced by Vitale models the cofiber of a map between stable one-types, and apply this to develop an algebraic model for the Postnikov data of a stable one-type."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/19/26-19abs.html", "title": "Opetopes and chain complexes", "authors": "Richard Steiner", "keywords": ["opetope", "augmented directed complex"], "abstract": "We give a simple algebraic description of opetopes in terms of chain  complexes, and we show how this description is related to combinatorial descriptions in terms of treelike structures. More generally, we show that the chain complexes associated to higher categories generate graphlike structures. The algebraic description gives a relationship between the opetopic approach and other approaches to higher category theory. It also gives an easy way to calculate the sources and targets of opetopes."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/18/26-18abs.html", "title": "Range categories II: Towards regularity", "authors": "J.R.B. Cockett, Xiuzhan Guo and Pieter Hofstra", "keywords": ["Categories of partial maps", "restriction category", "factorization systems"], "abstract": "In this paper, which is the second part of a study of partial map categories with images, we investigate the interaction between images and various other kinds of categorical structure and properties. In particular, we consider images in the context of partial products, meets and discreteness and survey a taxonomy of structures leading towards the partial map categories of regular categories. We also present a term logic for cartesian partial map categories with images and prove a soundness and completeness theorem for this logic. Finally, we exhibit several free constructions relating the different classes of categories under consideration."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/17/26-17abs.html", "title": "Range categories I: General theory", "authors": "J.R.B. Cockett, Xiuzhan Guo and Pieter Hofstra", "keywords": ["Categories of partial maps", "restriction category", "factorization systems"], "abstract": "In this two-part paper, we undertake a systematic study of abstract partial map categories in which every map has both a restriction (domain of definition) and a range (image). In this first part, we explore connections with related structures such as inverse categories and allegories, and establish two representational results. The first of these explains how every range category can be fully and faithfully embedded into a category of partial maps equipped with a suitable factorization system. The second is a generalization of a result from semigroup theory by Boris Schein, and says that every small range category satisfying the additional condition that every map is an epimorphism onto its range can be faithfully embedded into the category of sets and partial functions with the usual notion of range. Finally, we give an explicit construction of the free range category on a partial map category in terms of certain types of labeled trees."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/16/26-16abs.html", "title": "A presheaf interpretation of the generalized Freyd conjecture", "authors": "Anna Marie Bohmann and J. P. May", "keywords": ["Freyd conjecture", "generating hypothesis", "stable homotopy category"], "abstract": "We give a generalized version of the Freyd conjecture and a way to think about a possible proof.  The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new results, but at least one known example drops out very easily."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/15/26-15abs.html", "title": "Skew monoidales, skew warpings and quantum categories", "authors": "Stephen Lack and Ross Street", "keywords": ["bialgebroid", "fusion operator", "quantum category", "monoidal bicategory", "monoidale", "skew-monoidal category", "comonoid", "Hopf monad"], "abstract": "Kornel Szlachanyi recently used the term skew-monoidal category for a particular laxified version of monoidal category. He showed that bialgebroids $H$ with base ring $R$ could be characterized in terms of skew-monoidal structures on the category of one-sided $R$-modules for which the lax unit was $R$ itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal bicategory $\\cal M$. These are skew-monoidal categories when $\\cal M$ is $Cat$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories with base comonoid $C$ in a suitably complete braided monoidal category $\\CV$ are precisely skew monoidales in $Comod (\\cal V)$ with unit coming from the counit of $C$. Quantum groupoids (in the sense of Chikhladze et al rather than Day and Street)  are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping defined by Booker and Street to modify monoidal structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/14/26-14abs.html", "title": "Biequivalences in tricategories", "authors": "Nick Gurski", "keywords": ["biequivalence", "biadjoint biequivalence", "tricategory", "monoidal bicategory"], "abstract": "We show that every internal biequivalence in a tricategory $T$ is part of a biadjoint biequivalence.  We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/13/26-13abs.html", "title": "Injective hulls of partially ordered monoids", "authors": "J. Lambek, Michael Barr, John F. Kennison, R. Raphael", "keywords": ["partially ordered monoids", "injectives"], "abstract": "We find the injective hulls of partially ordered monoids in the category whose objects are po-monoids and submultiplicative order-preserving functions.  These injective hulls are with respect to a special class of monics called ``embeddings''.  We show as well that the injective objects with respect to these embeddings are precisely the quantales."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/12/26-12abs.html", "title": "Congruences of Morita equivalent small categories", "authors": "Valdis Laan", "keywords": ["category", "Morita equivalence", "congruence"], "abstract": "Two categories are called Morita equivalent if the categories of functors from these categories to the category of sets are equivalent. We prove that congruence lattices of Morita equivalent small categories are isomorphic."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/11/26-11abs.html", "title": "The coalgebraic structure of cell complexes", "authors": "Thomas Athorne", "keywords": ["relative cell complexes", "algebraic weak factorisation systems", "small object argument"], "abstract": "The relative cell complexes with respect to a generating set of cofibrations are an important class of morphisms in any model structure. In the particular case of the standard (algebraic) model structure on Top, we give a new expression of these morphisms by defining a category of relative cell complexes, which has a forgetful functor to the arrow category. This allows us to prove a conjecture of Richard Garner: considering the algebraic weak factorisation system given in that algebraic model structure between cofibrations and trivial fibrations, we show that the category of relative cell complexes is equivalent to the category of coalgebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/10/26-10abs.html", "title": "Notes on Bimonads and Hopf Monads", "authors": "Bachuki Mesablishvili and Robert Wisbauer", "keywords": ["Opmonoidal functors", "bimonads", "Hopf monads", "Galois entwinings"], "abstract": "For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguièeres and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads,). In a recent joint paper with S. Lack the same authors define the notion of a pre-Hopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the pre-Hopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/9/26-09abs.html", "title": "Graphical Methods for Tannaka duality of weak bialgebras and weak Hopf algebras", "authors": "Micah Blake McCurdy", "keywords": ["Tannaka duality", "Tannaka reconstruction", "bialgebras", "Hopf algebras", "weak bialgebras", "weak Hopf algebras", "separable Frobenius monoidal functors", "graphical methods"], "abstract": "Tannaka duality describes the relationship between algebraic objects in a given category and functors into that category; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful ``fibre functors'' to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish this goal, we make use of a graphical notation for functors between monoidal categories, using string diagrams with coloured regions. Not only does this notation extend our capacity to give simple proofs of complicated calculations, it makes plain some of the connections between Frobenius monoidal or separable Frobenius monoidal functors and the topology of the axioms defining certain algebraic structures. Finally, having generalized Tannaka duality to an arbitrary base category, we briefly discuss the functoriality of the construction as this base is varied."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/8/26-08abs.html", "title": "A characterization of representable intervals", "authors": "Michael A. Warren", "keywords": [], "abstract": "In this note we provide a characterization, in terms of additional algebraic structure, of those strict intervals (certain cocategory objects) in a symmetric monoidal closed category $\\cal E$ that are representable in the sense of inducing on $\\cal E$ the structure of a finitely bicomplete 2-category.  Several examples and connections with the homotopy theory of 2-categories are also discussed."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/7/26-07abs.html", "title": "Note on star-autonomous comonads", "authors": "Craig Pastro", "keywords": ["Star-autonomous", "autonomous", "rigid", "linearly distributive", "comonad", "Hopf"], "abstract": "We develop an alternative approach to star-autonomous comonads via linearly distributive categories. It is shown that in the autonomous case the notions  of star-autonomous comonad and Hopf comonad coincide."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/6/26-06abs.html", "title": "Syntactic characterizations of properties of classifying toposes", "authors": "Olivia Caramello", "keywords": ["Grothendieck topos", "site characterizations", "geometric logic", "presheaf topos"], "abstract": "We give characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (respectively connected and locally connected, atomic, compact, presheaf) topos, and exploit the existence of multiple sites of definition for a given topos to establish various results on quotients of theories of presheaf type."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/5/26-05abs.html", "title": "Closed categories vs. closed multicategories", "authors": "Oleksandr Manzyuk", "keywords": ["Closed category", "closed multicategory", "equivalence"], "abstract": "We prove that the 2-category of closed categories of Eilenberg and  Kelly is equivalent to a suitable full 2-subcategory of the  2-category of closed multicategories."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/4/26-04abs.html", "title": "Commutative monads as a theory of distributions", "authors": "Anders Kock", "keywords": ["monads", "distributions", "extensive quantities"], "abstract": "It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/3/26-03abs.html", "title": "On diagram-chasing in double complexes", "authors": "George M. Bergman", "keywords": ["double complex", "exact sequence", "diagram-chasing", "Salamander Lemma", "total homology", "triple complex"], "abstract": "We construct, for any double complex in an abelian category, certain ``short-distance'' maps, and an exact sequence involving these, instances of which can be pieced together to give the ``long-distance'' maps and exact sequences of results such as the Snake Lemma.  Further applications are given. We also note what the building blocks of an analogous study of triple complexes would be."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/2/26-02abs.html", "title": "On the iteration of weak wreath products", "authors": "Gabriella Böhm", "keywords": ["monad", "weak distributive law", "n-ary weak wreath product", "Yang-Baxter equation", "quantum spin chain"], "abstract": "Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street's weak wreath product construction. That is, for any 2-category $cal K$ and for any non-negative integer $n$, we introduce 2-categories $\\Wdl^{(n)}(\\cal K)$, of $(n+1)$-tuples of monads in $\\cal K$ pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance $\\Wdl^{(0)}(\\cal K)$ coincides with $\\Mnd(\\cal K)$, the usual 2-category of monads in $\\cal K$, and for other values of $n$, $\\Wdl^{(n)}(\\cal K)$ contains $\\Mnd^{n+1}(\\cK)$ as a full 2-subcategory.  For the local idempotent closure $\\overline \\cal K$ of $\\cal K$, extending the multiplication of the 2-monad $\\Mnd$, we equip these 2-categories with $n$ possible `weak wreath product' 2-functors $\\Wdl^{(n)}(\\ocK)\\to \\Wdl^{(n-1)}(\\overline \\cal K)$, such that all of their possible $n$-fold composites $\\Wdl^{(n)}(\\overline \\cal K)\\to \\Wdl^{(0)}(\\overline \\cal K)$ are equal;  that is, such that the weak wreath product is `associative'. Whenever idempotent 2-cells in $\\cal K$ split, this leads to pseudofunctors $\\Wdl^{(n)}(\\cal K)\\to \\Wdl^{(n-1)}(\\cal K)$ obeying the associativity property up-to isomorphism. We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras. We also construct a fully faithful embedding of $\\Wdl^{(n)}(\\overline \\cal K)$ into the 2-category of commutative $n+1$ dimensional cubes in $\\Mnd(\\overline \\cal K)$ (hence into the 2-category of commutative $n+1$ dimensional cubes in $\\cal K$ whenever $\\cal K$ has Eilenberg-Moore objects and its idempotent 2-cells split). Finally we give a sufficient and necessary condition on a monad in $\\overline \\cal K$ to be isomorphic to an $n$-ary weak wreath product."},
{"url": "http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html", "title": "Kan extensions and lax idempotent pseudomonads", "authors": "F. Marmolejo and R.J. Wood", "keywords": ["(co-) lax idempotent pseudomonads", "KZ-doctrines", "pseudo-distributive laws"], "abstract": "We show that colax idempotent pseudomonads and their algebras can be presented in terms of right Kan extensions. Dually, lax idempotent pseudomonads and their algebras can be presented in terms of left Kan extensions. We also show that a distributive law of a colax idempotent pseudomonad over a lax idempotent pseudomonad has a presentation in terms of Kan extensions."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/18/27-18abs.html", "title": "An equational metalogic for monadic equational systems", "authors": "Marcelo Fiore", "keywords": ["Monoidal action", "strong monad", "clones", "double dualization", "equational presentation", "free algebra", "equational logic", "soundness", "(strong)   completeness"], "abstract": "The paper presents algebraic and logical developments.  From the algebraic viewpoint, we introduce Monadic Equational Systems as an abstract enriched notion of equational presentation.  From the logical viewpoint, we provide Equational Metalogic as a general formal deductive system for the derivability of equational consequences. Relating the two, a canonical model theory for Monadic Equational Systems is given and for it the soundness of Equational Metalogic is established. This development  involves a study of clone and double-dualization structures. We also show that in the presence of free algebras %constructions the model theory of Monadic Equational Systems satisfies an internal strong-completeness property."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/17/27-17abs.html", "title": "Elementary quotient completion", "authors": "Maria Emilia Maietti and Giuseppe Rosolini", "keywords": ["quotient completion", "split fibration", "universal construction"], "abstract": "We extend the notion of exact completion on a category with weak finite limits to Lawvere's elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which is the universal solution to adding certain quotients. We note that the elementary quotient completion can be obtained as the composite of two other universal constructions: one adds effective quotients, the other forces extensionality of morphisms. We also prove that each construction preserves comprehension."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/13/27-13abs.html", "title": "Generalized Hopf modules for bimonads", "authors": "Marcelo Aguiar and Stephen U. Chase", "keywords": ["monad", "comonad", "bimonad", "Beck's theorem", "Hopf module", "Doi-Koppinen Hopf module", "Hopf Galois", "Sweedler's Fundamental Theorem", "Schneider's Structure Theorem", "Hilbert's Theorem 90"], "abstract": "Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler's Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodule-monad and a algebra-comonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler's result (to the setting of Hopf Galois extensions)."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/16/27-16abs.html", "title": "Composition of modules for lax functors", "authors": "Robert Paré", "keywords": ["double category", "lax functor", "module", "modulation", "representability"], "abstract": "We study the composition of modules between lax functors of weak double categories. We adapt the bicategorical notion of local cocompleteness to weak double categories, which the codomain of our lax functors will be assumed to satisfy. We introduce a notion of factorization of cells, which most weak double categories of interest possess, and which is sufficient to guarantee the strong representability of composites of modules between lax functors whose domain satisfies it."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/15/27-15abs.html", "title": "On actions and strict actions in homological categories", "authors": "Manfred Hartl and Bruno Loiseau", "keywords": ["action", "semi-direct product", "conjugation", "normal subobject", "ideal", "commutator", "homological category", "semi-abelian category", "algebra over monad"], "abstract": "Let $G$ be an object of a finitely cocomplete homological category $\\mathbb C$. We study actions of $G$ on objects $A$ of $\\mathbb C$  (defined by Bourn and Janelidze as being algebras over a certain monad $\\mathbb T_G$), with two objectives: investigating to which extent actions can be described in terms of smaller data, called action cores; and to single out those abstract action cores which extend to actions corresponding to semi-direct products of $A$ and $G$ (in a non-exact setting, not every action does). This amounts to exhibiting a subcategory of the category of the actions of $G$ on objects $A$ which is equivalent with the category of points in $\\mathbb C$ over $G$, and to describing it in terms of action cores.  This notion and its study are based on a preliminary investigation of co-smash products, in which cross-effects of functors in a general categorical context turn out to be a useful tool. The co-smash products also allow us to define higher categorical commutators, different from the ones of Huq, which are not generally expressible in terms of nested binary ones. We use strict action cores to show that any normal subobject of an object $E$ (i.e., the equivalence class of $0$ for some equivalence relation on $E$ in $\\mathbb C$) admits a strict conjugation action of $E$. If $\\mathbb C$ is semi-abelian, we show that for subobjects $X$, $Y$ of some object $A$, $X$ is proper in the supremum of $X$ and $Y$ if and only if $X$ is stable under the restriction to $Y$ of the conjugation action of $A$ on itself. This also amounts to an alternative proof of Bourn and Janelidze's category equivalence between points over $G$ in $\\mathbb C$ and actions of $G$ in the semi-abelian context.  Finally, we show that the two axioms of an algebra which characterize $G$-actions are equivalent with three others ones, in terms of action cores. These axioms are commutative squares involving only co-smash products. Two of them are associativity type conditions which generalize the usual properties of an action of one group on another, while the third is kind of a higher coherence condition which is a consequence of the other two in the category of groups, but probably not in general. As an application, we characterize abelian action cores, that is, action cores corresponding to Beck modules; here also the coherence condition follows from the others."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/12/27-12abs.html", "title": "Lawvere completeness as a topological property", "authors": "Serdar Sozubek", "keywords": ["Completeness", "compactness", "lax algebra", "module", "proper map", "injectivity", "fibrewise sober", "factorization system"], "abstract": "Lawvere's notion of completeness for quantale-enriched categories has been extended to the theory of lax algebras under the name of L-completeness. In this paper we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological concepts like separatedness, denseness, compactness and compactification with respect to L-complete morphisms. Moreover, we show that separated L-complete morphisms belong to a factorization system."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/14/27-14abs.html", "title": "Proper maps for lax algebras and the Kuratowski-Mrówka theorem", "authors": "Maria Manuel Clementino and Walter Tholen", "keywords": ["(T,V)-category", "compact space", "proper map", "Kuratowski-Mrówka Theorem"], "abstract": "The characterization of stably closed maps of topological spaces as the closed maps with compact fibres and the role of the Kuratowski-Mrówka' Theorem in this characterization are being explored in the general context of lax (T,V)-algebras, for a quantale V and a Set-monad T with a lax extension to V-relations. The general results are being applied in standard (topological and metric) and non-standard (labeled graphs) contexts."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/10/27-10abs.html", "title": "Descent in monoidal categories", "authors": "Bachuki Mesablishvili", "keywords": ["symmetric monoidal categories", "effective descent morphisms", "pure morphisms"], "abstract": "We consider a symmetric monoidal closed category $V = (V, \\otimes, I, [-,-])$ together with a regular injective object $Q$ such that the functor $[-, Q] : \\to V^{op}$ is comonadic and prove that in such a category, as in the monoidal category of abelian groups, a morphism of commutative monoids is an effective descent morphism for modules if and only if it is a pure monomorphism. Examples of this kind of monoidal categories are elementary toposes considered as cartesian closed monoidal categories, the module categories over a commutative ring object in a Grothendieck topos and Barr's star-autonomous categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/11/27-11abs.html", "title": "The *-autonomous category of uniform sup semi-lattices", "authors": "Michael Barr, John F. Kennison, and R. Raphael", "keywords": ["Uniform sup semi-lattices", "*-autonomous categories", "chu categories"], "abstract": "In 2001 Barr and Kleisli described *-autonomous structures on two full subcategories of topological abelian groups.  In this paper we do the same for sup semi-lattices except that uniform structures play the role that topology did in the earlier paper."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/9/27-09abs.html", "title": "Higher central extensions via commutators", "authors": "Diana Rodelo and Tim Van der Linden", "keywords": ["Higgins", "Huq", "Smith commutator", "higher central extension", "semi-abelian", "exact Mal'tsev category", "Hopf formula", "(co)homology"], "abstract": "We prove that all semi-abelian categories with the the Smith is Huq property satisfy the Commutator Condition(CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology with coefficients in the abelianisation functor, and an interpretation of cohomology with coefficients in an abelian object in terms of equivalence classes of higher central extensions. We also give a counterexample against (CC) in the semi-abelian category of (commutative) loops."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/8/27-08abs.html", "title": "The Ursini commutator as normalized Smith-Pedicchio commutator", "authors": "Sandra Mantovani", "keywords": ["commutator", "ideal determined category", "exact Mal'tsev category"], "abstract": "We introduce an intrinsic description of the Ursini commutator in any ideal determined category and we compare it with the Higgins  and Huq commutators. After describing also the Smith-Pedicchio commutator by means of canonical arrows from a coproduct, we compare the two notions, showing that  in any exact Mal'tsev normal category the Ursini commutator $[H,K]_{U}$ of two subobjects $H, K$ of $A$ is the normalization of the Smith-Pedicchio commutator of the equivalence relations generated by $H$ and $K$, extending the result valid for ideal determined varieties given by Ursini and Gumm."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html", "title": "Exact completions and small sheaves", "authors": "Michael Shulman", "keywords": ["exact completion", "site", "sheaf", "exact category", "pretopos", "topos"], "abstract": "We prove a general theorem which includes most notions of \"exact   completion\" as special cases.  The theorem is that \"κ-ary exact   categories\" are a reflective sub-2-category of \"κ-ary sites\",   for any regular cardinal κ.  A κ-ary exact category is an  exact   category with disjoint and universal κ-small coproducts, and a   κ-ary site is a site whose covering sieves are generated by   κ-small families and which satisfies a solution-set condition for   finite limits relative to κ.\nIn the unary case, this includes the exact completions of a regular   category, of a category with (weak) finite limits, and of a category   with a factorization system.  When κ=ω it includes the   pretopos completion of a coherent category.  And when κ=∞ is   the size of the universe, it includes the category of sheaves on a   small site, and the category of small presheaves on a locally small   and finitely complete category.  The ∞-ary exact completion of a   large nontrivial site gives a well-behaved \"category of small   sheaves\".\nAlong the way, we define a slightly generalized notion of \"morphism   of sites\" and show that κ-ary sites are equivalent to a type  of   \"enhanced allegory\".  This enables us to construct the exact   completion in two ways, which can be regarded as decategorifications   of \"representable profunctors\" (i.e. entire functional  relations)   and \"anafunctors\", respectively."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/6/27-06abs.html", "title": "Symmetry of regular diamonds, the Goursat property, and subtractivity", "authors": "Marino Gran, Zurab Janelidze, Diana Rodelo and Aldo Ursini", "keywords": ["Ideal of null morphisms", "Goursat category", "$3$-permutable variety", "subtractive category", "subtractive variety", "ideal", "clot", "ideal determined category", "good theory of ideals"], "abstract": "We investigate $3$-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of $E$-subtractive varieties (where $E$ is the set of constants in a variety) recently introduced by the fourth author. As an application, we show that ``ideals'' coincide with ``clots'' in any regular subtractive category, which can be considered as a pointed analogue of a known result for regular Goursat categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/5/27-05abs.html", "title": "Weakly Mal'tsev categories and strong relations", "authors": "Zurab Janelidze and Nelson Martins-Ferreira", "keywords": ["weakly Mal'tsev category", "Mal'tsev category", "difunctional relation", "factorization system"], "abstract": "We define a strong relation in a category $\\mathbb{C}$ to be a span which is ``orthogonal'' to the class of jointly epimorphic pairs of morphisms. Under the presence of finite limits, a strong relation is simply a strong monomorphism $R\\rightarrow X\\times Y$. We show that a category $\\mathbb{C}$ with pullbacks and equalizers is a weakly Mal'tsev category if and only if every reflexive strong relation in $\\mathbb{C}$ is an equivalence relation. In fact, we obtain a more general result which includes, as its another particular instance, a similar well-known characterization of Mal'tsev categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/3/27-03abs.html", "title": "On the axioms for adhesive and quasiadhesive categories", "authors": "Richard Garner and Stephen Lack", "keywords": ["adhesive category", "quasiadhesive category", "pushout", "exactness condition", "embedding theorem"], "abstract": "A category is adhesive if it has all pullbacks, all push-outs along monomorphisms, and all exactness conditions between pullbacks and pushouts along monomorphisms which hold in a topos. This condition can be modified by considering only pushouts along regular monomorphisms, or by asking only for the exactness conditions which hold in a quasitopos. We prove four characterization theorems dealing with adhesive categories and their variants."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/2/27-02abs.html", "title": "Exponentiability via double categories", "authors": "Susan Niefield", "keywords": ["exponentiable space", "function space", "lax slice", "specialization order"], "abstract": "For a small category $B$ and a double category $\\mathbb D$, let ${\\rm Lax}_N(B,\\mathbb D)$ denote the category whose objects are vertical normal lax functors $B\\to\\mathbb D$ and morphisms are horizontal lax transformations. It is well known that $Lax_N(B, \\mathbb Cat) \\simeq Cat/B$, where $\\mathbb Cat$ is the double category of small categories, functors, and profunctors.  We generalized this equivalence to certain double categories, in the case where $B$ is a finite poset.  Street showed that $Y\\to B$ is exponentiable in $Cat/B$ if and only if the corresponding normal lax functor $B\\to \\mathbb Cat$ is a pseudo-functor.  Using our generalized equivalence, we show that a morphism $Y\\to B$ is exponentiable in $ {\\mathbb D}_0/B$ if and only if the corresponding normal lax functor $B\\to\\mathbb D$ is a pseudo-functor plus an additional condition that holds for all $X\\to !B$ in $Cat$.  Thus, we obtain a single theorem which yields characterizations of certain exponentiable morphisms of small categories, topological spaces, locales, and posets."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/4/27-04abs.html", "title": "The core of adjoint functors", "authors": "Ross Street", "keywords": ["adjoint functor", "enriched category", "bicategory", "Kleisli cocompletion"], "abstract": "There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting."},
{"url": "http://www.tac.mta.ca/tac/volumes/27/1/27-01abs.html", "title": "Remarks on exactness notions pertaining to pushouts", "authors": "Richard Garner", "keywords": ["Exactness", "pushouts", "difunctional relation"], "abstract": "We call a finitely complete category diexact if every difunctional relation admits a pushout which is stable under pullback and itself a pullback. We prove three results relating to diexact categories: firstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only if it is Barr-exact, and every pair of monomorphisms admits a pushout which is stable and a pullback; and thirdly, that a small category with finite limits and pushouts of difunctional relations is diexact if and only if it admits a full structure-preserving embedding into a Grothendieck topos."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/33/28-33abs.html", "title": "Galois theories of commutative semigroups via semilattices", "authors": "Isabel A. Xarez and Joao J. Xarez", "keywords": ["Commutative semigroups", "semilattices", "admissible reflection", "covering morphisms", "stably-vertical morphisms", "normal morphisms", "inseparable-separable factorization"], "abstract": "The classes of stably-vertical, normal, separable, inseparable, purely inseparable and covering morphisms, defined in categorical Galois theory, are characterized for the reflection of the variety of commutative semigroups into its subvariety of semilattices. It is also shown that there is an inseparable-separable factorization, but there is no monotone-light factorization."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/32/28-32abs.html", "title": "The Gleason cover of a realizability topos", "authors": "Peter Johnstone", "keywords": ["realizability topos", "De Morgan topos", "Gleason cover"], "abstract": "Recently Benno~van~den~Berg introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the `constant object' functor from the topos of sets preserves finite coproducts, and that De Morgan's law is satisfied. In this paper we show that these properties are no accident: for any Schonfinkel algebra $\\Lambda$, the Herbrand realizability topos over $\\Lambda$ may be obtained as the Gleason cover (in the sense of Johnstone (1980)) of the ordinary realizability topos over $\\Lambda$. As a corollary, we obtain the functoriality of the Herbrand realizability construction on the category of Schonfinkel algebras and computationally dense applicative morphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/31/28-31abs.html", "title": "Tannaka--Krein duality for compact quantum homogeneous spaces. I. General theory", "authors": "Kenny De Commer and Makoto Yamashita", "keywords": ["compact quantum groups", "$C^*$-algebras", "Hilbert modules", "ergodic actions", "module categories"], "abstract": "An ergodic action of a compact quantum group $G$ on an operator algebra $A$ can be interpreted as a quantum homogeneous space for $G$. Such an action gives rise to the category of finite equivariant Hilbert modules over $A$, which has a module structure over the tensor category $Rep(G)$ of finite-dimensional representations of $G$.  We show that there is a one-to-one correspondence between the quantum $G$-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module $C^*$-categories over $Rep(G)$ up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/30/28-30abs.html", "title": "On theories of superalgebras of differentiable functions", "authors": "David Carchedi and Dmitry Roytenberg", "keywords": ["$C^\\infty$-ring", "Lawvere theory", "superalgebra", "supergeometry"], "abstract": "This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such a theory is called a super Fermat theory. Any category of superspaces and smooth functions has an associated such theory. This includes both real and complex supermanifolds, as well as algebraic superschemes. In particular, there is a super Fermat theory of $C^\\infty$-superalgebras. $C^\\infty$-superalgebras are the appropriate notion of supercommutative algebras in the world of $C^\\infty$-rings, the latter being of central importance both to synthetic differential geometry and to all existing models of derived smooth manifolds. A super Fermat theory is a natural generalization of the concept of a Fermat theory introduced by E. Dubuc and A. Kock. We show that any Fermat theory admits a canonical superization, however not every super Fermat theory arises in this way. For a fixed super Fermat theory, we go on to study a special subcategory of algebras called near-point determined algebras, and derive many of their algebraic properties."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/27/28-27abs.html", "title": "A double categorical model of weak 2-categories", "authors": "Simona Paoli and Dorette Pronk", "keywords": ["double categories", "strict 2-categories", "bicategories", "Tamsamani weak 2-categories", "pseudo-functors", "strictification"], "abstract": "We introduce the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories. We show that this model is suitably equivalent to bicategories and give an explicit description of the functors involved in this biequivalence. As an application we show that groupoidal weakly globular double categories model homotopy 2-types."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/29/28-29abs.html", "title": "Relative Mal'tsev categories", "authors": "Tomas Everaert, Julia Goedecke, Tamar Janelidze-Gray and Tim Van der Linden", "keywords": ["higher extension", "simplicial resolution", "Mal'tsev condition", "relative homological algebra", "arithmetical category"], "abstract": "We define relative regular Mal'tsev categories and give an overview of conditions which are equivalent to the relative Mal'tsev axiom. These include conditions on relations as well as conditions on simplicial objects. We also give various examples and counterexamples."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/28/28-28abs.html", "title": "Forms and exterior differentiation in Cartesian differential categories", "authors": "G.S.H. Cruttwell", "keywords": ["Cartesian differential categories", "Differential forms", "Exterior derivative", "de Rham cohomology"], "abstract": "Cartesian differential categories abstractly capture the notion of a differentiation operation.  In this paper, we develop some of the theory of such categories by defining differential forms and exterior differentiation in this setting.  We show that this exterior derivative, as expected, produces a cochain complex."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/25/28-25abs.html", "title": "Multitensor lifting and strictly unital higher category theory", "authors": "Michael Batanin, Denis-Charles Cisinski and Mark Weber", "keywords": ["multitensors", "strictly unital higher categories", "higher operads"], "abstract": "In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result - the lifting theorem for multitensors - enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak $n$-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak $n$-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/26/28-26abs.html", "title": "Multitensors as monads on categories of enriched graphs", "authors": "Mark Weber", "keywords": ["multitensors", "enriched graphs", "higher categories", "higher operads"], "abstract": "In this paper we unify the developments of Batanin [1998], Batanin-Weber [2011] and Cheng [2011] into a single framework in which the interplay between multitensors on a category $V$, and monads on the category $\\cal G V$ of graphs enriched in $V$, is taken as fundamental. The material presented here is the conceptual background for subsequent work: in Batanin-Cisinski-Weber [2013] the Gray tensor product of 2-categories and the Crans [1999] tensor product of Gray categories are exhibited as existing within our framework, and in Weber [2013] the explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/24/28-24abs.html", "title": "Complicial structures in the nerves of omega-categories", "authors": "Richard Steiner", "keywords": ["complicial set", "complicial identities", "omega-category"], "abstract": "It is known that strict omega-categories are equivalent through the nerve functor to complicial sets and to sets with complicial identities. It follows that complicial sets are equivalent to sets with complicial identities. We discuss these equivalences. In particular we give a conceptual proof that the nerves of omega-categories are complicial sets, and a direct proof that complicial sets are sets with complicial identities."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/23/28-23abs.html", "title": "The algebra of the nerves of omega-categories", "authors": "Richard Steiner", "keywords": ["complicial identities", "omega-category"], "abstract": "We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also construct an equivalence between the categories of strict omega-categories and of sets with complical identities."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/22/28-22abs.html", "title": "Tight spans, Isbell completions and semi-tropical modules", "authors": "Simon Willerton", "keywords": ["Metric spaces", "tropical algebra", "injective hull"], "abstract": "In this paper we consider generalized metric spaces in the sense of Lawvere and the categorical Isbell completion construction.  We show that this is an analogue of the tight span construction of classical metric spaces, and that the Isbell completion coincides with the directed tight span of Hirai and Koichi.  The notions of categorical completion and cocompletion are related to the existence of semi-tropical module structure, and it is shown that the Isbell completion (hence the directed tight span) has two different semi-tropical module structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html", "title": "Enriched indexed categories", "authors": "Michael Shulman", "keywords": ["monoidal category", "enriched category", "indexed category", "fibered category"], "abstract": "We develop a theory of categories which are simultaneously (1)   indexed over a base category $S$ with finite products, and (2)   enriched over an $S$-indexed monoidal category $V$.  This includes   classical enriched categories, indexed and fibered categories, and   internal categories as special cases.  We then describe the   appropriate notion of ``limit'' for such enriched indexed   categories, and show that they admit ``free cocompletions''   constructed as usual with a Yoneda embedding."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/20/28-20abs.html", "title": "Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions", "authors": "Lili Shen and Dexue Zhang", "keywords": ["Quantaloid", "Q-distributor", "complete Q-category", "Q-closure space", "Isbell adjunction", "Kan adjunction"], "abstract": "Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Q-categories factors through both of these functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/19/28-19abs.html", "title": "Sur les types d'homotopie modélisés par les $\\infty$-groupoides stricts", "authors": "Dimitri Ara", "keywords": ["strict $\\infty$-groupoid", "homotopy type", "chain complex", "homotopy category", "Eilenberg-Mac Lane space"], "abstract": "The purpose of this text is the study of the class of homotopy types which are modelized by strict $\\infty$-groupoids. We show that the homotopy category of simply connected strict $\\infty$-groupoids is equivalent to the derived category in homological degree $d \\ge 2$ of abelian groups. We deduce that the simply connected homotopy types modelized by strict $\\infty$-groupoids are precisely the products of Eilenberg-Mac Lane spaces. We also briefly study 3-categories with weak inverses. We finish by two questions about the problem suggested by the title of this text."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/18/28-18abs.html", "title": "Subgroupoids and quotient theories", "authors": "Henrik Forssell", "keywords": ["Grothendieck toposes", "sheaves on topological groupoids", "categorical logic"], "abstract": "Moerdijk's site description for equivariant sheaf toposes on open topological groupoids is used to give a proof for the (known, but apparently unpublished) proposition that if $H$ is a subgroupoid of an open topological groupoid $G$, then the topos of equivariant sheaves on $H$ is a subtopos of the topos of equivariant sheaves on $G$. This proposition is then applied to the study of quotient geometric theories and subtoposes. In particular, an intrinsic characterization is given of those subgroupoids that are definable by quotient theories."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html", "title": "Connections on non-Abelian gerbes and their holonomy", "authors": "Urs Schreiber and Konrad Waldorf", "keywords": ["Parallel transport", "surface holonomy", "path 2-groupoid", "gerbes", "2-bundles", "2-groups", "non-abelian differential cohomology", "non-abelian bundle gerbes"], "abstract": "We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the role of a band, or structure 2-group. Upon choosing certain examples of Lie 2-groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: non-abelian differential cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These relationships convey a well-defined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to non-abelian gerbes. Several new features of surface holonomy are exposed under its extension to non-abelian gerbes; for example, it carries an action of the mapping class group of the surface."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/16/28-16abs.html", "title": "Bounded Archimedean l-algebras and Gelfand-Neumark-Stone duality", "authors": "Guram Bezhanishvili, Patrick J. Morandi, Bruce Olberding", "keywords": ["Ring of continuous real-valued functions", "l-ring", "l-algebra", "uniform completeness", "Stone-Weierstrass theorem", "commutative $C^*$-algebra", "compact Hausdorff space", "Gelfand-Neumark-Stone duality"], "abstract": "By Gelfand-Neumark duality, the category $C^*Alg$ of commutative $C^*$-algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category $ubal$ of uniformly complete bounded Archimedean $\\ell$-algebras.  Consequently, $C^*Alg$ is equivalent to $ubal$, and this equivalence can be described through complexification.\nIn this article we study $ubal$ within the larger category $bal$ of bounded Archimedean $\\ell$-algebras.  We show that $ubal$ is the smallest nontrivial reflective subcategory of $bal$, and that $ubal$ consists of exactly those objects in $bal$ that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for $bal$.  It follows that $ubal$ is the unique nontrivial reflective epicomplete subcategory of $bal$.  We also show that each nontrivial reflective subcategory of $bal$ is both monoreflective and epireflective, and exhibit two other interesting reflective subcategories of $bal$ involving Gelfand rings and square closed rings.\nDually, we show that Specker ${\\mathbb R}$-algebras are precisely the co-epicomplete objects in $bal$.  We prove that the category $spec$ of Specker $\\mathbb R$-algebras is a mono-coreflective subcategory of $bal$ that is co-epireflective in a mono-coreflective subcategory of $bal$ consisting of what we term $\\ell$-clean rings, a version of clean rings adapted to the order-theoretic setting of $bal$.\nWe conclude the article by discussing the import of our results in the setting of complex $*$-algebras through complexification."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/15/28-15abs.html", "title": "Tensors, monads and actions", "authors": "Gavin J. Seal", "keywords": ["monoidal category", "monad", "Eilenberg--Moore category", "bimorphism", "action"], "abstract": "We exhibit sufficient conditions for a monoidal monad $T$ on a monoidal category $C$ to induce a monoidal structure on the Eilenberg--Moore category $C^T$ that represents bimorphisms. The category of actions in $C^T$ is then shown to be monadic over the base category $C$."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/12/28-12abs.html", "title": "n-tuple groupoids and optimally coupled factorizations", "authors": "Dany Majard", "keywords": ["factorization", "groupoid", "cubical categories"], "abstract": "In this paper, we prove that the category of vacant $n$-tuple groupoids is equivalent to the category of factorizations of groupoids by $n$ factors that satisfy some Yang-Baxter type equation. Moreover we extend this equivalence to the category of maximally exclusive $n$-tuple groupoids, which we define, by dropping one assumption. The paper concludes by a note on how these results could tell us more about some Lie groups of interest."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/11/28-11abs.html", "title": "Homotopy theories of diagrams", "authors": "J.F. Jardine", "keywords": ["model structures", "presheaves of categories", "diagrams"], "abstract": "Suppose that S is a space.  There is an injective and a projective model structure for the resulting category of spaces with S-action, and both are easily derived. These model structures are special cases of model structures for presheaf-valued diagrams $X$ defined on a fixed presheaf of categories E which is enriched in simplicial sets.\nVarying the parameter category object E (or parameter space S) along with the diagrams X up to weak equivalence requires model structures for E-diagrams having weak equivalences defined by homotopy colimits, and a generalization of Thomason's model structure for small categories to a model structure for presheaves of simplicial categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html", "title": "No-iteration pseudomonads", "authors": "F. Marmolejo and R.J. Wood", "keywords": ["Pseudomonad", "algebras"], "abstract": "We present the no-iteration version of the coherence conditions necessary to define a pseudomonad, and a description of the algebras for it in a similar fashion. We show that every no-iteration pseudomonad induces a pseudomonad, and that the corresponding algebras are equivalent. We also show that every pseudomonad induces a no-iteration pseudomonad, and again, that the corresponding algebras are equivalent. We conclude with an analysis of the algebras for the 2-monad $(-)^{\\mathbf{2}}$ on Cat in the light of the no-iteration description of the algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/13/28-13abs.html", "title": "Codensity and the ultrafilter monad", "authors": "Tom Leinster", "keywords": ["density", "codensity", "monad", "ultrafilter", "ultraproduct", "integration", "compact Hausdorff space", "double dual", "linearly compact vector space"], "abstract": "Even a functor without an adjoint induces a monad, namely, its codensity monad; this is subject only to the existence of certain limits.  We clarify the sense in which codensity monads act as substitutes for monads induced by adjunctions.  We also expand on an undeservedly ignored theorem of Kennison and Gildenhuys: that the codensity monad of the inclusion of (finite sets) into (sets) is the ultrafilter monad.  This result is analogous to the correspondence between measures and integrals.  So, for example, we can speak of integration against an ultrafilter.  Using this language, we show that the codensity monad of the inclusion of (finite-dimensional vector spaces) into (vector spaces) is double dualization.  From this it follows that compact Hausdorff spaces have a linear analogue: linearly compact vector spaces. Finally, we show that ultraproducts are categorically inevitable: the codensity monad of the inclusion of (finite families of sets) into (families of sets) is the ultraproduct monad."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/10/28-10abs.html", "title": "Diagonal model structures", "authors": "J. F. Jardine", "keywords": ["bisimplicial presheaves", "diagonal functor", "model structure"], "abstract": "The category of bisimplicial presheaves carries a model structure for which the weak equivalences are defined by the diagonal functor and the cofibrations are monomorphisms. This model structure has the most cofibrations of a large family of model structures with weak equivalences defined by the diagonal. The diagonal structure for bisimplicial presheaves specializes to a diagonal model structure for bisimplicial sets, for which the fibrations are the Kan fibrations."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/9/28-09abs.html", "title": "Geometric morphisms of realizability toposes", "authors": "Peter Johnstone", "keywords": ["realizability topos", "geometric morphism", "applicative morphism"], "abstract": "We show that every geometric morphism between realizability toposes satisfies the condition that its inverse image commutes with the `constant object' functors, which was assumed by John Longley in his pioneering study of such morphisms. We also provide the answer to something which was stated as an open problem on Jaap van Oosten's book on realizability toposes: if a subtopos of a realizability topos is (co)complete, it must be either the topos of sets or the degenerate topos. And we present a new and simpler condition equivalent to the notion of computational density for applicative morphisms of Schonfinkel algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/8/28-08abs.html", "title": "Tightly bounded completions", "authors": "Marta Bunge", "keywords": ["2-categories", "KZ-doctrines", "completions", "enriched category theory", "indexed categories", "distributors", "generalized functors", "Karoubi envelope", "Stack completion", "Cauchy completion", "Grothendieck completion"], "abstract": "By a `completion' on a 2-category K we mean here an idempotent pseudomonad on K. We are particularly interested in pseudomonads that arise from KZ-doctrines. Motivated by a question of Lawvere, we compare the Cauchy completion, defined in the setting of V-Cat for V a symmetric monoidal closed category, with the Grothendieck completion, defined in the setting of S-Indexed Cat for S a topos. To this end we introduce a unified setting (`indexed enriched category theory') in which to formulate and study certain properties of KZ-doctrines. We find that, whereas all of the KZ-doctrines that are relevant to this discussion (Karoubi, Cauchy, Stack, Grothendieck) may be regarded as `bounded', only the Cauchy and the Grothendieck completions are `tightly bounded' - two notions that we introduce and study in this paper. Tightly bounded KZ-doctrines are shown to be idempotent. We also show, in a different approach to answering the motivating question, that the Cauchy completion (defined using `distributors') and the Grothendieck completion (defined using `generalized functors') are actually equivalent constructions."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/7/28-07abs.html", "title": "Traced *-autonomous categories are compact closed", "authors": "Tamás Hajgató and Masahito Hasegawa", "keywords": ["symmetric monoidal closed categories", "traced monoidal categories", "*-autonomous categories", "compact closed categories"], "abstract": "We show that any traced $*$-autonomous category is compact closed."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/5/28-05abs.html", "title": "On the monad of internal groupoids", "authors": "Dominique Bourn", "keywords": ["Fibration of points", "monad of internal groupoids", "Mal'cev and protomodular categories", "split exact sequence", "algebraic exponentiation"], "abstract": "We deeply analyse the structural organisation of the fibration of points and of the monad of internal groupoids. From that we derive: 1) a new characterization of internal groupoids among reflexive graphs in the Mal'cev context; 2) a setting in which a Mal'cev category is necessarily a protomodular category."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/4/28-04abs.html", "title": "Semiunital semimonoidal categories (Applications to semirings and semicorings)", "authors": "Jawad Abuhlail", "keywords": ["Semimonoidal Categories", "Semiunits", "Monads", "Comonads", "Semirings", "Semimodules", "Semicorings", "Semicomodules"], "abstract": "The category $_{A}\\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor-like product $-\\boxtimes _{A}-,$ is semimonoidal but not monoidal. Although not a unit in $_{A}\\mathbb{S}% _{A},$ the base semialgebra $A$ has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories $(\\mathcal{V}% ,\\bullet ,\\mathbf{I})$ as a  framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call $\\mathbb{J}$-monads ($\\mathbb{J}$-comonads) with respect to the endo-functor $\\mathbb{J}:=\\mathbf{I}\\bullet -\\simeq -\\bullet \\mathbf{I}:\\mathcal{V}\\longrightarrow \\mathcal{V}.$ This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endo-functors. Applications to the semiunital semimonoidal variety $(_{A}\\mathbb{S}_{A},\\boxtimes _{A},A) $ provide us with examples of semiunital $A$-semirings (semicounital $A$-semicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules)."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/3/28-03abs.html", "title": "Duality for distributive spaces", "authors": "Dirk Hofmann", "keywords": ["Topological space", "approach space", "ultrafilter monad", "quantale-enriched  category", "module", "cocompleteness", "distributivity", "duality theory"], "abstract": "The main source of inspiration for the present paper is the work of R. Rosebrugh and R.J. Wood on constructively completely distributive lattices where the authors elegantly employ the concepts of adjunction and module. Both notions (suitably adapted) are available in topology too, which permits us to investigate topological, metric and other kinds of spaces in a similar spirit. We introduce here the notion of distributive space and algebraic space and show in particular that the category of distributive spaces and colimit preserving maps is dually equivalent to the idempotent split completion of a category of spaces and convergence relations between them. We explain the connection of this result to the well-known duality between topological spaces and frames, and deduce further duality theorems."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/6/28-06abs.html", "title": "Tannaka duality and convolution for duoidal categories", "authors": "Thomas Booker and Ross Street", "keywords": ["duoidal", "duoid", "bimonoid", "Tannaka duality", "monoidal category", "closed  category", "Hopf monoid"], "abstract": "Given a horizontal monoid $M$ in a duoidal category $\\cal F$, we examine the relationship between bimonoid structures on $M$ and monoidal structures on the category $\\cal F^{\\ast M}$ of right $M$-modules which lift the vertical monoidal structure of $\\cal F$. We obtain our result using a variant of the so-called Tannaka adjunction;  that is, an adjunction inducing the equivalence which expresses Tannaka duality. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts (``actegories''). The requirement of enrichment in $\\cal F$ itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors are monoidal, and so take monoids to monoids, unifies classical convolution in algebra and Day convolution for categories. Hopf bimonoids are defined leading to a lifting of closed structures on $\\cal F$ to $\\cal F^{\\ast M}$. We introduce the concept of warping monoidal structures and this permits the construction of new duoidal categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/2/28-02abs.html", "title": "Free products of higher operad algebras", "authors": "Mark Weber", "keywords": ["operads", "higher categories", "funny tensor product"], "abstract": "One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2-categories. In this paper we continue the developments of [Batanin-Weber, 2011], [Weber, 2011] and [Batanin-Cisinski-Weber, 2011] by understanding the natural generalisations of Gray's little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by a normalised n-operad in the sense of Batanin, an analogous tensor product which forms a symmetric monoidal closed structure on the category of algebras of the operad."},
{"url": "http://www.tac.mta.ca/tac/volumes/28/1/28-01abs.html", "title": "The monoidal structure of strictification", "authors": "Nick Gurski", "keywords": ["Gray tensor product", "strictification"], "abstract": "We study the monoidal structure of the standard strictification functor $st : Bicat \\rightarrow 2Cat$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/32/29-32abs.html", "title": "Iterated icons", "authors": "Eugenia Cheng and Nick Gurski", "keywords": ["symmetric monoidal bicategory", "icon", "enriched category"], "abstract": "We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells.  We show that when the monoidal bicategory in question is symmetric then this process can be iterated.  We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of `2-tuply monoidal categories' missing from our earlier studies of the Periodic Table.  We propose a generalisation to all $k$-tuply monoidal $n$-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/30/29-30abs.html", "title": "More on geometric morphisms between realizability toposes", "authors": "Eric Faber and Jaap van Oosten", "keywords": ["realizability toposes", "partial combinatory algebras", "geometric morphisms", "local operators"], "abstract": "Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose `lifts' to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize pcas whose realizability topos admits a geometric morphism to the effective topos."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/29/29-29abs.html", "title": "Bicategorical fibration structures and stacks", "authors": "Dorette A. Pronk and Michael A. Warren", "keywords": ["stacks", "fibrant objects", "homotopy bicategory", "bicategories of fractions", "algebraic stacks", "differentiable stacks", "topological stacks"], "abstract": "In this paper we introduce two notions - systems of fibrant objects and   fibration structures--- which will allow us to associate to a bicategory  $B$ a   homotopy bicategory $Ho(B)$ in such a way that $Ho(B)$   is the universal way to add pseudo-inverses to weak equivalences in  $B$.   Furthermore, $Ho(B)$ is locally small when $B$ is and  $Ho(B)$   is a 2-category when $B$ is.  We thereby resolve two of the   problems with known approaches to bicategorical localization.\nAs an important example, we describe a fibration structure on   the 2-category of prestacks on a site and prove that the resulting   homotopy bicategory is the 2-category of stacks. We also show how this  example   can be restricted to obtain algebraic, differentiable and topological (respectively)   stacks as homotopy categories of algebraic, differential and topological (respectively) prestacks."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/28/29-28abs.html", "title": "A Serre-Swan theorem for gerbe modules on étale Lie groupoids", "authors": "Christoph Schweigert, Christopher Tropp and Alessandro Valentino", "keywords": ["Gerbe modules", "Lie groupoids", "Serre-Swann theorem"], "abstract": "Given a torsion bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid M, we show that the corresponding category of gerbe modules is equivalent to the category of finitely generated projective modules over an Azumaya algebra on M. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/31/29-31abs.html", "title": "Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory", "authors": "Rory B. B. Lucyshyn-Wright", "keywords": ["completion", "closure", "density", "monad", "idempotent monad", "idempotent core", "idempotent approximation", "normed vector space", "adjunction", "reflective subcategory", "enriched category", "factorization system", "orthogonal subcategory", "sheaf", "sheafification", "Lawvere-Tierney topology", "monoidal category", "closed category"], "abstract": "Given a monad $T$ on a suitable enriched category $B$ equipped with a proper factorization system $(E,M)$, we define notions of $T$-completion, $T$-closure, and $T$-density.  We show that not only the familiar notions of completion, closure, and density in normed vector spaces, but also the notions of sheafification, closure, and density with respect to a Lawvere-Tierney topology, are instances of the given abstract notions.  The process of $T$-completion is equally the enriched idempotent monad associated to $T$ (which we call the idempotent core of $T$), and we show that it exists as soon as every morphism in $B$ factors as a $T$-dense morphism followed by a $T$-closed $M$-embedding.  The latter hypothesis is satisfied as soon as $B$ has certain pullbacks as well as wide intersections of $M$-embeddings.   Hence the resulting theorem on the existence of the idempotent core of an enriched monad entails Fakir's existence result in the non-enriched case, as well as adjoint functor factorization results of Applegate-Tierney and Day."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/27/29-27abs.html", "title": "On pointwise Kan extensions in double categories", "authors": "Seerp Roald Koudenburg", "keywords": ["double category", "equipment", "pointwise Kan extension", "exact cell", "tabulation"], "abstract": "In this paper we consider a notion of pointwise Kan extension in double categories that naturally generalises Dubuc's notion of pointwise Kan extension along enriched functors. We show that, when considered in equipments that admit opcartesian tabulations, it generalises Street's notion of pointwise Kan extension in $2$-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/26/29-26abs.html", "title": "Diagrammatic characterisation of enriched absolute colimits", "authors": "Richard Garner", "keywords": ["Absolute colimits", "enriched categories"], "abstract": "We provide a diagrammatic criterion for the existence of an absolute colimit in the context of enriched category theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/24/29-24abs.html", "title": "Stacks and sheaves of categories as fibrant objects, I", "authors": "Alexandru E. Stanculescu", "keywords": ["fibred category", "model category", "stack"], "abstract": "We show that the category of categories fibred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the fibrant objects are the stacks, as they were defined by J. Giraud. The generalized model category restricts to one on the full subcategory whose objects are the categories fibred in groupoids. We show that the category of sheaves of categories is a model category that is Quillen equivalent to the generalized model category for stacks and to the model category for strong stacks due to A. Joyal and M. Tierney."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/20/29-20abs.html", "title": "Continuous cohesion over sets", "authors": "Matias Menni", "keywords": ["topos", "Axiomatic Cohesion"], "abstract": "A pre-cohesive geometric morphism $p:\\cal E \\rightarrow \\cal S$ satisfies Continuity if the canonical  $p_! (X^{p^* S}) \\rightarrow (p_! X)^S$  is an iso for every $X$ in $\\cal E$ and $S$ in $\\cal S$.  We show that if $\\cal S = Set$ and $\\cal E$ is a presheaf topos then, $p$ satisfies Continuity if and only if it is a quality type.  Our proof of this characterization rests on a related result showing that Continuity and Sufficient Cohesion are incompatible for presheaf toposes. This incompatibility raises the question whether Continuity and Sufficient Cohesion are ever compatible for Grothendieck toposes.  We show that the answer is positive by building some examples."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/23/29-23abs.html", "title": "Constructing model categories with prescribed fibrant objects", "authors": "Alexandru E. Stanculescu", "keywords": ["model category", "enriched category"], "abstract": "We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith's result to left Bousfield localizations of categories of monoids in a suitable monoidal model category."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/22/29-22abs.html", "title": "Realizable homotopy colimits", "authors": "Beatriz Rodriguez Gonzalez", "keywords": ["Homotopy colimit", "Simplicial descent category", "Grothendieck derivator"], "abstract": "We show that the composition of a homotopically meaningful `geometric realization' (or simple functor) with the simplicial replacement produces all homotopy colimits and Kan extensions in a relative category which is closed under coproducts. Examples (and its duals) include model categories, $\\Delta$-closed classes and other concrete examples such as complexes on (AB4) abelian categories, (filtered) commutative dg algebras and mixed Hodge complexes. The resulting homotopy colimits satisfy the expected properties as cofinality and Fubini, and are moreover colimits in a suitable 2-category of relative categories. Conversely, the existence of homotopy colimits satisfying these properties guarantees that $hocolim_{\\Delta^o}$ is a simple functor."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/21/29-21abs.html", "title": "On deformations of pasting diagrams, II", "authors": "Tej Shrestha and D. N. Yetter", "keywords": ["pasting diagrams", "pasting schemes", "deformation theory"], "abstract": "We continue the development of the infinitesimal deformation theory of pasting diagrams of $k$-linear categories begun in TAC, Vol 22, #2.  In that article the standard result that all obstructions are cocycles was established only for the elementary, composition-free parts of pasting diagrams.  In the present work we give a proof for pasting diagrams in general.  As tools we use the method developed by Shrestha  of simultaneously representing formulas for obstructions, along with the corresponding cocycle and cobounding conditions by suitably labeled polygons, giving a rigorous exposition of the previously heuristic method; and deformations of pasting diagrams in which some cells are required to be deformed trivially."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/19/29-19abs.html", "title": "Sequential multicategories", "authors": "Claudio Pisani", "keywords": ["Sequential", "representable", "exponentiable and cartesian multicategories", "preadditive", "additive and finite product categories", "Boardman-Vogt tensor product"], "abstract": "We study the monoidal closed category of symmetric multicategories, especially in relation with its cartesian structure and with sequential multicategories (whose arrows are sequences of concurrent arrows in a given category). Then we consider cartesian multicategories in a similar perspective and develop some peculiar items such as algebraic products. Several classical facts arise as a consequence of this analysis when some of the multicategories involved are representable."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/18/29-18abs.html", "title": "Enriched factorization systems", "authors": "Rory B. B. Lucyshyn-Wright", "keywords": ["factorization systems", "factorisation systems", "enriched categories", "strong monomorphisms", "strong epimorphisms", "monoidal categories", "closed categories"], "abstract": "In a paper of 1974, Brian Day employed a notion of factorization system in the context of enriched category theory, replacing the usual diagonal lifting property with a corresponding criterion phrased in terms of hom-objects.  We set forth the basic theory of such enriched factorization systems.  In particular, we establish stability properties for enriched prefactorization systems, we examine the relation of enriched to ordinary factorization systems, and we provide general results for obtaining enriched factorizations by means of wide (co)intersections.  As a special case, we prove results on the existence of enriched factorization systems involving enriched strong monomorphisms or strong epimorphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/17/29-17abs.html", "title": "Extending obstructions to noncommutative functorial spectra", "authors": "Benno van den Berg and Chris Heunen", "keywords": ["Ring spectra", "Kochen-Specker Theorem"], "abstract": "Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of $n$-by-$n$ matrices for $n \\geq 3$. This obstruction also applies to other spectra such as those named after Zariski, Stone, and Pierce. We extend these no-go results to functors with values in (ringed) topological spaces, (ringed) toposes, schemes, and quantales.  The possibility of spectra in other categories is discussed."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/14/29-14abs.html", "title": "Toward categorical risk measure theory", "authors": "Takanori Adachi", "keywords": ["conditional expectation", "Radon-Nikodym derivative", "monetary value measure", "sheaf", "Grothendieck topology"], "abstract": "We introduce a category that represents varying risk as well as ambiguity. We give a generalized conditional expectation as a presheaf for this category, which not only works as a traditional conditional expectation given a $\\sigma$-field but also is compatible with change of measure. Then, we reformulate dynamic monetary value measures as a presheaf for the category. We show how some axioms of dynamic monetary value measures in the classical setting are deduced as theorems in the new formulation, which is evidence that the axioms are correct. Finally, we point out the possibility of giving a theoretical criteria with which we can pick up appropriate sets of axioms required for monetary value measures to be good, using a topology-as-axioms paradigm."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/13/29-13abs.html", "title": "Projective lines as groupoids with projection structure", "authors": "Anders Kock", "keywords": ["projective geometry", "groupoids", "geometric algebra"], "abstract": "The coordinate projective line over a field is seen as a groupoid with a further `projection' structure. We investigate conversely to what extent such an, abstractly given, groupoid may be coordinatized by a suitable field constructed out of the geometry."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/16/29-16abs.html", "title": "A Bayesian characterization of relative entropy", "authors": "John C. Baez and Tobias Fritz", "keywords": ["relative entropy", "Kullback-Leibler divergence", "measures of information", "categorical probability theory"], "abstract": "We give a new characterization of relative entropy, also known as the Kullback--Leibler divergence. We use a number of interesting categories related to probability theory.  In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function $f \\maps X \\to Y$ together with a stochastic right inverse $s \\maps Y \\to X$.  The function $f$ can be thought of as a measurement process, while $s$ provides a hypothesis about the state of the measured system given the result of a measurement.  Given this data we can define the entropy of the probability distribution on $X$ relative to the `prior' given by pushing the probability distribution on $Y$ forwards along $s$.  We say that $s$ is `optimal' if these distributions agree.  We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid $[0,\\infty]$ which vanishes when $s$ is optimal must be a scalar multiple of this relative entropy.  Our proof is independent of all earlier characterizations, but inspired by the work of Petz."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/15/29-15abs.html", "title": "Topological functors as total categories", "authors": "Richard Garner", "keywords": ["Topological functors", "total categories", "enriched categories", "quantaloids", "MacNeille completion"], "abstract": "A notion of central importance in categorical topology is that of topological functor.  A faithful functor $\\cal E \\to \\cal B$ is called topological if it admits cartesian liftings of all (possibly large)   families of arrows; the basic example is the forgetful functor $Top \\to Set$. A topological functor $\\cal E \\to 1$ is the same thing as a (large) complete preorder, and the general topological functor $\\cal E \\to \\cal B$ is intuitively thought of as a \"complete preorder relative to $\\cal B$\". We make this intuition precise by considering an enrichment base $\\cal Q_\\cal B$ such that $\\cal Q_\\cal B$-enriched categories are faithful functors into $\\cal B$, and show that, in this context, a faithful functor is topological if and only if it is total (=totally cocomplete) in the sense of Street-Walters. We also consider the MacNeille completion of a faithful functor to a topological one, first described by Herrlich, and show that it may be obtained as an instance of Isbell's generalised notion of MacNeille completion for enriched categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/12/29-12abs.html", "title": "On the infinity category of homotopy Leibniz algebras", "authors": "David Khudaverdyan, Norbert Poncin, Jian Qiu", "keywords": ["Homotopy algebra", "categorified algebra", "higher category", "quasi-category", "Kan complex", "Maurer-Cartan equation", "composition of homotopies", "Leibniz algebra"], "abstract": "We discuss various concepts of $\\infty$-homotopies, as well as the relations between them (focussing on the Leibniz type). In particular $\\infty$-$n$-homotopies appear as the $n$-simplices of the nerve of a complete Lie ${\\infty}$-algebra. In the nilpotent case, this nerve is known to be a Kan complex. We argue that there is a quasi-category of $\\infty$-algebras and show that for truncated $\\infty$-algebras, i.e. categorified algebras, this $\\infty$-categorical structure projects to a strict 2-categorical one. The paper contains a shortcut to $(\\infty,1)$-categories, as well as a review of Getzler's proof of the Kan property. We make the latter concrete by applying it to the 2-term $\\infty$-algebra case, thus recovering the concept of homotopy of Baez and Crans, as well as the corresponding composition rule \\cite{SS07}. We also answer a question of Shoikhet about composition of $\\infty$-homotopies of $\\infty$-algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/11/29-11abs.html", "title": "Duality in non-abelian algebra I. From cover relations to Grandis  ex2-categories", "authors": "Zurab Janelidze and Thomas Weighill", "keywords": ["biform", "cover relation", "ex2-category", "factorization system", "faithful  amnestic functor", "form", "Grothendieck fibration", "ideal of null morphisms", "subobject", "universalizer"], "abstract": "The aim of this series of papers is to develop a self-dual categorical approach to some topics in non-abelian algebra, which is based on replacing the framework of a category with that of a category equipped with a functor to it. The present paper gives some preliminary steps in this direction, where several known structures on a category, which arise in the categorical treatment of these topics, are viewed as such functors; as a result, we obtain some new conceptual links between these structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/9/29-09abs.html", "title": "The theory and practice of Reedy categories", "authors": "Emily Riehl and Dominic Verity", "keywords": ["reedy categories", "homotopy limits and colimits", "weighted limits and colimits"], "abstract": "The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but difficult to find in the literature. While the results presented here are not new, our approach to their proofs is somewhat novel. Specifically, we reduce much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushout-product) constructions. The general theory is developed in parallel with examples, which we use to prove that familiar formulae for homotopy limits and colimits indeed have the desired properties."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/25/29-25abs.html", "title": "The weakly globular double category of fractions of a category", "authors": "Simona Paoli, Dorette Pronk", "keywords": ["double categories", "strict 2-categories", "bicategories", "pseudo-functors", "companions", "conjoints", "localizations", "bicategories of fractions"], "abstract": "This paper introduces the construction of a weakly globular double  category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/8/29-08abs.html", "title": "Twisted actions of categorical groups", "authors": "Saikat Chatterjee,  Amitabha Lahiri, and  Ambar N. Sengupta", "keywords": ["Representations", "Double Categories", "Categorical Groups", "2-Groups", "Semidirect Products"], "abstract": "We develop a theory of twisted actions of categorical groups using a notion of semidirect product of categories. We work through numerous examples to demonstrate the power of these notions. Turning to representations, which are actions that respect vector space structures, we establish an analog of Schur's lemma in this context.  Keeping new terminology to a minimum, we concentrate on examples exploring the essential new notions introduced."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/6/29-06abs.html", "title": "Analytic spectrum of rig categories", "authors": "Frederic Paugam", "keywords": ["Rig categories", "global analytic geometry", "generalized rings", "Arakelov compactifications"], "abstract": "We define the analytic spectrum of a rig category $(A,\\oplus,\\otimes)$, and equip it with a sheaf of categories of rational functions. If the category is additive, we define a sheaf of categories of analytic functions. We relate this construction to Berkovich's analytic spaces, to Durov's generalized schemes and to Haran's F-schemes. We use these relations to define analytic versions of Arakelov compactifications of affine arithmetic varieties."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/5/29-05abs.html", "title": "Mapping Spaces of Gray-Categories", "authors": "Björn Gohla", "keywords": ["Higher gauge theory", "Gray-categories"], "abstract": "We define a mapping space for Gray-enriched categories   adapted to higher gauge theory. Our construction differs   significantly from the canonical mapping space of enriched   categories in that it is much less rigid. The two essential   ingredients are a path space construction for Gray-categories and   a kind of comonadic resolution of the 1-dimensional structure of a   given Gray-category obtained by lifting the resolution of   ordinary categories along the canonical fibration of GrayCat   over Cat."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/7/29-07abs.html", "title": "A Galois theory for monoids", "authors": "Andrea Montoli, Diana Rodelo and Tim Van der Linden", "keywords": ["categorical Galois theory", "homogeneous split epimorphism", "special homogeneous surjection", "central extension", "group completion", "Grothendieck group"], "abstract": "We show that the adjunction between monoids and groups obtained via the Grothendieck group construction is admissible, relatively to surjective homomorphisms, in the sense of categorical Galois theory. The central extensions with respect to this Galois structure turn out to be the so-called special homogeneous surjections."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/3/29-03abs.html", "title": "On the image of the almost strict Morse n-category under almost strict  n-functors", "authors": "Sonja Hohloch", "keywords": ["n-category", "Morse theory", "functors", "moduli spaces"], "abstract": "In an earlier work, we constructed the almost strict Morse n-category $\\mathcal X$ which extends Cohen and Jones and Segal's flow category. In this article, we define two other almost strict n-categories $\\mathcal V$ and $\\mathcal W$ where $\\mathcal V$ is based on homomorphisms between real vector spaces and $\\mathcal W$ consists of tuples of positive integers. The Morse index and the dimension of the Morse moduli spaces give rise to almost strict n-category functors $\\mathcal F : \\mathcal X \\to \\mathcal V$ and $\\mathcal G : \\mathcal X \\to \\mathcal W$."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/4/29-04abs.html", "title": "Obvious natural morphisms of sheaves are unique", "authors": "Ryan Cohen Reich", "keywords": ["commutative diagrams", "coherence theorem", "string diagrams", "pullback", "pushforward"], "abstract": "We prove that a large class of natural transformations (consisting   roughly of those constructed via composition from the ``functorial''   or ``base change'' transformations) between two functors of the form   $... f^* g_* ...$ actually has only one element, and thus that   any diagram of such maps necessarily commutes.  We identify the   precise axioms defining what we call a ``geofibered category'' that   ensure that such a coherence theorem exists.  Our results apply to   all the usual sheaf-theoretic contexts of algebraic geometry.  The   analogous result that would include any other of the six functors   remains unknown."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/2/29-02abs.html", "title": "Erratum to `Towards a homotopy theory of higher dimensional transition   systems'", "authors": "Philippe Gaucher", "keywords": ["higher dimensional transition system", "locally presentable   category", "topological category", "combinatorial model category", "left determined model category", "Bousfield localization", "bisimulation"], "abstract": "Counterexamples for Proposition~8.1 and   Proposition~8.2 in the article Theor. Appl. Categ. 25(2011), pp 295-341 are given. They are used in the paper only to prove   Corollary~8.3. A proof of this corollary is given without them. The   proof of the fibrancy of some cubical transition systems is fixed."},
{"url": "http://www.tac.mta.ca/tac/volumes/29/1/29-01abs.html", "title": "Some stability properties of epimorphism classes", "authors": "Dali Zangurashvili", "keywords": ["(effective) descent morphism", "balanced morphism", "factorization system", "stability under pullback/pushout"], "abstract": "It is proved that in any pointed category with pullbacks, coequalizers and regular epi-mono factorizations, the class of regular epimorphisms is stable under pullback along the so-called balanced effective descent morphisms. Here ``balanced'' can be omitted if the category is additive. A balanced effective descent morphism is defined as an effective descent morphism $p:E\\rightarrow B$ such that any subobject of $E$ is a pullback of some morphism along $p$. It is shown that, in any category with pullbacks and coequalizers, the class of effective descent morphisms is stable under pushout if and only if any regular epimorphism is an effective descent morphism. Moreover, it is shown that the class of descent morphisms is stable under pushout if and only if the class of regular epimorphisms is stable under pullback."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/56/30-56abs.html", "title": "On the 3-representations of groups and the 2-categorical traces", "authors": "Wei Wang", "keywords": ["the 3-representation of a group in a 3-category", "the 2-categorical trace", "the 3-cocycle condition", "the induced strict 2-categorical actions", "the 3-character", "2-categorification"], "abstract": "To 2-categorify the theory of group representations, we introduce the notions of the 3-representation of a group in a strict 3-category and the strict 2-categorical action of a group on a strict 2-category. We also 2-categorify the concept of the trace by introducing the 2-categorical trace of a 1-endomorphism in a strict 3-category. For a 3-representation $\\rho$ of a group G and an element f of G, the 2-categorical trace $Tr_2\\rho_f$ is a category. Moreover, the centralizer of f in G acts categorically on this 2-categorical trace. We construct the induced strict 2-categorical action of a finite group, and show that the 2-categorical trace $Tr_2$ takes an induced strict 2-categorical action into an induced categorical action of the initia groupoid. As a corollary, we get the 3-character formula of the induced strict 2-categorical action."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/53/30-53abs.html", "title": "Profinite topological spaces", "authors": "G. Bezhanishvili, D. Gabelaia, M. Jibladze, P. J. Morandi", "keywords": ["Profinite space", "spectral space", "stably compact space", "bitopological space", "ordered topological space", "Priestley space"], "abstract": "It is well known that profinite $T_0$-spaces are exactly the spectral spaces. We generalize this result to the category of all topological spaces by showing that the following conditions are equivalent:  (1) $(X,\\tau)$ is a profinite topological space. (2) The $T_0$-reflection of $(X,\\tau)$ is a profinite $T_0$-space. (3) $(X,\\tau)$ is a quasi spectral space.  (4) $(X,\\tau)$ admits a stronger Stone topology $\\pi$ such that $(X,  \\tau,\\pi)$ is a bitopological quasi spectral space"},
{"url": "http://www.tac.mta.ca/tac/volumes/30/52/30-52abs.html", "title": "Mutation pairs and triangulated quotients", "authors": "Zengqiang Lin and Minxiong Wang", "keywords": ["Quotient category", "mutation pair", "pseudo-triangulated category", "triangulated category"], "abstract": "We introduce the notion of mutation pairs in pseudo-triangulated categories. Given such a mutation pair, we prove that the corresponding quotient category carries a natural triangulated structure under certain conditions. This result unifies many previous constructions of quotient triangulated categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/55/30-55abs.html", "title": "Groupoids in categories with pretopology", "authors": "Ralf Meyer and Chenchang Zhu", "keywords": ["Grothendieck topology", "cover", "groupoid", "groupoid action", "groupoid sheaf", "principal bundle", "Hilsum--Skandalis morphism", "anafunctor", "bicategory", "comorphism", "infinite dimensional groupoid"], "abstract": "We survey the general theory of groupoids, groupoid actions,   groupoid principal bundles, and various kinds of morphisms between   groupoids in the framework of categories with   pretopology. The categories of topological spaces and finite or   infinite dimensional manifolds are examples of such categories.  We   study extra assumptions on pretopologies that are needed for this   theory.  We check these extra assumptions in several categories with   pretopologies.\nFunctors between groupoids may be localised at equivalences in two   ways.  One uses spans of functors, the other bibundles (commuting   actions) of groupoids.  We show that both approaches give equivalent   bicategories.  Another type of groupoid morphism, called an actor,   is closely related to functors between the categories of groupoid   actions.  We also generalise actors using bibundles, and show that   this gives another bicategory of groupoids."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/54/30-54abs.html", "title": "Algebraically coherent categories", "authors": "Alan S. Cigoli, James R. A. Gray and Tim Van der Linden", "keywords": ["Coherent functor", "Smith", "Huq", "Higgins commutator", "semi-abelian", "locally  algebraically cartesian closed category", "category of interest"], "abstract": "We call a finitely complete category algebraically coherent if the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give examples of categories satisfying this condition; for instance, coherent categories and categories of interest in the sense of Orzech. We study equivalent conditions in the context of semi-abelian categories, as well as some of its consequences: including amongst others, strong protomodularity, and normality of Higgins commutators for normal subobjects, and in the varietal case, fibre-wise algebraic cartesian closedness."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/51/30-51abs.html", "title": "Algebras of open dynamical systems on the operad of wiring diagrams", "authors": "Dmitry Vagner, David I. Spivak, and Eugene Lerman", "keywords": ["Operads", "Monoidal Categories", "Wiring Diagrams", "Dynamical Systems"], "abstract": "In this paper, we use the language of operads to study open dynamical systems. More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams. We define the symmetric monoidal category $W$, from which we may construct an operad $OW$, whose objects are black boxes with input and output ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic rules for interconnection. We then define two $W$-algebras} $G$ and $L$, which associate semantic content to the structures in $W$. Respectively, they correspond to general and to linear systems of differential equations, in which an internal state is controlled by inputs and produces outputs. As an example, we use these algebras to formalize the classical problem of systems of tanks interconnected by pipes, and hence make explicit the algebraic relationships among systems at different levels of granularity."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/50/30-50abs.html", "title": "Internal algebra classifiers as codescent objects of crossed internal categories", "authors": "Mark Weber", "keywords": ["internal algebras", "codescent objects", "crossed internal categories"], "abstract": "Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects arising from our monad-theoretic framework, as certain 2-categorical colimits called codescent objects. We isolate the extra structure present on these simplicial objects which enable their codescent objects to be computed. These are the crossed internal categories of the title, and generalise the crossed simplicial groups of Loday and Fiedorowicz. The most general results of this article are concerned with how to compute such codescent objects in 2-categories of internal categories, and on isolating conditions on the monad-theoretic situation which enable these results to apply. Combined with earlier work of the author in which operads are seen as polynomial 2-monads, our results are then applied to the theory of non-symmetric, symmetric and braided operads. In particular, the well-known construction of a PROP from an operad is recovered, as an illustration of our techniques."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/48/30-48abs.html", "title": "Limits of abstract elementary classes", "authors": "M. Lieberman, J. Rosicky", "keywords": ["accessible category", "abstract elementary class", "PIE limit"], "abstract": "We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of ``limit type,\" which generalizes the approach of Mariano, Zambrano and Villaveces away from the syntactically oriented framework of institutions.  Moreover, we provide a broader view of this closure phenomenon, considering a variety of categories of accessible categories with additional structure, and relaxing the assumption that the morphisms be concrete functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/49/30-49abs.html", "title": "Operads as polynomial 2-monads", "authors": "Mark Weber", "keywords": ["operads", "polynomial functors"], "abstract": "In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. Moreover it enables us to characterise operads as categorical polynomial monads in a canonical way. This point of view reveals categorical polynomial monads as a unifying environment for operads, Cat-operads and clubs. We recover the standard construction of a monad from an operad in a 2-categorical way from our associated 2-monad as a coidentifier of 2-monads, and understand the algebras of both as weak morphisms of operads into a Cat-operad of categories. Algebras of operads within general symmetric monoidal categories arise from our new associated 2-monad in a canonical way. When the operad is sigma-free, we establish a Quillen equivalence, with respect to the model structures on algebras of 2-monads found by Lack, between the strict algebras of our associated 2-monad, and those of the standard one."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/47/30-47abs.html", "title": "The waves of a total category", "authors": "R.J. Wood", "keywords": ["adjunction", "totally cocomplete", "totally distributive", "taxon", "i-module", "proarrow equipment"], "abstract": "For any total category $K$, with defining adjunction $\\sup \\ladj Y : K \\rightarrow set^{K^{op}}$, the expression $W(a)(k)= set^{set^{K^{op}}}(K(a,\\sup -),[k,-])$, where $[k,-]$ is evaluation at $k$, provides a well-defined functor $W : K \\rightarrow \\hat{K} = set^{K^{op}}$. Also, there are natural transformations $\\beta : W\\sup \\rightarrow 1_{\\hat{K}}$ and $\\gamma : \\sup W \\rightarrow 1_K$ satisfying $\\sup\\beta =\\gamma\\sup$ and $\\beta W =W\\gamma$. A total $K$ is totally distributive if $\\sup$ has a left adjoint. We show that $K$ is totally distributive iff $\\gamma$ is invertible iff $W \\ladj \\sup$. The elements of $W(a)(k)$ are called waves from $k$ to $a$.\nWrite $\\tilde{K}(k,a)$ for $W(a)(k)$. For any total $K$ there is an associative composition of waves. Composition becomes an arrow $\\bullet : \\tilde{K}\\circ_{K}\\tilde{K} \\rightarrow \\tilde{K}$. Also, there is an augmentation $\\tilde{K}(-,-) \\rightarrow K(-,-)$ corresponding to a natural $\\delta : W \\rightarrow Y$ constructed via $\\beta$. We show that if $K$ is totally distributive then $\\bullet$ is invertible and then $\\tilde{K}$ supports an idempotent comonad structure. In fact, $\\tilde{K} \\circ_{K} \\tilde{K} = \\tilde{K} \\circ_{\\tilde{K}} \\tilde{K}$ so that $\\bullet$ is the coequalizer of $\\bullet K$ and $K \\bullet$, making $\\tilde{K}$ a taxon in the sense of Koslowski. For a small taxon $T$, the category of interpolative modules $iMod(1,T)$ is totally distributive. Here we show, for any totally distributive $K$, that there is an equivalence $K \\rightarrow iMod(1,\\tilde{K})$."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/46/30-46abs.html", "title": "Homotopy unital $A_\\infty$-morphisms with several entries", "authors": "Volodymyr Lyubashenko", "keywords": ["$A_\\infty$-algebra", "$A_\\infty$-morphism", "multicategory", "multifunctor", "operad", "operad module", "polymodule cooperad"], "abstract": "We show that morphisms from n homotopy unital $A_\\infty$-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad $A_\\infty^\\hu$, whose algebras are homotopy unital $A_\\infty$-algebras. The operad $A_\\infty$ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of $A_\\infty^\\hu$\\n-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/45/30-45abs.html", "title": "$A_\\infty$-morphisms with several entries", "authors": "Volodymyr Lyubashenko", "keywords": ["$A_\\infty$-algebra", "$A_\\infty$-morphism", "multicategory", "multifunctor", "operad", "operad module", "polymodule cooperad"], "abstract": "We show that morphisms from n $A_\\infty$-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad $A_\\infty$, whose algebras are conventional $A_\\infty$-algebras. The composition of $A_\\infty$-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures. Under these notions lie two examples of Cat-operads: that of graded modules and of complexes."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/44/30-44abs.html", "title": "Proximity biframes and compactifications of completely regular ordered spaces", "authors": "Guram Bezhanishvili and Patrick J. Morandi", "keywords": ["Frame", "biframe", "bispace", "ordered space", "compactification", "proximity"], "abstract": "We generalize the concept of a strong inclusion on a biframe to that of a proximity on a biframe, which is related to the concept of a strong bi-inclusion on a frame introduced by Picado and Pultr. We also generalize the concept of a bi-compactification of a biframe to that of a compactification of a biframe, and prove that the poset of compactifications of a biframe L is isomorphic to the poset of proximities on L. As a corollary, we obtain Schauerte's characterization of bi-compactifications of a biframe. In the spatial case this yields Blatter and Seever's characterization of compactifications of completely regular ordered spaces and a characterization of bi-compactifications of completely regular bispaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/43/30-43abs.html", "title": "Transformation double categories associated to 2-group actions", "authors": "Jeffrey C. Morton and Roger Picken", "keywords": ["2-group", "categorical group", "crossed module", "action", "double category", "adjoint action"], "abstract": "Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in set-theoretic terms. This paper examines the analogous situation for structures described in category-theoretic terms, where symmetry is expressed as the action of a 2-group G (equivalently, a categorical group) on a category C. It describes the construction of a transformation groupoid in diagrammatic terms, and considers this construction internal to Cat, the category of categories. The result is a double category C//G which describes the local symmetries of C. We define this and describe some of its structure, with the adjoint action of G on itself as a guiding example."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/42/30-42abs.html", "title": "Gauge invariant surface holonomy and monopoles", "authors": "Arthur J. Parzygnat", "keywords": ["Surface holonomy", "gauge theory", "2-groups", "crossed modules", "higher-dimensional algebra", "monopoles", "gauge-invariance", "non-abelian 2-bundles", "iterated integrals"], "abstract": "There are few known computable examples of non-abelian surface holonomy. In this paper, we give several examples whose structure 2-groups are covering 2-groups and show that the surface holonomies can be computed via a simple formula in terms of paths of 1-dimensional holonomies inspired by earlier work of Chan Hong-Mo and Tsou Sheung Tsun on magnetic monopoles. As a consequence of our work and that of Schreiber and Waldorf, this formula gives a rigorous meaning to non-abelian magnetic flux for magnetic monopoles. In the process, we discuss gauge covariance of surface holonomies for spheres for any 2-group, therefore generalizing the notion of the reduced group introduced by Schreiber and Waldorf. Using these ideas, we also prove that magnetic monopoles form an abelian group."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/38/30-38abs.html", "title": "Intercategories", "authors": "Marco Grandis and Robert Paré", "keywords": ["interchange law", "intercategory", "triple category", "2-category", "double category", "lax and colax functor", "pseudocategory"], "abstract": "We introduce a 3-dimensional categorical structure which we call intercategory. This is a kind of weak triple category with three kinds of arrows, three kinds of 2-dimensional cells and one kind of 3-dimensional cells. In one dimension, the compositions are strictly associative and unitary, whereas in the other two, these laws only hold up to coherent isomorphism. The main feature is that the interchange law between the second and third compositions does not hold, but rather there is a non-invertible comparison cell which satisfies some coherence conditions. We introduce appropriate morphisms of intercategory, of which there are three types, and cells relating these. We show that these fit together to produce a strict triple category of intercategories."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/39/30-39abs.html", "title": "Stacks of", "authors": "Ettore Aldrovandi", "keywords": ["Categorical ring", "ann category", "ring-like stack", "crossed bimodule", "butterfly", "Shukla", "Barr", "André-Quillen cohomology"], "abstract": "We show that ann-categories admit a presentation by crossed bimodules, and prove that morphisms between them can be expressed by special kinds spans between the presentations. More precisely, we prove the groupoid of morphisms between two ann-categories is equivalent to that of bimodule butterflies between the presentations.  A bimodule butterfly is a specialization of a butterfly, i.e. a special kind of span or fraction, between the underlying complexes"},
{"url": "http://www.tac.mta.ca/tac/volumes/30/35/30-35abs.html", "title": "Generalized covering space theories", "authors": "Jeremy Brazas", "keywords": ["Fundamental group", "generalized covering map", "coreflective hull", "unique  path lifting property"], "abstract": "In this paper, we unify various approaches to generalized covering space theory by introducing a categorical framework in which coverings are defined purely in terms of unique lifting properties. For each category C of path-connected spaces having the unit disk as an object, we construct a category of C-coverings over a given space X that embeds in the category of $\\pi_1(X,x_0)$-sets via the usual monodromy action on fibers. When C is extended to its coreflective hull H(C), the resulting category of based H(C)-coverings is complete, has an initial object, and often characterizes more of the subgroup lattice of $\\pi_1(X,x_0)$ than traditional covering spaces.  We apply our results to three special coreflective subcategories: (1) The category of $\\Delta$-coverings employs the convenient category of $\\Delta$-generated spaces and is universal in the sense that it contains every other generalized covering category as a subcategory. (2) In the locally path-connected category, we preserve notion of generalized covering due to Fischer and Zastrow and characterize the topology of such coverings using the standard whisker topology. (3) By employing the coreflective hull Fan of the category of all contractible spaces, we characterize the notion of continuous lifting of paths and identify the topology of Fan-coverings as the natural quotient topology inherited from the path space."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/37/30-37abs.html", "title": "A C-system defined by a universe category", "authors": "Vladimir Voevodsky", "keywords": ["contextual category", "universe category", "C-system"], "abstract": "This is the third paper in a series. In it we construct a C-system CC(C,p) starting from a category C together with a morphism  $p:\\tilde{U} \\to U$, a choice of pull-back squares based on $p$ for all morphisms to $U$ and a choice of a final object of C.  Such a quadruple is called a universe category. We then define universe category functors and construct homomorphisms of C-systems CC(C,p) defined by universe category functors.  In the sections before the last section we give, for any C-system CC, three different constructions of pairs ((C,p),H) where (C,p) is a universe category and $H : CC \\to CC(C,p)$ is an isomorphism.    In the last section we construct for any (set) category C with a choice of a final object and fiber products a C-system and an equivalence between C and the precategory underlying CC."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/36/30-36abs.html", "title": "Building a Model Category out of cofibrations and fibrations", "authors": "Seunghun Lee", "keywords": ["Model Category", "Weak Factorization Systems", "Two out of Three Property"], "abstract": "The purpose of this note is to understand the two out of three property of the model category in terms of the weak factorization systems. We will show that if a category with classes of trivial cofibrations, cofibrations, trivial fibrations, and fibrations is given a simplicial structure similar to that of the simplicial model category, then the full subcategory of cofibrant and fibrant objects has the two out of three property, and we will give a list of necessary and sufficient conditions in terms of the simplicial structure for the associated canonical \"weak equivalence class\" to have the two out of three property."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/34/30-34abs.html", "title": "A note on transport of algebraic structures", "authors": "Henrik Holm", "keywords": ["Algebraic structure", "algebraic theory", "completion of metric space", "equational class", "Stone-Cech compactification", "universal covering space", "universal locally connected refinement"], "abstract": "We study transport of algebraic structures and prove a theorem which subsumes results of Comfort and Ross on topological group structures on Stone-Cech compactifications, of Chevalley and of Gil de Lamadrid and Jans on topological group and ring structures on universal covering spaces, and of Gleason on topological group structures on universal locally connected refinements."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/33/30-33abs.html", "title": "Decorated cospans", "authors": "Brendan Fong", "keywords": ["cospan", "decorated cospan", "hypergraph category", "well-supported compact closed category", "separable algebra", "Frobenius algebra", "Frobenius monoid"], "abstract": "Let $C$ be a category with finite colimits, writing its coproduct +, and let $(D, \\otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal functor $F : (C,+) \\to (D, \\otimes)$, and of producing a strong monoidal functor between such categories from a monoidal natural transformation between such functors. The objects of these categories, our so-called `decorated cospan categories', are simply the objects of $C$, while the morphisms are pairs comprising a cospan $X \\rightarrow N \\leftarrow Y$ in $C$ together with an element $1 \\to FN$ in $D$. Moreover, decorated cospan categories are hypergraph categories - each object is equipped with a special commutative Frobenius monoid - and their functors preserve this structure."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/32/30-32abs.html", "title": "Cyclic homology arising from adjunctions", "authors": "Niels Kowalzig, Ulrich Krähmer, and Paul Slevin", "keywords": ["cyclic homology", "Hopf algebroids", "Hopf monads", "distributive laws"], "abstract": "Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel'd distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Böhm and Stefan.  In fact, their approach generates two duplicial objects and morphisms between them which are mutual inverses if and only if the duplicial objects are cyclic. A 2-categorical perspective on the process of twisting coefficients is provided and the role of the two notions of bimonad studied in the literature is clarified."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/31/30-31abs.html", "title": "Reduced smooth stacks?", "authors": "Giorgio Trentinaglia", "keywords": ["Lie groupoids", "effective orbifolds", "categories of fractions"], "abstract": "An arbitrary Lie groupoid gives rise to a groupoid of germs of local diffeomorphisms over its base manifold, known as its effect. The effect of any bundle of Lie groups is trivial. All quotients of a given Lie groupoid determine the same effect. It is natural to regard the effects of any two Morita equivalent Lie groupoids as being ``equivalent''. In this paper we shall describe a systematic way of comparing the effects of different Lie groupoids. In particular, we shall rigorously define what it means for two arbitrary Lie groupoids to give rise to ``equivalent'' effects. For effective orbifold groupoids, the new notion of equivalence turns out to coincide with the traditional notion of Morita equivalence. Our analysis is relevant to the presentation theory of proper smooth stacks."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/28/30-28abs.html", "title": "Skew-monoidal reflection and lifting theorems", "authors": "Stephen Lack and Ross Street", "keywords": ["skew monoidal category", "reflective subcategory", "warping", "comonad"], "abstract": "This paper extends the Day Reflection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore coalgebras for a comonad."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/30/30-30abs.html", "title": "Finite categories with pushouts", "authors": "D. Tambara", "keywords": ["pushout", "coequalizer", "hom-functor", "familially representable functor", "nearly  representable functor"], "abstract": "Let C be a finite category.  For an object X of C one has the hom-functor Hom(-,X) of C to Set. If G is a subgroup of Aut(X), one has the quotient functor Hom(-,X)/G. We show that any finite product of hom-functors of C is a sum of hom-functors if and only if C has pushouts and coequalizers and that any finite product of hom-functors of C is a sum of functors of the form \\Hom(-,X)/G if and only if C has pushouts. These are variations of the fact that a finite category has products if and only if it has coproducts."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/29/30-29abs.html", "title": "Deligne groupoid revisited", "authors": "Paul Bressler, Alexander Gorokhovsky, Ryszard Nest and Boris Tsygan", "keywords": ["groupoid", "$L_\\infty$-algebra", "simplicial nerve"], "abstract": "We show that for a differential graded Lie algebra g whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of g-valued differential forms introduced by V.~Hinich."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/27/30-27abs.html", "title": "On the second cohomology categorical group and a Hochschild-Serre 2-exact sequence", "authors": "A.R. Garzon and E.M. Vitale", "keywords": ["(symmetric) categorical group", "2-exactness", "cohomology categorical group", "extension"], "abstract": "We introduce the second cohomology categorical group of a categorical group G with coefficients in a symmetric G-categorical group and we show that it classifies extensions of G with symmetric kernel and a functorial section. Moreover, from an essentially surjective homomorphism of categorical groups we get 2-exact sequences a la Hochschild-Serre connecting the categorical groups of derivations and the first and the second cohomology categorical groups."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/23/30-23abs.html", "title": "Reflexivity and dualizability in categorified linear algebra", "authors": "Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd", "keywords": ["locally presentable", "dualizable", "cocomplete", "cocontinuous"], "abstract": "The \"linear dual\" of a cocomplete linear category $C$ is the category of all cocontinuous linear functors $C \\to Vect$.  We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing).   Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens.  Along the way, we prove that the category $QCoh(X)$ of quasi-coherent sheaves on a stack $X$ is not dualizable if $X$ is the classifying stack of a semisimple algebraic group in positive characteristic or if $X$ is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if $X$ is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/26/30-26abs.html", "title": "Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness", "authors": "F. W. Lawvere and M. Menni", "keywords": ["Axiomatic cohesion", "Topos theory"], "abstract": "We introduce an apparent strengthening of Sufficient Cohesion that we call  Stable Connected Codiscreteness (SCC) and show that if $p: E --> S$ is cohesive and satisfies SCC then the internal axiom of choice holds in $S$. Moreover, in this case, $p^!: S --> E$ is equivalent to the inclusion $E_{\\neg\\neg} --> E$."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/25/30-25abs.html", "title": "Actions in modified categories of interest with application to crossed modules", "authors": "Y. Boyaci, J. M. Casas, T. Datuashvili and E. O. Uslu", "keywords": ["split extension classifier", "category of interest", "associative algebra", "crossed module", "cat^1-associative algebra", "equivalence of categories", "actor", "universal strict general actor", "bimultiplier"], "abstract": "The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories $XAss \\simeq Cat^1-Ass$ we consider this problem in $Cat^1-Ass$. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of category, which will be called modified category of interest. Applying the results obtained in this direction and the equivalence of categories we find a condition under which there exists the split extension classifier of a crossed module and give the corresponding construction."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/24/30-24abs.html", "title": "Categories in control", "authors": "John C. Baez and Jason Erbele", "keywords": ["control theory", "graphical calculus", "Frobenius algebra", "bialgebra", "dagger-compact category", "signal-flow diagram"], "abstract": "Control theory uses `signal-flow diagrams' to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted.  These diagrams can be seen as string diagrams for the symmetric monoidal category FinVectk of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces.  For any field k we give a presentation of FinVectk in terms of the generators used in signal-flow diagrams.  A broader class of signal-flow diagrams also includes `caps' and `cups' to model feedback.  We show these diagrams can be seen as string diagrams for the symmetric monoidal category FinRelk, where objects are still finite-dimensional vector spaces but the morphisms are linear relations.  We also give a presentation for FinRelk.  The relations say, among other things, that the 1-dimensional vector space k has two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid.  This sort of structure, but with tensor product replacing direct sum, is familiar from the `ZX-calculus' obeyed by a finite-dimensional Hilbert space with two mutually unbiased bases."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/22/30-22abs.html", "title": "An algebraic definition of ($\\infty$,n)-categories", "authors": "Camell Kachour", "keywords": ["($\\infty$,n)-categories", "weak $\\infty$-groupoids", "homotopy types"], "abstract": "In this paper we define a sequence of monads   $T^{(\\infty,n)} (n\\in\\mathbb{N})$ on the category $\\infty-Gr$ of   $\\infty$-graphs. We conjecture that algebras for $\\T^{(\\infty,0)}$, which are   defined in a purely algebraic setting, are models of   $\\infty$-groupoids. More generally, we conjecture that  $T^{(\\infty,n)}$-algebras   are models for $(\\infty,n)$-categories. We prove that our  $(\\infty,0)$-categories are   bigroupoids when truncated at level 2."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/41/30-41abs.html", "title": "On reflective subcategories of locally presentable categories", "authors": "J. Adamek and J. Rosicky", "keywords": ["locally presentable category", "reflective subcategory", "elementary equivalence"], "abstract": "Are all subcategories of locally finitely presentable categories that are closed under limits and $\\lambda$-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the case $\\lambda = \\aleph_0$ the answer is affirmative also for all iso-full subcategories, i. e., those containing with every pair of objects all isomorphisms between them. We discuss a possible generalization of this from $\\aleph_0$ to an arbitrary $\\lambda$."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/40/30-40abs.html", "title": "Segal group actions", "authors": "Matan Prasma", "keywords": ["Model category", "Segal space", "group action", "equivariant homotopy theory"], "abstract": "We define a model category structure on a slice category of simplicial spaces, called the \"Segal group action\" structure, whose fibrant-cofibrant objects may be viewed as representing spaces $X$ with an action of a fixed Segal group (i.e. a group-like, reduced Segal space). We show that this model structure is Quillen equivalent to the projective model structure on $G$-spaces, $S^BG}$, where $G$ is a simplicial group corresponding to the Segal group.  One advantage of this model is that if we start with an ordinary group action $X\\in S^BG$ and apply a weakly monoidal functor of spaces $L: S \\to S$ (such as localization or completion) on each simplicial degree of its associated Segal group action, we get a new Segal group action of $LG$ on $LX$ which can then be rigidified via the above-mentioned Quillen equivalence."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/21/30-21abs.html", "title": "Categories enriched over a quantaloid: Algebras", "authors": "Qiang Pu and Dexue Zhang", "keywords": ["Quantaloid", "Q-category", "complete Q-category", "completely distributive Q-category", "Q-powerset", "Eilenberg-Moore algebra", "monadicity"], "abstract": "Given a small quantaloid Q with a set of objects Q_0, it is proved that complete skeletal Q-categories, completely distributive skeletal Q-categories, and Q-powersets of Q-typed sets are all monadic over athe slice category of Set over Q_0."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/20/30-20abs.html", "title": "A model structure on internal categories in simplicial sets", "authors": "Geoffroy Horel", "keywords": ["internal categories", "complete Segal spaces", "infinity categories"], "abstract": "We put a model structure on the category of categories internal to simplicial sets. The weak equivalences in this model structure are preserved and reflected by the nerve functor to bisimplicial sets with the complete Segal space model structure.  This model structure is shown to be a model for the homotopy theory of infinity categories. We also study the homotopy theory of internal presheaves over an internal category."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/19/30-19abs.html", "title": "The accessibility rank of weak equivalences", "authors": "G. Raptis and J. Rosický", "keywords": ["model category", "weak equivalence", "accessible category"], "abstract": "We study the accessibility properties of trivial cofibrations and weak equivalences in a combinatorial model category and prove an estimate for the accessibility rank of weak equivalences. In particular, we show that the class of weak equivalences between simplicial sets is finitely accessible."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/17/30-17abs.html", "title": "Extensive categories and the size of an orbit", "authors": "Ernie Manes", "keywords": ["extensive category", "Burnside-Frobenius lemma", "conjugacy class of subgroups"], "abstract": "It is well known how to compute the number of orbits of a group action. A related problem, apparently not in the literature, is to determine the number of elements in an orbit.  The theory that addresses this question leads to orbital extensive categories and to combinatorial aspects of such categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/18/30-18abs.html", "title": "Cartesian differential storage categories", "authors": "R. Blute, J.R.B. Cockett, and R.A.G. Seely", "keywords": ["monoidal categories", "differential categories", "Kleisli categories", "differential  operators"], "abstract": "Monoidal differential categories provide the framework for categorical models of differential linear logic. The coKleisli category of any monoidal differential category is always a Cartesian differential category.  Cartesian differential categories, besides arising in this manner as coKleisli categories, occur in many different and quite independent ways.  Thus, it was not obvious how to pass from Cartesian differential categories back to monoidal differential categories.\nThis paper provides natural conditions under which the linear maps of a Cartesian differential category form a monoidal differential category. This is a question of some practical importance as much of the machinery of modern differential geometry is based on models which implicitly allow such a passage, and thus the results and tools of the area tend to freely assume access to this structure.\nThe purpose of this paper is to make precise the connection between the two types of differential categories.  As a prelude to this, however, it is convenient to have available a general theory which relates the behaviour of \"linear\" maps in Cartesian categories to the structure of Seely categories.   The latter were developed to provide the categorical semantics for (fragments of) linear logic which use a \"storage\" modality.  The general theory of storage, which underlies the results mentioned above, is developed in the opening sections of the paper and is then applied to the case of differential categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/16/30-16abs.html", "title": "Polynomials in categories with pullbacks", "authors": "Mark Weber", "keywords": ["polynomial functors", "2-monads"], "abstract": "The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks.  The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street's theory of fibrations within 2-categories is explored. Johnstone's notion of \"bagdomain data\" is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html", "title": "Notes on commutation of limits and colimits", "authors": "Marie Bjerrum, Peter Johnstone, Tom Leinster, William F. Sawin", "keywords": ["Limit", "colimit", "filtered colimit", "group action", "simple group", "Galois connection"], "abstract": "We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all (small) colimits. On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is contained in that of pseudo-filtered colimits."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/14/30-14abs.html", "title": "Slicing sites and semireplete factorization systems", "authors": "Thorsten Palm", "keywords": ["factorization system", "slice categories", "cartesian morphisms", "slicing site"], "abstract": "A factorization system (E, M) on a category A gives rise to the covariant category-valued pseudofunctor P of A sending each object to its slice category over M. This article characterizes the P so obtained as follows: their object images have terminal objects, and they admit bicategorically cartesian liftings, up to equivalence, of slice-category projections. It is clear that, and how, (E, M) can be recovered from such a P. The correspondence thus described is actually the second of three similar ones between certain (E, M) and certain P that the article presents. In the first one the characterization of the P has all ultimately bicategorical ingredients replaced with their categorical analogues. A category A with such a P is precisely what the author has called a `slicing site'. In the article's terms the associated (E, M) are again factorization systems - but the concept conveyed extends the standard one by not obliging isomorphisms to belong to either factor class -, namely those that are `right semireplete' (isomorphisms do belong to M and `left semistrict' (morphisms in M are monic relative to E). The third correspondence subsumes the other two; here the (E, M) are all right-semireplete factorization systems."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/13/30-13abs.html", "title": "Operadic definitions of weak n-category: coherence and comparisons", "authors": "Thomas Cottrell", "keywords": ["n-category", "operad", "higher-dimensional category"], "abstract": "This paper concerns the relationships between notions of weak n-category defined as algebras for n-globular operads, as well as their coherence properties.  We focus primarily on the definitions due to Batanin and Leinster.\nA correspondence between the contractions and systems of compositions used in Batanin's definition, and the unbiased contractions used in Leinster's definition, has long been suspected, and we prove a conjecture of Leinster that shows that the two notions are in some sense equivalent. We then prove several coherence theorems which apply to algebras for any operad with a contraction and system of compositions or with an unbiased contraction; these coherence theorems thus apply to weak $n$-categories in the senses of Batanin, Leinster, Penon and Trimble.\nWe then take some steps towards a comparison between Batanin weak n-categories and Leinster weak n-categories.  We describe a canonical adjunction between the categories of these, giving a construction of the left adjoint, which is applicable in more generality to a class of functors induced by monad morphisms.  We conclude with some preliminary statements about a possible weak equivalence of some sort between these categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/12/30-12abs.html", "title": "Normalizers, centralizers and action accessibility", "authors": "J. R. A. Gray", "keywords": ["action accessible", "protomodular", "Barr exact", "normality", "centrality", "normalizer", "centralizer"], "abstract": "We give several reformulations of action accessibility in the sense of D. Bourn and G. Janelidze. In particular we prove that a pointed exact protomodular category is action accessible if and only if for each normal monomorphism $\\kappa:X\\to A$ the normalizer of $< \\kappa,\\kappa>: X\\to A\\times A$ exists. This clarifies the connection between normalizers and action accessible categories established in a joint paper of D. Bourn and the author, in which it is proved that for pointed exact protomodular categories the existence of normalizers implies action accessibility. In addition we prove a pointed exact protomodular category with coequalizers is action accessible if centralizers of normal monomorphisms exist, and the normality of unions holds."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/11/30-11abs.html", "title": "A cocategorical obstruction to tensor products of Gray-categories", "authors": "John Bourke and Nick Gurski", "keywords": ["Gray-category", "monoidal biclosed category", "cocategory"], "abstract": "It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure.  We make this precise by establishing undesirable properties that any such monoidal biclosed structure must have.  In particular we show that there does not exist any tensor product making the model category of Gray-categories into a monoidal model category."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/10/30-10abs.html", "title": "Lax formal theory of monads, monoidal approach to bicategorical  structures and generalized operads", "authors": "Dimitri Chikhladze", "keywords": ["Bicategories", "equipments", "formal theory of monads", "generalized multicategories", "lax categorification", "tricategories}"], "abstract": "Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads.  Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to bicategorical structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/8/30-08abs.html", "title": "Limit closures of classes of commutative rings", "authors": "Michael Barr, John F. Kennison, R. Raphael", "keywords": ["limit closure", "reflection", "domain"], "abstract": "We study and, in a number of cases, classify completely the limit closures in the category of commutative rings of subcategories of integral domains."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/9/30-09abs.html", "title": "Partial-sup lattices", "authors": "Toby Kenney", "keywords": ["Partial Sup Lattice", "Complete Distributivity"], "abstract": "The study of sup lattices teaches us the important distinction between the algebraic part of the structure (in this case suprema) and the coincidental part of the structure (in this case infima). While a sup lattice happens to have all infima, only the suprema are part of the algebraic structure.\nExtending this idea, we look at posets that happen to have all suprema (and therefore all infima), but we will only declare some of them to be part of the algebraic structure (which we will call joins). We find that a lot of the theory of complete distributivity for sup lattices can be extended to this context. There are a lot of natural examples of completely join-distributive partial lattice complete partial orders, including for example, the lattice of all equivalence relations on a set X, and the lattice of all subgroups of a group G. In both cases we define the join operation as union. This is a partial operation, because for example, the union of subgroups of a group is not necessarily a subgroup. However, sometimes it is, and keeping track of this can help with topics such as the inclusion-exclusion principle.\nAnother motivation for the study of sup lattices is as a simplified model for the study of presheaf categories. The construction of downsets is a form of the Yoneda embedding, and the study of downset lattices can be a useful guide for the study of presheaf categories. In this context, partial lattices can be viewed as a simplified model for the study of sheaf categories.\n"},
{"url": "http://www.tac.mta.ca/tac/volumes/30/6/30-06abs.html", "title": "Bicategorical homotopy pullbacks", "authors": "A.M. Cegarra, B.A. Heredia, J.Remedios", "keywords": [], "abstract": "The homotopy theory of higher categorical structures has become a relevant part of the machinery of algebraic topology and algebraic K-theory, and this paper contains contributions to the study of the relationship between Benabou's bicategories and the homotopy types of their classifying spaces. Mainly, we state and prove an extension of Quillen's Theorem B by showing, under reasonable necessary conditions, a bicategory-theoretical interpretation of the homotopy-fibre product of the continuous maps induced on classifying spaces by a diagram of bicategories $A\\to B \\leftarrow A'$. Applications are given for the study of homotopy pullbacks of monoidal categories and of crossed modules."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/7/30-07abs.html", "title": "Characteristic subobjects in semi-abelian categories", "authors": "Alan S. Cigoli and Andrea Montoli", "keywords": ["characteristic subobject", "semi-abelian categories", "commutators", "centralisers"], "abstract": "We extend to semi-abelian categories the notion of characteristic subobject, which is widely used in group theory and in the theory of Lie algebras. Moreover, we show that many of the classical properties of characteristic subgroups of a group hold in the general semi-abelian context, or in stronger ones."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/5/30-05abs.html", "title": "Algebraic Kan extensions in double categories", "authors": "Seerp Roald Koudenburg", "keywords": ["double monad", "algebraic Kan extension", "free bicommutative Hopf monoid"], "abstract": "We study Kan extensions in three weakenings of the Eilenberg-Moore double category associated to a double monad, that was introduced by Grandis and Paré. To be precise, given a normal oplax double monad T on a double category K, we consider the double categories consisting of pseudo T-algebras, `weak' vertical T-morphisms, horizontal T-morphisms and T-cells, where `weak' means either `lax', `colax' or `pseudo'.  Denoting these double categories by Alg_w(T), where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (pointwise)  Kan extensions can be lifted along the forgetful double functor Alg_w(T) --> K. As an application we recover and generalise a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors. As an application of Getzler's result we prove, in suitable symmetric monoidal categories, the existence of bicommutative Hopf monoids that are freely generated by cocommutative comonoids."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/4/30-04abs.html", "title": "Un théorème A de Quillen pour les 2-foncteurs lax", "authors": "Jonathan Chiche", "keywords": ["homotopy theory", "2-categories", "Quillen's Theorem A"], "abstract": "We generalize Quillen's Theorem A to triangles of lax 2-functors which commute up to transformation. It follows from a special case of this result that 2-categories are models for homotopy types."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/3/30-03abs.html", "title": "Weak braided monoidal categories and their homotopy colimits", "authors": "Mirjam Solberg", "keywords": ["Weak braided monoidal categories", "homotopy colimits", "double loop spaces"], "abstract": "We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories.  This provides a more flexible way to realize $E_2$ spaces categorically."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/2/30-02abs.html", "title": "Model categories with simple homotopy categories", "authors": "Jean-Marie Droz and Inna Zakharevich", "keywords": ["model category", "graph"], "abstract": "In the present article we describe constructions of model structures on general bicomplete categories. We are motivated by the following question: given a category C with a suitable subcategory wC, when is there a model structure on C with wC as the subcategory of weak equivalences?  We begin exploring this question in the case where wC = F^{-1}(iso D) for some functor F : C --> D.  We also prove properness of our constructions under minor assumptions and examine an application to the category of infinite graphs."},
{"url": "http://www.tac.mta.ca/tac/volumes/30/1/30-01abs.html", "title": "On strong homotopy for quasi-schemoids", "authors": "Katsuhiko Kuribayashi", "keywords": ["Association scheme", "small category", "schemoids", "homotopy"], "abstract": "A quasi-schemoid is a small category with a particular partition of the set of morphisms.  We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. The homotopy set of self-homotopy equivalences on a quasi-schemoid is used as a homotopy invariant in the study.  The main theorem enables us to deduce that the homotopy invariant for the quasi-schemoid induced by a finite group is isomorphic to the automorphism group of the given group.  %These considerations are the first step to develop homotopy theory for quasi-schemoids."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/38/31-38abs.html", "title": "Cosheafification", "authors": "Andrei V. Prasolov", "keywords": ["Cosheaves", "smooth precosheaves", "cosheafification", "pro-category", "cosheaf homology", "locally presentable categories", "accessible categories"], "abstract": "It is proved that for any small Grothendieck site X, there exists a coreflection (called \\emph{cosheafification}) from the category of precosheaves on X with values in a category $K$, to the full subcategory of cosheaves, provided either $K$ or $K^{op}$ is locally presentable. If $K$ is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category $Pro(K)$ of pro-objects in $K$. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in $Pro(K)$ is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/37/31-37abs.html", "title": "A functorial approach to Dedekind completions and the representation of vector lattices and", "authors": "G. Bezhanishvili, P. J. Morandi, B. Olberding", "keywords": ["Vector lattice", "$\\ell$-algebra", "uniform completion", "Dedekind completion", "compact Hausdorff space", "extremally disconnected space", "continuous real-valued function", "normal real-valued function", "proximity", "representation"], "abstract": "Unlike the uniform completion, the Dedekind completion of a vector lattice is not functorial. In order to repair the lack of functoriality of Dedekind completions, we enrich the signature of vector lattices with a proximity relation, thus arriving at the category pdv of proximity Dedekind vector lattices. We prove that the Dedekind completion induces a functor from the category bav of bounded archimedean vector lattices to pdv, which in fact is an equivalence. We utilize the results of Dilworth to show that every proximity Dedekind vector lattice D is represented as the normal real-valued functions on the compact Hausdorff space associated with D. This yields a contravariant adjunction between pdv and the category KHaus of compact Hausdorff spaces, which restricts to a dual equivalence between KHaus and the proper subcategory of pdv consisting of those proximity Dedekind vector lattices in which the proximity is uniformly closed. We show how to derive the classic Yosida Representation, Kakutani-Krein Duality, Stone-Gelfand-Naimark Duality, and Stone-Nakano Theorem from our approach."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/36/31-36abs.html", "title": "Products of families of types and $(\\Pi,\\lambda)$-structures on C-systems", "authors": "Vladimir Voevodsky", "keywords": ["Type theory", "contextual category", "dependent product"], "abstract": "In this paper we continue, following the pioneering works by J. Cartmell and T. Streicher, the study of the most important structures on C-systems, the structures that correspond, in the case of the syntactic C-systems, to the $(\\Pi,\\lambda,app,\\beta,\\eta)$-system of inference rules.\nOne such structure was introduced by J. Cartmell and later studied by T. Streicher under the name of the products of families of types.\nWe introduce the notion of a $(\\Pi,\\lambda)$-structure and construct a bijection, for a given C-system, between the set of $(\\Pi,\\lambda)$-structures and the set of Cartmell-Streicher structures. In the following paper we will show how to construct, and in some cases fully classify, the $(\\Pi,\\lambda)$-structures on the C-systems that correspond to universe categories.\nThe first section of the paper provides careful proofs of many of the properties of general C-systems.  Methods of the paper are fully constructive, that is, neither the axiom of excluded middle nor the axiom of choice are used."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html", "title": "Monads on dagger categories", "authors": "Chris Heunen and Martti Karvonen", "keywords": ["Dagger category", "Frobenius monad", "Kleisli algebra", "Eilenberg-Moore algebra"], "abstract": "The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law.  Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleisli and Frobenius-Eilenberg-Moore algebras, which again have a dagger.  We characterize the Frobenius law as a coherence property between dagger and closure, and characterize strong such monads as being induced by Frobenius monoids."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/34/31-34abs.html", "title": "On Evrard's homotopy fibrant replacement of a functor", "authors": "Boris Shoikhet", "keywords": ["Homotopy fibrant replacement of a functor", "Quillen theorem B", "Evrard's  theorem"], "abstract": "We provide a more economical refined version of Evrard's categorical cocylinder factorization of a functor. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred functor which satisfies the (dual) assumption of Quillen's Theorem B."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/33/31-33abs.html", "title": "Compositories and gleaves", "authors": "Cecilia Flori and Tobias Fritz", "keywords": ["sheaf theory", "uniqueness of gluing", "nerve of a category", "higher span", "conditional product distribution", "Lawvere metric space", "relational database"], "abstract": "Sheaves are objects of a local nature: a global section is determined by how it looks locally. Hence, a sheaf cannot describe mathematical structures which contain global or nonlocal geometric information. To fill this gap, we introduce the theory of ``gleaves'', which are presheaves equipped with an additional ``gluing operation'' of compatible pairs of local sections. This generalizes the conditional product structures of Dawid and Studeny, which correspond to gleaves on distributive lattices. Our examples include the gleaf of metric spaces and the gleaf of joint probability distributions. A result of Johnstone shows that a category of gleaves can have a subobject classifier despite not being cartesian closed.\nGleaves over the simplex category $\\Delta$, which we call compositories, can be interpreted as a new kind of higher category in which the composition of an m-morphism and an n-morphism along a common k-morphism face results in an (m+n-k)-morphism. The distinctive feature of this composition operation is that the original morphisms can be recovered from the composite morphism as initial and final faces. Examples of compositories include nerves of categories and compositories of higher spans."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/32/31-32abs.html", "title": "On the representations of 2-groups in Baez-Crans 2-vector spaces", "authors": "Benjamin A. Heredia and Josep Elgueta", "keywords": ["2-groups (categorical groups)", "2-vector spaces", "Representations", "2-categories"], "abstract": "We study the theory of representations of a 2-group G in Baez-Crans 2-vector spaces over a field k of arbitrary characteristic, and the corresponding 2-vector spaces of intertwiners. We also characterize the irreducible and indecomposable representations. Finally, it is shown that when the 2-group is finite and the base field k is of characteristic zero or coprime to the orders of the homotopy groups of G, the theory essentially reduces to the theory of k-linear representations of the first homotopy group of G, the remaining homotopy invariants of G playing no role."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/31/31-31abs.html", "title": "Tannaka theory over sup-lattices and descent for topoi", "authors": "Eduardo J. Dubuc and Martin Szyld", "keywords": ["Tannaka", "Galois", "Sup-lattice", "Locale", "Topos"], "abstract": "We consider locales B as algebras in the tensor category sl of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections q : shB  --> E  in Galois theory and a Tannakian recognition theorem over sl for the sl-functor Rel (q^*) : Rel(E) -->  Rel(shB) \\cong (B-Mod)_0  into the sl-category of discrete B-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of  the localic groupoid G   associated by Joyal-Tierney to q, and do an exhaustive comparison with the Deligne Tannakian construction of the Hopf algebroid L associated to Rel(q^*), and show they are isomorphic, that is, L \\cong O(G).  On the other hand, we show that the sl-category of relations of the classifying topos of any localic groupoid G, is equivalent to the sl-category of L-comodules with discrete subjacent B-module, where L = O(G).}\nWe are forced to work over an arbitrary base topos because, contrary to the neutral case which can be developed completely over Sets, here change of base techniques are unavoidable."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/30/31-30abs.html", "title": "Projectivity, continuity and adjointness: quantales, Q-posets and  Q-modules", "authors": "Susan Niefield", "keywords": ["projective", "flat", "completely distributive", "totally continuous", "quantale module"], "abstract": "Projectivity, continuity and adjointness: quantales, Q-posets and  Q-modules\n\nIn this paper, projective modules over a quantale are characterized by distributivity, continuity, and adjointness conditions. It is then shown that a morphism Q --> A of commutative quantales is coexponentiable if and only if the corresponding Q-module is projective, and hence, satisfies these equivalent conditions."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/29/31-29abs.html", "title": "Enriched Yoneda lemma", "authors": "Vladimir Hinich", "keywords": ["enriched categories", "Yoneda embedding", "left-tensored categories"], "abstract": "We present a version of the enriched Yoneda lemma for conventional (not $\\infty$-) categories. We do not require the base monoidal category M to be closed or symmetric monoidal. In the case M has colimits and the monoidal structure in M preserves colimits in each argument, we prove that the Yoneda embedding A to P_M(A) is a universal functor from A to a category with colimits, left-tensored over M."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/28/31-28abs.html", "title": "The Fermat functors", "authors": "Enxin Wu", "keywords": ["Fermat reals", "the adding infinitesimal functor", "the deleting infinitesimal functor"], "abstract": "In this paper, we use some basic quasi-topos theory to study two functors: one adding infinitesimals of Fermat reals to diffeological spaces (which generalize smooth manifolds including singular spaces and infinite-dimensional spaces), and the other deleting infinitesimals on Fermat spaces. We study the properties of these functors, and calculate some examples. These serve as fundamentals for developing differential geometry on diffeological spaces using infinitesimals in a future paper."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/27/31-27abs.html", "title": "Strict $\\omega$-categories are monadic over polygraphs", "authors": "Francois Metayer", "keywords": ["$\\omega$-categories", "polygraphs", "monads"], "abstract": "We give a direct proof that the category of strict $\\omega$-categories is monadic over the category of polygraphs."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/26/31-26abs.html", "title": "Compact closed bicategories", "authors": "Michael Stay", "keywords": ["compact", "closed", "bicategory", "span 18D05", "18D15"], "abstract": "A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual ``zig-zag'' identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law.  We give several examples of compact closed bicategories, then review previous work.  In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly.\nWe prove that given a 2-category T with finite products and weak pullbacks, the bicategory of objects of C, spans, and isomorphism classes of maps of spans is compact closed.  As corollaries, the bicategory of spans of sets and certain bicategories of ``resistor networks'' are compact closed."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/25/31-25abs.html", "title": "Homotopy locally presentable enriched categories", "authors": "Stephen Lack and Jiri Rosicky", "keywords": ["monoidal model category", "enriched model category", "weighted homotopy colimit", "locally presentable category"], "abstract": "We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where all objects of V are cofibrant."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/24/31-24abs.html", "title": "The local Joyal model structure", "authors": "Nicholas J. Meadows", "keywords": ["simplicial presheaves", "quasi-categories", "model structure duality", "trace", "derivator", "absolute colimit"], "abstract": "The Joyal model structure on simplicial sets  is extended to a model structure on the  simplicial presheaves on a small site, in which the cofibrations are monomorphisms and the weak equivalences are local (or stalkwise) Joyal equivalences.  The model structure is shown to be left proper."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/23/31-23abs.html", "title": "The linearity of traces in monoidal categories and bicategories", "authors": "Kate Ponto and Michael Shulman", "keywords": ["duality", "trace", "derivator", "absolute colimit"], "abstract": "We show that in any symmetric monoidal category, if a weight for   colimits is absolute, then the resulting colimit of any diagram of   dualizable objects is again dualizable.  Moreover, in this case, if   an endomorphism of the colimit is induced by an endomorphism of the   diagram, then its trace can be calculated as a linear combination of   traces on the objects in the diagram.  The formal nature of this   result makes it easy to generalize to traces in homotopical contexts   (using derivators) and traces in bicategories.  These   generalizations include the familiar additivity of the Euler   characteristic and Lefschetz number along cofiber sequences, as well   as an analogous result for the Reidemeister trace, but also the   orbit-counting theorem for sets with a group action, and a general   formula for homotopy colimits over EI-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/22/31-22abs.html", "title": "Classical and relative realizability", "authors": "Jaap van Oosten and Tingxiang Zou", "keywords": ["realizability toposes", "partial combinatory algebras", "geometric morphisms", "local operators", "abstract Krivine structures", "non-localic Boolean toposes"], "abstract": "We show that every abstract Krivine structure in the sense of Streicher can be obtained, up to equivalence of the resulting tripos, from a filtered opca (A,A') and a subobject of 1 in the relative realizability topos RT(A',A); the topos is always a Boolean subtopos of RT(A',A). We exhibit a range of non-localic Boolean subtriposes of the Kleene-Vesley tripos."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/21/31-21abs.html", "title": "Representation and character theory of finite categorical groups", "authors": "Nora Ganter and Robert Usher", "keywords": ["categorical groups", "representation theory", "inertia groupoid", "drinfeld double"], "abstract": "We study the gerbal representations of a finite group G or, equivalently, module categories over Ostrik's category $Vec_G^\\alpha$ for a 3-cocycle $\\alpha$. We adapt Bartlett's string diagram formalism to this situation to prove that the categorical character of a gerbal representation is a representation of the inertia groupoid of a categorical group.  We interpret such a representation as a module over the twisted Drinfeld double $D^\\alpha(G)$."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/20/31-20abs.html", "title": "Linear structures on locales", "authors": "Pedro Resende and Joao Paulo Santos", "keywords": ["Quotient vector bundles", "locales", "Banach bundles", "lower Vietoris topology", "Fell topology"], "abstract": "We define a notion of morphism for quotient vector bundles that yields both a category $QVBun$ and a contravariant global sections functor $C:QVBun^{op} \\to Vect$ whose restriction to trivial vector bundles with fiber F coincides with the contravariant functor $Top^{op} \\to Vect$ of F-valued continuous functions. Based on this we obtain a linear extension of the adjunction between the categories of topological spaces and locales: (i) a linearized topological space is a spectral vector bundle, by which is meant a mildly restricted type of quotient vector bundle; (ii) a linearized locale is a locale $\\Delta$ equipped with both a topological vector space A and a $\\Delta$-valued support map for the elements of A satisfying a continuity condition relative to the spectrum of $\\Delta$ and the lower Vietoris topology on $Sub A$;  (iii) we obtain an adjunction between the full subcategory of spectral vector bundles $QVBun_\\Sigma$ and the category of linearized locales $LinLoc$, which restricts to an equivalence of categories between sober spectral vector bundles and spatial linearized locales. The spectral vector bundles are classified by a finer topology on $Sub A$, called the open support topology, but there is no notion of universal spectral vector bundle for an arbitrary topological vector space A."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/19/31-19abs.html", "title": "The snail lemma", "authors": "Enrico M. Vitale", "keywords": ["snail lemma", "snake lemma", "protomodular category", "strong homotopy kernel"], "abstract": "The classical snake lemma produces a six terms exact sequence starting from a commutative square with one of the edge being a regular epimorphism. We establish a new diagram lemma, that we call snail lemma, removing such a condition. We also show that the snail lemma subsumes the snake lemma and we give an interpretation of the snail lemma in terms of strong homotopy kernels. Our results hold in any pointed regular protomodular category."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/18/31-18abs.html", "title": "The morphism axiom for n-angulated categories", "authors": "Emilie Arentz-Hansen, Petter Andreas Bergh and Marius Thaule", "keywords": ["Triangulated categories", "n-angulated categories", "morphism axiom"], "abstract": "We show that the morphism axiom for n-angulated categories is   redundant."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/17/31-17abs.html", "title": "Right delocalization of model categories", "authors": "Bruce R Corrigan-Salter", "keywords": ["localization", "delocalization", "model categories", "diagram"], "abstract": "Model categories have long been a useful tool in homotopy theory, allowing many generalizations of results in topological spaces to other categories. Giving a localization of a model category provides an additional model category structure on the same base category, which alters what objects are being considered equivalent by increasing the class of weak equivalences.  In some situations, a model category where the class of weak equivalences is restricted from the original one could be more desirable.  In this situation we need the notion of a delocalization.  In this paper, right Bousfield delocalization is defined, we provide examples of right Bousfield delocalization as well as an existence theorem.  In particular, we show that given two model category structures $\\MO$ and $\\MT$ we can define an additional model category structure $\\MO \\cap \\MT$ by defining the class of weak equivalences to be the intersection of the $\\MO$ and $\\MT$ weak equivalences.  In addition we consider the model category on diagram categories over a base category (which is endowed with a model category structure) and show that delocalization is often preserved by the diagram model category structure."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/16/31-16abs.html", "title": "Relative internal actions", "authors": "James Richard Andrew Gray and Tamar Janelidze-Gray", "keywords": ["relative semi-abelian category", "relative homological category", "semi-abelian category", "homological category", "category of relative points", "relative internal actions"], "abstract": "For a relative exact homological category (C,E), we define relative points over an arbitrary object in C, and show that they form an exact homological category. In particular, it follows that the full subcategory of nilpotent objects in an exact homological category is an exact homological category. These nilpotent objects are defined with respect to a Birkhoff subcategory in C as defined by T. Everaert and T. Van der Linden. In addition, we introduce relative internal actions and show that, just as in the classical case, there is an equivalence of categories between the category of relative points over an object and the category of relative internal actions for the same object."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/15/31-15abs.html", "title": "Partial linearity and partial natural Mal'tsevness", "authors": "Dominique Bourn", "keywords": ["Fibration of points", "Mal'tsev", "protomodular", "naturally Mal'tsev and additive categories", "internal groupoids.}"], "abstract": "We introduced in a previous article a notion of Mal'tsevness relative to a specific class $\\Sigma$ of split epimorphisms. We investigate here the induced relative notion of natural Mal'tsevness, with a special attention to the example of quandles."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/14/31-14abs.html", "title": "Construction of categorical bundles from local data", "authors": "Saikat Chatterjee,  Amitabha Lahiri, and  Ambar N. Sengupta", "keywords": ["Categorical Groups", "2-Groups", "Categorical geometry", "Principal  bundles"], "abstract": "A categorical principal bundle is a structure comprised of categories that is analogous to a classical principal bundle; examples arise from geometric contexts involving bundles over path spaces. We show how a categorical principal bundle can be constructed from local data specified through transition functors and natural transformations."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/13/31-13abs.html", "title": "A cotriple construction of a simplicial algebra used in the definition of higher Chow groups", "authors": "Jason Polak", "keywords": ["cotriple", "simplicial algebra"], "abstract": "We present a brief and simple cotriple description of the simplicial algebra used in Bloch's construction of the higher Chow groups."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/12/31-12abs.html", "title": "Spectra of compact regular frames", "authors": "Guram Bezhanishvili, David Gabelaia, Mamuka Jibladze", "keywords": ["Frame", "the spectrum of a frame", "compact regular frame", "compact Hausdorff space", "Gleason cover", "zero-dimensional frame", "extremally disconnected frame", "scattered frame"], "abstract": "By Isbell duality, each compact regular frame L is isomorphic to the frame of opens of a compact Hausdorff space X. In this note we study the spectrum Spec(L) of prime filters of a compact regular frame L. We prove that X is realized as the minimum of Spec(L) and the Gleason cover of X as the maximum of Spec(L). We also characterize zero-dimensional, extremally disconnected, and scattered compact regular frames by means of Spec(L)."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/11/31-11abs.html", "title": "Stacks and sheaves of categories as fibrant objects, II", "authors": "Alexandru E. Stanculescu", "keywords": ["Grothendieck topology", "fibred category", "stack", "model category"], "abstract": "We revisit what we call the fibred topology on a fibred category over a site and we prove a few basic results concerning this topology. We give a general result concerning the invariance of a 2-category of stacks under change of base."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/10/31-10abs.html", "title": "Some insights on bicategories of fractions: representations and compositions of 2-morphisms", "authors": "Matteo Tommasini", "keywords": ["bicategories of fractions", "bicalculus of fractions", "pseudofunctors"], "abstract": "In this paper we investigate the construction of bicategories of fractions originally described by D.~Pronk: given any bicategory $C$ together with a suitable class of morphisms $W$, one can construct a bicategory $C[W^{-1}]$, where all the morphisms of $W$ are turned into internal equivalences, and that is universal with respect to this property. Most of the descriptions leading to this construction were long and heavily based on the axiom of choice. In this paper we considerably simplify the description of the equivalence relation on 2-morphisms and the constructions of associators, vertical and horizontal compositions in $C[W^{-1}]$, thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and compositions will also play a crucial role in two forthcoming papers about pseudofunctors and equivalences between bicategories of fractions."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/9/31-09abs.html", "title": "On biadjoint triangles", "authors": "Fernando Lucatelli Nunes", "keywords": ["adjoint triangles", "descent objects", "Kan extensions", "pseudomonads", "biadjunctions"], "abstract": "We prove a biadjoint triangle theorem and its strict version, which are 2-dimensional analogues of the adjoint triangle theorem of Dubuc. Similarly to the 1-dimensional case, we demonstrate how we can apply our results to get the pseudomonadicity characterization (due to Le Creurer, Marmolejo and Vitale).\nFurthermore, we study applications of our main theorems in the context of the 2-monadic approach to coherence. As a direct consequence of our strict biadjoint triangle theorem, we give the construction (due to Lack) of the left 2-adjoint to the inclusion of the strict algebras into the pseudoalgebras.\nIn the last section, we give two brief applications on lifting biadjunctions and pseudo-Kan extensions."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/8/31-08abs.html", "title": "A characterization of central extensions in the variety of quandles", "authors": "Valérian Even, Marino Gran and Andrea Montoli", "keywords": ["Quandle", "symmetric quandle", "abelian object", "Mal'tsev variety", "central extension", "categorical Galois theory"], "abstract": "The category of symmetric quandles is a Mal'tsev variety whose subvariety of abelian symmetric quandles is the category of abelian algebras. We give an algebraic description of the quandle extensions that are central for the adjunction between the variety of quandles and its subvariety of abelian symmetric quandles."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/7/31-07abs.html", "title": "Transfinite limits in topos theory", "authors": "Moritz Kerz", "keywords": ["topos theory", "pro-etale topology"], "abstract": "For a coherent site we construct a canonically associated enlarged coherent site, such that cohomology of bounded below complexes is preserved by the enlargement. In the topos associated to the enlarged site transfinite compositions of epimorphisms are epimorphisms and a weak analog of the concept of the algebraic closure exists.  The construction is a variant of the work of Bhatt and Scholze on the pro-etale topology."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/3/31-03abs.html", "title": "On the magnitude of a finite dimensional algebra", "authors": "Joseph Chuang, Alastair King and Tom Leinster", "keywords": ["algebra", "magnitude", "indecomposable projective", "simple module", "Cartan matrix", "Euler form", "Cartan determinant conjecture"], "abstract": "There is a general notion of the magnitude of an enriched category, defined subject to hypotheses.  In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic and dimension.  Here we establish its significance in an algebraic context.  Specifically, in the representation theory of an associative algebra $A$, a central role is played by the indecomposable projective $A$-modules, which form a category enriched in vector spaces. We show that the magnitude of that category is a known homological invariant of the algebra: writing $\\chi_A$ for the Euler form  of $A$ and $S$ for the direct sum of the simple $A$-modules, it is $\\chi_A(S,S)$."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/6/31-06abs.html", "title": "Relative symmetric monoidal closed categories I: Autoenrichment and change of base", "authors": "Rory B. B. Lucyshyn-Wright", "keywords": ["monoidal category", "closed category", "enriched category", "enriched monoidal category", "monoidal functor", "monoidal adjunction", "2-category", "2-functor", "2-fibration", "pseudomonoid"], "abstract": "Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena.  In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category $V$ its associated $V$-enriched category $underline{V}$ extends to a 2-functor valued in an op-2-fibred 2-category of symmetric monoidal closed categories enriched over various bases.  For a fixed $V$, we show that this induces a 2-functorial passage from symmetric monoidal closed categories over $V$ (i.e., equipped with a morphism to $V$) to symmetric monoidal closed $V$-categories over $underline{V}$.  As a consequence, we find that the enriched adjunction determined a symmetric monoidal closed adjunction can be obtained by applying a 2-functor and, consequently, is an adjunction in the 2-category of symmetric monoidal closed $V$-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/5/31-05abs.html", "title": "Enriched algebraic theories and monads for a system of arities", "authors": "Rory B. B. Lucyshyn-Wright", "keywords": ["algebraic theory", "Lawvere theory", "universal algebra", "monad", "enriched category theory", "free cocompletion"], "abstract": "Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category $V$ with respect to a specified system of arities $j:J  \\hookrightarrow V$.  Lawvere's notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted $V$-enriched $J$-cotensor theory, or $J$-theory for short.  For suitable choices of $V$ and $J$, such $J$-theories include the enriched algebraic theories of Borceux and Day, the enriched Lawvere theories of Power, the equational theories of Linton's 1965 work, and the $V$-theories of Dubuc, which are recovered by taking $J = V$ and correspond to arbitrary $V$-monads on $V$.  We identify a modest condition on $j$ that entails that the $V$-category of $T$-algebras exists and is monadic over $V$ for every $J$-theory $T$, even when $T$ is not small and $V$ is neither complete nor cocomplete.  We show that $j$ satisfies this condition if and only if $j$ presents $V$ as a free cocompletion of $J$ with respect to the weights for left Kan extensions along $j$, and so we call such systems of arities eleutheric.  We show that $J$-theories for an eleutheric system may be equivalently described as (i) monads in a certain one-object bicategory of profunctors on $J$, and (ii) $V$-monads on $V$ satisfying a certain condition.  We prove a characterization theorem for the categories of algebras of $J$-theories, considered as $V$-categories $A$ equipped with a specified $V$-functor $A \\rightarrow V$."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/4/31-04abs.html", "title": "A two-dimensional Birkhoff's theorem", "authors": "Matej Dostal", "keywords": ["2-dimensional universal algebra", "equational subcategories"], "abstract": "Birkhoff's variety theorem from universal algebra characterises equational subcategories of varieties. We give an analogue of Birkhoff's theorem in the setting of enrichment in categories. For a suitable notion of an equational subcategory we characterise these subcategories by their closure properties in the ambient algebraic category."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/2/31-02abs.html", "title": "The heart of a combinatorial model category", "authors": "Zhen Lin Low", "keywords": ["cofibrant generation", "closed model category", "weak factorization system", "locally presentable category", "ind-object", "filtered colimit"], "abstract": "We show that every small model category that satisfies certain size conditions can be completed to yield a combinatorial model category, and conversely, every combinatorial model category arises in this way. We will also see that these constructions preserve right properness and compatibility with simplicial enrichment. Along the way, we establish some technical results on the index of accessibility of various constructions on accessible categories, which may be of independent interest."},
{"url": "http://www.tac.mta.ca/tac/volumes/31/1/31-01abs.html", "title": "The Euler characteristic of an enriched category", "authors": "Kazunori Noguchi and Kohei Tanaka", "keywords": ["Euler characteristic", "enriched categories", "monoidal model categories"], "abstract": "We develop the homotopy theory of Euler characteristic (magnitude) of a category enriched in a monoidal model category. If a monoidal model category $V$ is equipped with an Euler characteristic that is compatible with weak equivalences and fibrations in $V$, then our Euler characteristic of $V$-enriched categories is also compatible with weak equivalences and fibrations in the canonical model structure on the category of $V$-enriched categories. In particular, we focus on the case of topological categories; i.e., categories enriched in the category of topological spaces. As its application, we obtain the ordinary Euler characteristic of a cellular stratified space $X$ by computing the Euler characteristic of the face category $C(X)$."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/46/32-46abs.html", "title": "A characterization of the Huq commutator", "authors": "Vaino Tuhafeni Shaumbwa", "keywords": ["Commuting morphisms", "Commutator", "Normal", "Ideal-determined", "Unital category"], "abstract": "We study a categorical commutator, introduced by Huq, defined for a pair of coterminal morphisms. We show that in a normal unital category C with finite colimits, the normal closure of the regular image of the Huq commutator of a pair of subobjects under an arbitrary morphism is the same as the Huq commutator of their respective regular images. Then we use this property to characterize the Huq commutator as the largest commutator satisfying certain properties."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/45/32-45abs.html", "title": "Stability properties characterising n-permutable categories", "authors": "Pierre-Alain Jacqmin, Diana Rodelo", "keywords": ["Mal'tsev category", "Goursat category", "n-permutable category", "embedding theorem", "unconditional exactness property"], "abstract": "The purpose of this paper is two-fold. A first and more concrete aim is to characterise n-permutable categories through certain stability properties of regular epimorphisms. These characterisations allow us to recover the ternary terms and the (n+1)-ary terms describing n-permutable varieties of universal algebras.\nA second and more abstract aim is to explain two proof techniques, by using the above characterisation as an opportunity to provide explicit new examples of their use:  - an embedding theorem for n-permutable categories which allows us to follow the varietal proof to show that an n-permutable category has certain properties;  - the theory of unconditional exactness properties which allows us to remove the assumption of the existence of colimits, in particular when we use the approximate co-operations approach to show that a regular category is n-permutable."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/42/32-42abs.html", "title": "Mal'tsev objects, $R_1$-spaces and ultrametric spaces", "authors": "Thomas Weighill", "keywords": ["Mal'tsev object", "Mal'tsev category", "$R_1$-space", "ultrametric space"], "abstract": "In this paper we introduce a notion of Mal'tsev object, and the dual notion of co-Mal'tsev object, in a general category. In particular, a category C is a Mal'tsev category if and only if every object in C is a Mal'tsev object. We show that for a well-powered regular category C which admits coproducts, the full subcategory of Mal'tsev objects is coreflective in C. We show that the co-Mal'tsev objects in the category of topological spaces and continuous maps are precisely the $R_1$-spaces, and that the co-Mal'tsev objects in the category of metric spaces and short maps are precisely the ultrametric spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/44/32-44abs.html", "title": "Approximate categorical structures", "authors": "Abdelkrim Aliouche and Carlos Simpson", "keywords": ["metric", "$2$-metric space", "category", "functor", "Yoneda embedding", "bimodule", "path", "triangle"], "abstract": "We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of 2-metric space. We prove an embedding theorem giving sufficient conditions for an approximate categorical structure to come from an inclusion into a metrized category."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/43/32-43abs.html", "title": "Action representability via spans and prefibrations", "authors": "J. R. A. Gray", "keywords": ["action representable", "semi-abelian", "split extension", "normalizer", "prefibration", "regular span"], "abstract": "We give several reformulations of action representability of a category as well as action representability of its category of morphisms. In particular we show that for a semi-abelian category C, its category of morphisms is action representable if and only if the functor from the category of split extensions in C to C, sending a split extension to its kernel, is a prefibration. To obtain these reformulations we show that certain conditions are equivalent for right regular spans of categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/40/32-40abs.html", "title": "Operads and phylogenetic trees", "authors": "John C. Baez and Nina Otter", "keywords": ["operads", "trees", "phylogenetic trees", "Markov processes"], "abstract": "We construct an operad Phyl whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of Com, the operad for commutative semigroups, and $[0,\\infty)$, the operad with unary operations corresponding to nonnegative real numbers, where composition is addition.  We show that there is a homeomorphism between the space of n-ary operations of Phyl and $\\T_n\\times [0,\\infty)^{n+1}$, where $\\T_n$ is the space of metric n-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of Phyl.  These always extend to coalgebras of the larger operad Com + $[0,\\infty]$, since Markov processes on finite sets converge to an equilibrium as time approaches infinity.  We show that for any operad O, its coproduct with $[0,\\infty]$ contains the operad W(O) constructed by Boardman and Vogt.  To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/39/32-39abs.html", "title": "Kan's combinatorial spectra and their sheaves revisited", "authors": "Ruian Chen, Igor Kriz and Ales Pultr", "keywords": ["combinatorial spectra", "spectral sheaves"], "abstract": "We define a right Cartan-Eilenberg structure on the category of Kan's combinatorial spectra, and the category of sheaves of such spectra, assuming some conditions. In both structures, we use the geometric concept of homotopy equivalence as the strong equivalence. In the case of sheaves, we use local equivalence as the weak equivalence. This paper is the first step in a larger-scale program of investigating sheaves of spectra from a geometric viewpoint."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/38/32-38abs.html", "title": "Fibered multiderivators and (co)homological descent", "authors": "Fritz Hörmann", "keywords": ["Derivators", "fibered derivators", "multiderivators", "fibered multicategories", "Grothendieck's six-functor-formalism", "cohomological descent", "homological descent", "fundamental localizers", "well-generated triangulated categories", "equivariant derived categories"], "abstract": "The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The main motivation is a descent theory for Grothendieck's six operations."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/37/32-37abs.html", "title": "Hopf polyads, Hopf categories and Hopf group monoids viewed as Hopf monads", "authors": "Gabriella Böhm", "keywords": ["monoidal bicategory", "monoidale", "Hopf monad", "Hopf polyad", "Hopf category", "Hopf group algebra"], "abstract": "We associate, in a functorial way, a monoidal bicategory Span|V to any monoidal bicategory V. Two examples of this construction are of particular interest: Hopf polyads of Bruguieres can be seen as Hopf monads in Span|Cat while Hopf group monoids in the spirit of Zunino and Turaev in a braided monoidal category V, and Hopf categories of Batista-Caenepeel-Vercruysse over V both turn out to be Hopf monads in Span|V. Hopf group monoids and Hopf categories are Hopf monads on a distinguished type of monoidales fitting the framework of Bohm-Lack. These examples are related by a monoidal pseudofunctor V -> Cat."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/35/32-35abs.html", "title": "Localization of enriched categories and cubical sets", "authors": "Tyler Lawson", "keywords": ["Enriched localization", "invertibility hypothesis"], "abstract": "The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category Cat_S of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is a consequence of the other axioms."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/34/32-34abs.html", "title": "A Dold-Kan theorem for simplicial Lie algebras", "authors": "P. Carrasco and  A.M. Cegarra", "keywords": ["Dold-Kan theorem", "simplicial Lie algebra", "chain complex", "Moore complex", "hypercrossed complex"], "abstract": "We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded chain complexes of Lie algebras $L=(L_n,\\partial_n)$ endowed with an additional structure by means of a suitable set of bilinear maps $L_r\\times L_s\\rightarrow L_n$. The Moore complex of any simplicial Lie algebra acquires such a structure and, in this way, we prove a Dold-Kan type equivalence between the category of simplicial Lie algebras and the category of hypercrossed complexes of Lie algebras. Several consequences of examining particular classes of hypercrossed complexes of Lie algebras are presented."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/36/32-36abs.html", "title": "Coexponentiability and  Projectivity: Rigs, Rings, and Quantales", "authors": "S.B. Niefield and R.J. Wood", "keywords": ["monoidal category", "projective module", "rig"], "abstract": "We show that a commutative monoid A is coexponentiable in  CMon(V) if and only if $-\\otimes A : V \\to V$  has a left adjoint, when V is a cocomplete symmetric monoidal closed category with finite biproducts and in which every object is a quotient of a free. Using a general characterization of the latter, we show that an algebra over a rig or ring R is coexponentiable if and only if it is finitely generated and projective as an R-module. Omitting the finiteness condition, the same result (and proof) is obtained for algebras over a quantale."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/33/32-33abs.html", "title": "Combinatorics of past-similarity in higher dimensional transition systems", "authors": "Philippe Gaucher", "keywords": ["left determined model category", "combinatorial model category", "discrete model structure", "higher dimensional transition system", "causal structure", "bisimulation"], "abstract": "The key notion to understand the left determined Olschok model category of star-shaped Cattani-Sassone transition systems is past-similarity. Two states are past-similar if they have homotopic pasts. An object is fibrant if and only if the set of transitions is closed under past-similarity. A map is a weak equivalence if and only if it induces an isomorphism after the identification of all past-similar states. The last part of this paper is a discussion about the link between causality and homotopy."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/32/32-32abs.html", "title": "Free objects over posemigroups in the category PoSgr_v", "authors": "Shengwei Han, Changchun Xia, Xiaojuan Gu", "keywords": ["partially ordered semigroup", "injective hull", "quantale", "quantale completion", "free object"], "abstract": "As we all know, the complete lattice I_D(S) of all D-ideals of a meet-semilattice S is precisely the injective hull of S in the category of meet-semilattices. In this paper, we consider sm-ideals of posemigroups which can be regarded as a generalization of D-ideals of meet-semilattices. Unfortunately, the quantale R(S) of all sm-ideals of a posemigroup S is in general not an injective hull of S. However, R(S) can be seen as a new type of quantale completions of S. Further, we can see that R(S) is also a free object over S in the category PoSgr_v of posemigroups with sm-distributive join homomorphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/31/32-31abs.html", "title": "Classifiers for monad morphisms and adjunction morphisms", "authors": "Dimitri Zaganidis", "keywords": ["monad", "adjunction", "2-category", "Artin-Mazur codiagonal"], "abstract": "We provide an explicit model for the free 2-category containing n composable adjunction morphisms, comparable to the Schanuel and Street model for the free adjunction. We can extract from it an explicit model for the free 2-category containing n composable lax monad morphisms. A careful proof is given, which goes through presentations of the hom-categories of our model. We use one of these hom-categories as an indexing category to construct an extended Artin-Mazur codiagonal, whose underlying bisimplicial set has the classical Artin-Mazur codiagonal as its first column."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/30/32-30abs.html", "title": "The canonical 2-gerbe of a holomorphic vector bundle", "authors": "Markus Upmeier", "keywords": ["Holomorphic Gerbes", "Second Chern class", "Complex manifolds", "Holomorphic vector bundles"], "abstract": "For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between holomorphic and smooth gerbes. For example, we introduce an Atiyah class for gerbes and prove a Koszul-Malgrange type theorem."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/29/32-29abs.html", "title": "A bicategory of decorated cospans", "authors": "Kenny Courser", "keywords": ["bicategory", "decorated cospan", "network", "symmetric monoidal"], "abstract": "If C is a category with pullbacks then there is a bicategory with the same objects as C, spans as morphisms, and maps of spans as 2-morphisms, as shown by Benabou. Fong has developed a theory of `decorated cospans', which are cospans in C equipped with extra structure. This extra structure arises from a symmetric lax monoidal functor F : C --> D; we use this functor to `decorate' each cospan with apex N in C with an element of F(N). Using a result of Shulman, we show that when C has finite colimits, decorated cospans are morphisms in a symmetric monoidal bicategory. We illustrate our construction with examples from electrical engineering and the theory of chemical reaction networks."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/27/32-27abs.html", "title": "Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings", "authors": "Ettore Aldrovandi", "keywords": ["Categorical ring", "biextension", "bimonoidal", "ring-like stack", "butterfly", "multi-extension", "multi-category", "multi-functor", "Mac Lane cohomology"], "abstract": "We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to n-variables, and prove that, in a manner analogous to that of butterflies, these multi-extensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory.  As an application, we study a bimonoidal category or stack, treating the multiplicative structure as a bimonoidal functor with respect to the additive one. In the context of the multilinear functor calculus, we view the bimonoidal structure as an instance of the general notion of pseudo-monoid. We show that when the structure is ring-like, i.e. the pseudo-monoid is a stack whose fibers are categorical rings, we can recover the classification by the third Mac~Lane cohomology of a ring with values in a bimodule."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/28/32-28abs.html", "title": "A Urysohn type lemma for groupoids", "authors": "Madalina Roxana Buneci", "keywords": ["groupoid", "Urysohn-type lemma", "metrization theorem"], "abstract": "Starting from the observation that through groupoids we can express in a unified way the notions of fundamental system of entourages of a uniform structure on a space X, respectively the system of neighborhoods of the unity of a topological group that determines its topology, we introduce in this paper a notion of G-uniformity for a groupoid G. The topology induced by a G-uniformity turns G into a topological locally transitive groupoid.  We also prove a Urysohn type lemma for groupoids and obtain metrization theorems for groupoids unifying in two ways the Alexandroff-Urysohn Theorem and Birkhoff-Kakutani Theorem."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/26/32-26abs.html", "title": "Connections in tangent categories", "authors": "J.R.B. Cockett and G.S.H. Cruttwell", "keywords": ["Tangent categories", "connections"], "abstract": "Connections are an important tool of differential geometry.  This paper investigates their definition and structure in the abstract setting of tangent categories.  At this level of abstraction we derive several classically important results about connections, including the Bianchi identities, identities for curvature and torsion, almost complex structure, and parallel transport."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/41/32-41abs.html", "title": "Topological properties of non-Archimedean approach spaces", "authors": "Eva Colebunders and Karen Van Opdenbosch", "keywords": ["Lax algebra", "quantale", "non-Archimedean approach space", "quasi-ultrametric space", "initially dense object", "topological properties in  $(\\beta", "P_\\wedge$-Cat", "compactification"], "abstract": "In this paper we give an isomorphic description of the category of non-Archimedian approach spaces as a category of lax algebras for the ultrafilter monad and an appropriate quantale. Non-Archimedean approach spaces are characterised as those approach spaces having a tower consisting of topologies. We study topological properties p, for p compactness and Hausdorff separation along with low-separation properties, regularity, normality and extremal disconnectedness and link these properties to the condition that all or some of the level topologies in the tower have p. A compactification technique is developed based on Shanin's method."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/25/32-25abs.html", "title": "A unifying approach to the acyclic models method and other lifting lemmas", "authors": "Leonard Guetta", "keywords": ["homological algebra", "acyclic models", "lifting theorems", "comparison theorems"], "abstract": "We prove a fundamental lemma of homological algebra and show how it sets a framework for many different lifting (or comparison) theorems of homological algebra and algebraic topology. Among these are different versions of the acyclic models method."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/24/32-24abs.html", "title": "Metric spaces and SDG", "authors": "Anders Kock", "keywords": ["metric spaces", "synthetic differential geometry"], "abstract": "We present an axiomatic theory, based on the notions of  metric space and  space with a (first order) neighbour relation. The axiomatics implies a synthetic proof of Huygens' principle of  wave fronts, as envelopes of a family of spheres. A model of the axiomatics is presented in terms of synthetic differential geometry (SDG)."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/23/32-23abs.html", "title": "Levi categories and graphs of groups", "authors": "Mark V. Lawson and Alistair R. Wallis", "keywords": ["Graphs of groups", "self-similar groupoid actions", "cancellative categories"], "abstract": "We define a Levi category to be a weakly orthogonal category equipped with a suitable length functor and prove two main theorems about them. First, skeletal cancellative Levi categories are precisely the categorical versions of graphs of groups with a given orientation. Second, the universal groupoid of a skeletal cancellative Levi category is the fundamental groupoid of the corresponding graph of groups. These two results can be viewed as a co-ordinate-free refinement of a classical theorem of Philip Higgins."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/22/32-22abs.html", "title": "A property of effectivization and its uses in categorical logic", "authors": "Vasileios Aravantinos-Sotiropoulos and Panagis Karazeris", "keywords": ["regular category", "effectivization", "pretopos", "conceptual completeness", "quotient"], "abstract": "We show that a fully faithful and covering regular functor between regular categories induces a fully faithful (and covering) functor between their respective effectivizations. Such a functor between effective categories is known to be an equivalence. We exploit this result in order to give a constructive proof of conceptual completeness for regular logic. We also use it in analyzing what it means for a morphism between effective categories to be a quotient in the 2-category of effective categories and regular functors between them."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/21/32-21abs.html", "title": "Lax distributive laws for topology, II", "authors": "Hongliang Lai, Lili Shen and Walter Tholen", "keywords": ["quantaloid", "monad", "presheaf monad", "copresheaf monad", "double presheaf monad", "double copresheaf monad", "lax distributive law", "lax $\\lambda$-algebra", "lax monad extension", "Q-closure space", "Q-interior space"], "abstract": "For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf 2-monad P and the copresheaf 2-monad P^{\\dagger}, as well as by their two composite 2-monads, and establish that they all laxly distribute over P. These four 2-monads therefore admit lax extensions to the category Q-Dist of Q-categories and their distributors. We characterize the corresponding (T,Q)-categories in each of the four cases, leading us to both known and novel categorical structures."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/20/32-20abs.html", "title": "Closure operators in abelian categories and spectral spaces", "authors": "Abhishek Banerjee", "keywords": ["Spectral spaces", "closure operators", "abelian categories"], "abstract": "We give several new ways of constructing spectral spaces starting with objects in abelian categories satisfying certain conditions which apply, in particular, to Grothendieck categories. For this, we consider the spaces of invariants of closure operators acting on subobjects of a given object. The key to our results is a newly discovered criterion of Finocchiaro that uses ultrafilters to identify spectral spaces along with subbases of quasi-compact open sets."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/19/32-19abs.html", "title": "A presentation of bases for parametrized iterativity", "authors": "Jiri Adamek, Stefan Milius and Jiri Velebil", "keywords": ["parametrized monads", "bases", "algebra", "presentation"], "abstract": "Finitary monads on a locally finitely presentable category A are well-known to possess a presentation by signatures and equations. Here we prove that, analogously, bases on A, i.e., finitary functors from A to the category of finitary monads on A, possess a presentation by parametrized signatures and equations."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/18/32-18abs.html", "title": "Frobenius algebras and homotopy fixed points of group actions on bicategories", "authors": "Jan Hesse, Christoph Schweigert, and Alessandro Valentino", "keywords": ["symmetric Frobenius algebras", "homotopy fixed points", "group actions  on bicategories"], "abstract": "We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras, bimodules and intertwiners. The results are motivated by the 2-dimensional Cobordism Hypothesis for oriented manifolds, and can hence be interpreted in the realm of Topological Quantum Field Theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/16/32-16abs.html", "title": "Algebraic databases", "authors": "Patrick Schultz, David I. Spivak, Christina Vasilakopoulou, and Ryan Wisnesky", "keywords": ["Databases", "algebraic theories", "proarrow equipments", "collage construction", "data migration"], "abstract": "Databases have been studied category-theoretically for decades. While mathematically elegant, previous categorical models have typically struggled withrepresenting concrete data such as integers or strings.  In the present work, we propose an extension of the earlier set-valued functor model, making use of multi-sorted algebraic theories (a.k.a. Lawvere theories) to incorporate concrete data in a principled way. This approach easily handles missing information (null values), and also allows constraints and queries to make use of operations on data, such as multiplication or comparison of numbers, helping to bridge the gap between traditional databases and programming languages.  We also show how all of the components of our model - including schemas, instances, change-of-schema functors, and queries fit into a single double categorical structure called a proarrow equipment (a.k.a.  framed bicategory)."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/17/32-17abs.html", "title": "The two out of three property in ind-categories and a convenient model category of spaces", "authors": "Ilan Barnea", "keywords": ["Ind-categories", "model categories", "cofibration categories", "simplicially enriched categories", "compact Hausdorff spaces"], "abstract": "The author and Tomer Schlank studied a much weaker homotopical structure than a model category, which we called a \"weak cofibration category\". We showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The main purpose of this paper is to give sufficient intrinsic conditions on a weak cofibration category for this two out of three property to hold. We consider an application to the category of compact metrizable spaces, and thus obtain a model structure on its ind-category. This model structure is defined on a category that is closely related to a category of topological spaces and has many convenient formal properties. A more general application of these results, to the (opposite) category of separable $C^*$-algebras, appears in a paper by the author, Michael Joachim and Snigdhayan Mahanta."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/13/32-13abs.html", "title": "Building a model category out of multiplier ideal sheaves", "authors": "Seunghun Lee", "keywords": ["Multiplier Ideal sheaf", "Model Category", "Pro-Category"], "abstract": "We will construct a Quillen model structure out of the multiplier ideal sheaves on a smooth quasi-projective variety using earlier works of Isaksen and Barnea and Schlank. We also show that fibrant objects of this model category are made of kawamata log terminal pairs in birational geometry."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/14/32-14abs.html", "title": "A categorical approach to Picard-Vessiot theory", "authors": "Andreas Maurischat", "keywords": ["Tannakian categories", "Picard-Vessiot theory", "Galois theory"], "abstract": "Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restricts to the categorical properties which all the categories of differential modules resp.~difference modules etc.~share, it gives unified proofs for all these Galois theories (and maybe more general ones)."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/15/32-15abs.html", "title": "Epireflective subcategories and formal closure operators", "authors": "Mathieu Duckerts-Antoine, Marino Gran, and Zurab Janelidze", "keywords": ["category of morphisms", "category of epimorphisms", "category of  monomorphisms", "cartesian lifting", "closure operator", "codomain functor", "cohereditary operator", "domain functor", "epimorphism", "epireflective subcategory", "form", "minimal operator", "monomorphism", "normal category", "pointed endofunctor", "reflection", "reflective subcategory", "regular category", "subobject", "quotient"], "abstract": "On a category $\\mathscr{C}$ with a designated (well-behaved) class $\\mathcal{M}$ of monomorphisms, a closure operator in the sense of D.~Dikranjan and E.~Giuli is a pointed endofunctor of $\\mathcal{M}$, seen as a full subcategory of the arrow-category $\\mathscr{C}^\\mathbf{2}$ whose objects are morphisms from the class $\\mathcal{M}$, which ``commutes'' with the codomain functor $\\mathsf{cod}\\colon \\mathcal{M}\\to \\mathscr{C}$. In other words, a closure operator consists of a functor $C\\colon \\mathcal{M}\\to\\mathcal{M}$ and a natural transformation $c\\colon 1_\\mathcal{M}\\to C$ such that $\\mathsf{cod} \\cdot C=C$ and $\\mathsf{cod}\\cdot c=1_\\mathsf{cod}$. In this paper we adapt this notion to the domain functor $\\mathsf{dom}\\colon \\mathcal{E}\\to\\mathscr{C}$, where $\\mathcal{E}$ is a class of epimorphisms in $\\mathscr{C}$, and show that such closure operators can be used to classify $\\mathcal{E}$-epireflective subcategories of $\\mathscr{C}$, provided $\\mathcal{E}$ is closed under composition and contains isomorphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/12/32-12abs.html", "title": "Monoid-like definitions of cyclic operad", "authors": "Jovana Obradovic", "keywords": ["operads", "cyclic operads", "species of structures", "monoid", "microcosm principle"], "abstract": "Guided by the microcosm principle of Baez-Dolan and by the algebraic definitions of operads of Kelly and Fiore, we introduce two ``monoid-like'' definitions of cyclic operads, one for the original, ``exchangable-output'' characterisation of Getzler-Kapranov, and the other for the alternative ``entries-only'' characterisation, both within the category of Joyal's species of structures. Relying on a result of Lamarche on descent for species, we use these \"monoid-like\" definitions to prove the equivalence between the ``exchangable-output'' and ``entries-only'' points of view on cyclic operads."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/9/32-09abs.html", "title": "Classifying tangent structures using Weil algebras", "authors": "Poon Leung", "keywords": ["Tangent Structure", "Weil algebra"], "abstract": "At the heart of differential geometry is the construction of the tangent bundle of a manifold. There are various abstractions of this construction, and of particular interest here is that of Tangent Structures. Tangent Structure is defined via giving an underlying category M and a tangent functor T along with a list of natural transformations satisfying a set of axioms, then detailing the behaviour of T in the category End(M). However, this axiomatic definition at first seems somewhat disjoint from other approaches in differential geometry. The aim of this paper is to present a perspective that addresses this issue. More specifically, this paper highlights a very explicit relationship between the axiomatic definition of Tangent Structure and the Weil algebras (which have a well established place in differential geometry)."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/11/32-11abs.html", "title": "Corelations are the prop for extraspecial commutative Frobenius monoids", "authors": "Brandon Coya and Brendan Fong", "keywords": ["corelation", "extra law", "Frobenius monoid", "prop", "PROP"], "abstract": "Just as binary relations between sets may be understood as jointly monic spans, so too may equivalence relations on the disjoint union of sets be understood as jointly epic cospans. With the ensuing notion of composition inherited from the pushout of cospans, we call these equivalence relations corelations. We define the category of corelations between finite sets and prove that it is equivalent to the prop for extraspecial commutative Frobenius monoids.  Dually, we show that the category of relations is equivalent to the prop for special commutative bimonoids. Throughout, we emphasise how corelations model interconnection."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/10/32-10abs.html", "title": "KZ-monadic categories and their logic", "authors": "Jiri Adamek and Lurdes Sousa", "keywords": ["order-enriched category", "Kan-injectivity", "KZ-monad", "Kan-injectivity logic", "locally ranked category"], "abstract": "Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed.  We present a sound logic for Kan-injectivity consequences and prove that in ``reasonable\"  categories (such as $\\Pos$ or $\\Top_0$)  it is also complete for every set H of  squares."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/8/32-08abs.html", "title": "Witt vectors and truncation posets", "authors": "Vigleik Angeltveit", "keywords": ["Witt vectors", "truncation posets", "Tambara functors"], "abstract": "One way to define Witt vectors starts with a truncation set $S \\subset N$. We generalize Witt vectors to truncation posets, and show how three types of maps of truncation posets can be used to encode the following six structure maps on Witt vectors: addition, multiplication, restriction, Frobenius, Verschiebung and norm."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/7/32-07abs.html", "title": "A note on injective hulls of posemigroups", "authors": "Changchun Xia, Shengwei Han, Bin Zhao", "keywords": ["partially ordered semigroup", "quantale", "quantic nucleus", "injective hull"], "abstract": "In this note, we prove the existence of $E_\\leq$-injective hulls in the category $PoSgr_\\leq$ of posemigroups and their submultiplicative order-preserving maps; here $E_\\leq$ denotes the class of those morphisms $h : A \\to B$ for which $h(a_1)...h(a_n)\\leq h(a)$ always implies $a_1...a_n\\leq a$. The result of this note subsumes the results given by Lambek et al. (2012) and by Zhang and Laan (2014)."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/5/32-05abs.html", "title": "Bourn-normal monomorphisms in regular Mal'tsev categories", "authors": "Giuseppe Metere", "keywords": ["normal monomorphism", "Mal'tsev category", "fibred category"], "abstract": "Normal monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this subject often focus on protomodular settings, where normality becomes a property. This paper clarifies the connections between internal equivalence relations and Bourn-normal monomorphisms in regular Mal'tesv categories with pushouts of split monomorphisms along arbitrary morphisms, whereas a full description is achieved for quasi-pointed regular Mal'tsev categories with pushouts of split monomorphisms along arbitrary morphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/6/32-06abs.html", "title": "Homotopy theory for algebras over polynomial monads", "authors": "M. A. Batanin and C. Berger", "keywords": ["Quillen model category", "polynomial monad", "coloured operad", "graph"], "abstract": "We study the existence and left properness of transferred model structures for ``monoid-like'' objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular, higher operads, properads and PROP's. All these structures can be realised as algebras over polynomial monads.  We give a general condition for a polynomial monad which ensures the existence and (relative) left properness of a transferred model structure for its algebras. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame polynomial. Many important monads are shown to be tame polynomial."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/4/32-04abs.html", "title": "The $(\\Pi,\\lambda)$-structures on the C-systems defined by universe categories", "authors": "Vladimir Voevodsky", "keywords": ["Type theory", "Contextual category", "Universe category", "dependent product", "product of families of types"], "abstract": "We define the notion of a $(P,\\tilde{P})$-structure on a universe $p$ in a locally cartesian closed category category with a binary product structure and construct a $(\\Pi,\\lambda)$-structure on the C-systems  $CC(C,p)$ from a $(P,\\tilde{P})$-structure on $p$.  We then define homomorphisms of C-systems with $(\\Pi,\\lambda)$-structures and functors of universe categories with $(P,\\tilde{P})$-structures and show that our construction is functorial relative to these definitions."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/3/32-03abs.html", "title": "C-systems defined by universe categories: presheaves", "authors": "Vladimir Voevodsky", "keywords": ["Type theory", "Contextual category", "Universe category"], "abstract": "The main result of this paper may be stated as a construction of \"almost representations\" $\\mu_n$ and $\\tilde{\\mu}_n$ for the presheaves $\\Ob_n$ and $\\tilde{Ob}_n$ on the C-systems defined by locally cartesian closed universe categories with binary product structures and the study of the behavior of these \"almost representations\" with respect to the universe category functors.  In addition, we study a number of constructions on presheaves on C-systems and on universe categories that are used in the proofs of our main results, but are expected to have other applications as well."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/2/32-02abs.html", "title": "Simplicial Nerve of an $A_\\infty$-category", "authors": "Giovanni Faonte", "keywords": ["$A_\\infty$-categories", "nerve", "higher categories", "pretriangulated dg-categories"], "abstract": "We introduce a functor called the simplicial nerve of an $A_\\infty$-category defined on the category of $A_\\infty$-categories with values in simplicial sets. We show that the nerve of an $A_\\infty$-category is an $(\\infty,1)$-category in the sense of J. Lurie. This construction generalizes the nerve construction for differential graded categories given by Lurie. We prove that if a differential graded category is pretriangulated in the sense of A.I. Bondal and M. Kapranov then its nerve is a stable $(\\infty,1)$-category in the sense of J. Lurie."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/42/33-42abs.html", "title": "On finitely aligned left cancellative small categories, Zappa-Szep products and Exel-Pardo algebras", "authors": "Erik Bedos, S. Kaliszewski, John Quigg, and Jack Spielberg", "keywords": ["Groups", "graphs", "self-similarity", "category of paths", "left cancellative small categories", "Zappa-Szep products", "Toeplitz algebras", "Cuntz-Krieger algebras"], "abstract": "We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with fin\\-itely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szep product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/39/33-39abs.html", "title": "Coarse-graining open Markov processes", "authors": "John C. Baez and Kenny Courser", "keywords": ["double category", "cospan", "Markov process", "coarse-graining", "network", "black box"], "abstract": "Coarse-graining is a standard method of extracting a simpler Markov process from a more complicated one by identifying states.  Here we extend coarse-graining to `open' Markov processes: that is, those where probability can flow in or out of certain states called `inputs' and `outputs'.  One can build up an ordinary Markov process from smaller open pieces in two basic ways: composition, where we identify the outputs of one open Markov process with the inputs of another, and tensoring, where we set two open Markov processes side by side.  In previous work, Fong, Pollard and the first author showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category.  Here we go further by constructing a symmetric monoidal double category where the 2-morphisms include ways of coarse-graining open Markov processes.  We also extend the already known `black-boxing' functor from the category of open Markov processes to our double category.  Black-boxing sends any open Markov process to the linear relation between input and output data that holds in steady states, including nonequilibrium steady states where there is a nonzero flow of probability through the process.  To extend black-boxing to a functor between double categories, we need to prove that black-boxing is compatible with coarse-graining."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/41/33-41abs.html", "title": "The localic isotropy group of a topos", "authors": "Simon Henry", "keywords": ["Topos", "Isotropy", "localic groups"], "abstract": "It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos $T$ is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of the topos. We show that this group is in fact the group of points of a localic group object, called the localic isotropy group, which also acts on every object, and in fact also on every internal locale and on every $T$-topos. This new localic isotropy group has better functoriality and stability property than the original version and sheds some light on the phenomenon of higher isotropy observed for the ordinary isotropy group. We prove in particular using a localic version of the isotropy quotient that any geometric morphism can be factored uniquely as a connected atomic geometric morphism followed by a so called ``essentially anisotropic'' geometric morphism, and that connected atomic morphisms are exactly the quotients by open isotropy actions, hence providing a form of Galois theory for general (unpointed) connected atomic geometric morphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/40/33-40abs.html", "title": "Category theory for genetics I:mutations and sequence alignments", "authors": "Remy Tuyeras", "keywords": ["Multiple sequence alignment", "mutation mechanism", "right Kan extension", "category", "limit sketch", "pedigrad"], "abstract": "The present article is the first of a series whose goal is to define a logical formalism in which it is possible to reason about genetics. In this paper, we introduce the main concepts of our language whose domain of discourse consists of a class of limit-sketches and their associated models. While our program will aim to show that different phenomena of genetics can be modeled by changing the category in which the models take their values, in this paper, we study models in the category of sets to capture mutation mechanisms such as insertions, deletions, substitutions, duplications and inversions. We show how the proposed formalism can be used for constructing multiple sequence alignments with an emphasis on mutation mechanisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/37/33-37abs.html", "title": "Linear Distributivity With Negation, Star-Autonomy, and Hopf Monads", "authors": "Masahito Hasegawa and Jean-Simon P. Lemay", "keywords": ["monoidal categories", "linearly distributive categories", "$*$-autonomous categories", "comonoidal monads", "Hopf monads"], "abstract": "We show that a Hopf monad on a *-autonomous category lifts *-autonomous structure to the category of algebras precisely when there is an algebra structure on the dualizing object. Our proof is based on Pastro's characterization of *-autonomous (co)monads as linearly distributive (co)monads with negation."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/36/33-36abs.html", "title": "Polynomials, fibrations and distributive laws", "authors": "Tamara von Glehn", "keywords": ["polynomial functor", "fibration", "pseudo-distributive law", "lax-idempotent monad", "locally cartesian closed category", "2-bicategory"], "abstract": "We study the structure of the category of polynomials in a locally cartesian closed category. Formalizing the conceptual view that polynomials are constructed from sums and products, we characterize this category in terms of the composite of the pseudomonads which freely add fibred sums and products to fibrations. The composite pseudomonad structure corresponds to a pseudo-distributive law between these two pseudomonads, which exists if and only if the base category is locally cartesian closed."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/35/33-35abs.html", "title": "A tangent category alternative to the Faa di Bruno construction", "authors": "Jean-Simon P. Lemay", "keywords": ["Cartesian Differential Categories", "Cofree Cartesian Differential  Categories", "Tangent Categories", "Higher-Order Chain Rule"], "abstract": "The Faa di Bruno construction, introduced by Cockett and Seely, constructs a comonad Faa whose coalgebras are precisely Cartesian differential categories. In other words, for a Cartesian left additive category X, Faa(X) is the cofree Cartesian differential category over X. Composition in these cofree Cartesian differential categories is based on the Faa di Bruno formula, and corresponds to composition of differential forms. This composition, however, is somewhat complex and difficult to work with. In this paper we provide an alternative construction of cofree Cartesian differential categories inspired by tangent categories. In particular, composition defined here is based on the fact that the chain rule for Cartesian differential categories can be expressed using the tangent functor, which simplifies the formulation of the higher order chain rule."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/34/33-34abs.html", "title": "$T_0$ topological spaces and $T_0$ posets in the topos of M-sets", "authors": "M.M. Ebrahimi, M. Mahmoudi, and A.H. Nejah", "keywords": ["Topos", "M-set", "M-topological space", "M-poset", "M-continuous map", "$T_{0}$ M-topological space", "$T_0$ M-poset"], "abstract": "In this paper, we introduce the concept of a topological space in the topos M-Set of M-sets, for a monoid M. We do this by replacing the notion of open \"subset\" by open \"subobject\" in the definition of a topology. We prove that the resulting category has an open subobject classifier, which is the counterpart of the Sierpinski space in this topos. We also study the relation between the given notion of topology and the notion of a poset in this universe. In fact, the counterpart of the specialization pre-order is given for topological spaces in M-Set, and it is shown that, similar to the classic case, for a special kind of topological spaces in M-Set, namely $T_0$ ones, it is a partial order. Furthermore, we obtain the universal $T_0$ space, and give the adjunction between topological spaces and $T_0$ posets, in M-Set."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/32/33-32abs.html", "title": "On a higher structure on operadic deformation complexes", "authors": "Boris Shoikhet", "keywords": ["Deformation theory", "operads", "higher structures"], "abstract": "In this paper, we prove that there is a canonical homotopy (n+1)-algebra structure on the shifted operadic deformation complex $\\Def(e_n\\to\\mathcal{P})[-n]$ for any operad $\\mathcal{P}$ and a map of operads $f\\colon e_n\\to\\mathcal{P}$. This result generalizes a result of Tamarkin, who considered the case $\\mathcal{P}=\\End_\\Op(X)$. Another more computational proof of the same result was recently sketched by Calaque and Willwacher.\nOur method combines the one of Tamarkin, with the categorical algebra on the category of symmetric sequences, introduced by Rezk and further developed by Kapranov-Manin and Fresse. We define suitable deformation functors on n-coalgebras, which are considered as the \"non-commutative\" base of deformation, prove their representability, and translate properties of the functors to the corresponding properties of the representing objects. A new point, which makes the method more powerful, is to consider the argument of our deformation theory as an object of the category of symmetric sequences of dg vector spaces, not as just a single dg vector space ."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/30/33-30abs.html", "title": "Coalgebroids in monoidal bicategories and their comodules", "authors": "Ramon Abud Alcala", "keywords": ["coalgebroid", "comodules", "monoidale", "skew monoidale", "oplax action", "monoidal bicategory", "bicategory", "quantum category", "bialgebroid"], "abstract": "Quantum categories have been recently studied because of their relation to bialgebroids, small categories, and skew monoidales. This is the first of a series of papers based on the author's PhD thesis in which we examine the theory of quantum categories developed by Day, Lack, and Street.  A quantum category is an opmonoidal monad on the monoidale associated to a biduality $R\\dashv R^{o}$, or enveloping monoidale, in a monoidal bicategory of modules $\\Mod(V})$ for a monoidal category $V$. Lack and Street proved that quantum categories are in equivalence with right skew monoidales whose unit has a right adjoint in $\\Mod(V)$. Our first important result is similar to that of Lack and Street. It is a characterisation of opmonoidal arrows on enveloping monoidales in terms of a new structure named oplax action. We then provide three different notions of comodule for an opmonoidal arrow, and using a similar technique we prove that they are equivalent. Finally, when the opmonoidal arrow is an opmonoidal monad, we are able to provide the category of comodules for a quantum category with a monoidal structure such that the forgetful functor is monoidal."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/31/33-31abs.html", "title": "Tube representations and twisting of graded categories", "authors": "", "keywords": ["monoidal category", "quantum double", "tube algebra"], "abstract": "We study deformation of tube algebra under twisting of graded monoidal categories.  When a tensor category $C$ is graded over a group $\\Gammaa$, a torus-valued 3-cocycle on $\\Gammaa$ can be used to deform the associator of $C$.  We show that it induces a 2-cocycle on the groupoid of the adjoint action of $\\Gammaa$.  Combined with the natural Fell bundle structure of tube algebra over this groupoid, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/29/33-29abs.html", "title": "Crossed products of crossed modules of Hopf monoids", "authors": "J.N. Alonso Alvarez, J.M. Fernandez Vilaboa, and R. Gonzalez Rodriguez", "keywords": ["Hopf monoid", "crossed module", "entwining structure"], "abstract": "In this paper we consider a crossed product of two crossed modules of Hopf monoids in a strict symmetric monoidal category ${\\mathcal C}$ and give necessary and sufficient conditions to get a new crossed module of Hopf monoids in ${\\mathcal C}$. Moreover we introduce the notion of projection of crossed modules of Hopf monoids and show that with additional hypotheses, any of these projections defines a new crossed module of Hopf monoids and allows us to construct an example of this kind of crossed product. Finally, we develop the explicit computations of a crossed product associated to a projection of crossed modules in groups."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/28/33-28abs.html", "title": "On the geometric notion of connection and its expression in tangent categories", "authors": "Rory B. B. Lucyshyn-Wright", "keywords": ["connection", "tangent category", "linear connection", "affine connection", "vector bundle", "differential bundle"], "abstract": "Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and combinatorics.  Generalizing the notion of \\textit{(linear) connection} on a smooth vector bundle, Cockett and Cruttwell have defined a notion of connection on a differential bundle in an arbitrary tangent category.  Herein, we establish equivalent formulations of this notion of connection that reduce the amount of specified structure required.  Further, one of our equivalent formulations substantially reduces the number of axioms imposed, and others provide useful abstract conceptualizations of connections.  In particular, we show that a connection on a differential bundle $E$ over $M$ is equivalently given by a single morphism $K$ that induces a suitable decomposition of $TE$ as a biproduct.  We also show that a connection is equivalently given by a vertical connection $K$ for which a certain associated diagram is a limit diagram."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/27/33-27abs.html", "title": "The Ehresmann-Schein-Nambooripad Theorem for inverse categories", "authors": "Darien DeWolf and Dorette Pronk", "keywords": ["Inverse semigroup", "inverse category", "inductive groupoid", "locally complete inductive groupoid", "inverse semicategory"], "abstract": "The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors - a natural generalization of inverse semigroups and semigroup homomorphisms - and extend the ESN Theorem to an equivalence between this category and the category of locally complete inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally inductive groupoids and to an equivalence between the category of inverse categories with oplax functors and the category of locally complete inductive groupoids and ordered functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/32/1/32-01abs.html", "title": "A structure theorem for quasi-Hopf bimodule coalgebras", "authors": "Daniel Bulacu", "keywords": ["monoidal equivalence", "(bi)comodule algebra", "bimodule coalgebra", "structure theorem"], "abstract": "Let H be a quasi-Hopf algebra. We show that any H-bimodule coalgebra C for which there exists an H-bimodule coalgebra morphism n : C -> H is isomorphic to what we will call a smash product coalgebra. To this end, we use an explicit monoidal equivalence between the category of two-sided two-cosided Hopf modules over H and the category of left Yetter-Drinfeld modules over H. This categorical method allows also to reobtain the structure theorem for a quasi-Hopf (bi)comodule algebra given by Panaite and Van Oystaeyen, and by Dello et al."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/38/33-38abs.html", "title": "A compositional framework for passive linear networks", "authors": "John C. Baez and Brendan Fong", "keywords": ["passive linear network", "electric circuit", "principle of minimum power", "black box", "decorated cospan", "compact closed category", "hypergraph category", "Lagrangian relation"], "abstract": "Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where a morphism is a circuit of this sort with marked input and output terminals.  In this category, composition describes the process of attaching the outputs of one circuit to the inputs of another.  We construct a functor, dubbed the `black box functor', that takes a circuit, forgets its internal structure, and remembers only its external behavior.  Two circuits have the same external behavior if and only if they impose same relation between currents and potentials at their terminals.  The space of these currents and potentials naturally has the structure of a symplectic vector space, and the relation imposed by a circuit is a Lagrangian linear relation.  Thus, the black box functor goes from our category of circuits to a category with Lagrangian linear relations as morphisms.  We prove that this functor is symmetric monoidal and indeed a hypergraph functor.  We assume the reader is familiar with category theory, but not with circuit theory or symplectic linear algebra."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/25/33-25abs.html", "title": "Props in network theory", "authors": "John C. Baez, Brandon Coya, Franciscus Rebro", "keywords": ["circuit", "functorial semantics", "network", "PROP", "symmetric monoidal category"], "abstract": "Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components.  We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition.  In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop.  A network with $m$ inputs and $n$ outputs is a morphism from $m$ to $n$, putting networks together in series is composition, and setting them side by side is tensoring.  Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources.  Each kind of circuit corresponds to a mathematically natural prop.  We describe the `behavior' of these circuits using morphisms between props.  In particular, we give a new construction of the black-boxing functor of Fong and the first author; unlike the original construction, this new one easily generalizes to circuits with nonlinear components.  We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory.  Technically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/26/33-26abs.html", "title": "Spark complexes on good effective orbifold atlases categorically", "authors": "Cheng-Yong Du, Lili Shen and Xiaojuan Zhao", "keywords": ["good effective orbifold atlas", "compatible system", "spark complex", "spark homomorphism", "spark homotopy", "spark character"], "abstract": "Good atlases are defined for effective orbifolds, and a spark complex is constructed on each good atlas. It is proved that this process is 2-functorial with compatible systems playing as morphisms between good atlases, and that the spark character 2-functor factors through this 2-functor."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/24/33-24abs.html", "title": "Spheres as Frobenius objects", "authors": "Djordje Baralic, Zoran Petric and Sonja Telebakovic", "keywords": ["symmetric monoidal category", "commutative Frobenius object", "oriented manifold", "cobordism", "normal form", "coherence", "topological quantum field theory", "Brauerian representation"], "abstract": "Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension d=1, all the spheres are commutative Frobenius objects in categories whose arrows are (d+1)-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres - they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/22/33-22abs.html", "title": "Decorated corelations", "authors": "Brendan Fong", "keywords": ["decorated cospan", "corelation", "Frobenius monoid", "hypergraph category", "well-supported compact closed category"], "abstract": "Let $C$ be a category with finite colimits, and let $(E, M)$ be a  factorisation system on $C$ with $M$ stable under pushout.  Writing $C;M^{\\op}$ for the symmetric monoidal category with morphisms cospans of the form $\\stackrel{c}\\to \\stackrel{m}\\leftarrow$, where $c \\in C$ and $m \\in M$, we give a method for constructing a category from a symmetric lax monoidal functor $F : (C; \\mc M^{\\op},+) \\to (Set,\\times)$. A morphism in this category, termed a decorated corelation, comprises (i) a cospan $X \\to N \\leftarrow Y$ in $C$ such that the canonical copairing $X+Y \\to N$ lies in $E$, together with (ii)  an element of $FN$.  Functors between decorated corelation categories can be constructed from natural transformations between the decorating functors $F$.  This provides a general method for constructing hypergraph categories - symmetric monoidal categories in which each object is a special commutative Frobenius monoid in a coherent way - and their functors. Such categories are useful for modelling network languages, for example circuit diagrams, and such functors are useful for modelling their semantics."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/23/33-23abs.html", "title": "A parallel section functor for 2-vector bundles", "authors": "Christoph Schweigert and Lukas Woike", "keywords": ["parallel section", "homotopy fixed points", "higher representation", "higher vector bundle", "groupoid", "topological field theory"], "abstract": "We associate to a 2-vector bundle over an essentially finite groupoid a 2-vector space of parallel sections, or, in representation theoretic terms, of higher invariants, which can be described as homotopy fixed points. Our main result is the extension of this assignment to a symmetric monoidal 2-functor $Par : 2VecBunGrpd \\to 2Vect$. It is defined on the symmetric monoidal bicategory $2VecBunGrpd$ whose morphisms arise from spans of groupoids in such a way that the functor $Par$ provides pull-push maps between 2-vector spaces of parallel sections of 2-vector bundles. The direct motivation for our construction comes from the orbifoldization of extended equivariant topological field theories."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/21/33-21abs.html", "title": "Theory of interleavings on categories with a flow", "authors": "V. de Silva, E. Munch, A. Stefanou", "keywords": ["Topological Data Analysis", "Persistent Homology", "Category Theory", "Lawvere Metric Spaces"], "abstract": "The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by Chazal et al. as a metric on the class of persistence modules parametrized over the real line. Bubenik et al. subsequently extended the definition to categories of functors on a poset, the objects in these categories being regarded as `generalized persistence modules'. These metrics typically depend on the choice of a lax semigroup of endomorphisms of the poset. The purpose of the present paper is to develop a more general framework for the notion of interleaving distance using the theory of `actegories'. Specifically, we extend the notion of interleaving distance to arbitrary categories equipped with a flow, i.e. a lax monoidal action by the monoid $[0,\\infty)$. In this way, the class of objects in such a category acquires the structure of a Lawvere metric space. Functors that are colax $[0,\\infty)$-equivariant yield maps that are 1-Lipschitz. This leads to concise proofs of various known stability results from TDA, by considering appropriate colax $[0,\\infty)$-equivariant functors. Along the way, we show that several common metrics, including the Hausdorff distance and the $L^\\infty$-norm, can be realized as interleaving distances in this general perspective."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/33/33-33abs.html", "title": "New exactness conditions involving split cubes in protomodular categories", "authors": "J. R. A. Gray and N. Martins-Ferreira", "keywords": ["protomodular category", "split cube", "van Kampen Theorem", "non-pointed coincidence of commutators", "normality inside unions", "partially multiplicative graph", "descent"], "abstract": "We introduce and compare several new exactness conditions involving what we call split cubes. These conditions are motivated by their special cases, some which become familiar, in the pointed context, once we reformulate them with split cubes replaced with split extensions."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/20/33-20abs.html", "title": "Higher isotropy", "authors": "J. Funk, P. Hofstra, and S. Khan", "keywords": ["utomorphism groups", "algebraic invariants of categories", "toposes"], "abstract": "This paper is about an invariant of small categories called \\emph{isotropy}. Every small category C has associated with it a presheaf of groups on C, called its isotropy group, which in a sense solves the problem of making the assignment C |->  Aut(C) functorial. Consequently, every category has a canonical congruence that annihilates the isotropy; however, it turns out that the resulting quotient may itself have non-trivial isotropy. This phenomenon, which we term higher order isotropy, is the subject of our investigation. We show that with each category C we may associate a sequence of groups called its higher isotropy groups, and that these give rise to a sequence of quotients of C. This sequence leads us to an ordinal invariant for small categories, which we call isotropy rank: the isotropy rank of a small category is the ordinal at which the sequence of quotients stabilizes. Our main results state that each small category has a well-defined isotropy rank, and moreover, that for each small ordinal one may construct a small category with precisely that rank. It happens that isotropy rank of a small category is an instance of the same concept for Grothendieck toposes, for which corresponding statements hold. Most of the technical work in the paper is concerned with the development of tools that allow us to compute (higher) isotropy groups of categories in terms of those of certain suitable subcategories."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/19/33-19abs.html", "title": "Stability for inner fibrations revisited", "authors": "Danny Stevenson", "keywords": ["$\\infty$-categories", "joins of simplicial sets", "inner anodyne morphisms", "cocartesian fibrations"], "abstract": "In this paper we prove a stability result for inner fibrations in terms of the wide, or fat join operation on simplicial sets.  We also prove some additional results on inner anodyne morphisms that may be of independent interest."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/18/33-18abs.html", "title": "Regularity vs. constructive complete (co)distributivity", "authors": "Hongliang Lai and Lili Shen", "keywords": ["Quantaloid", "Girard quantaloid", "Quantale", "Girard quantale", "Regular $\\mathcal{Q}$-distributor", "Complete distributivity", "Kan adjunction"], "abstract": "It is well known that a relation $\\phi$ between sets is regular if, and only if, $K\\phi$ is completely distributive (cd), where $K\\phi$ is the complete lattice consisting of fixed points of the Kan adjunction induced by $\\phi$. For a small quantaloid Q, we investigate the Q-enriched version of this classical result, i.e., the regularity of Q-distributors versus the constructive complete distributivity (ccd) of Q-categories, and prove that ``the dual of $K\\phi$ is (ccd) implies $\\phi$ is regular implies $K\\phi$ is (ccd)'' for any Q-distributor $\\phi$. Although the converse implications do not hold in general, in the case that Q is a commutative integral quantale, we show that these three statements are equivalent for any $\\phi$ if, and only if, Q is a Girard quantale."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/17/33-17abs.html", "title": "Double power monad preserving adjunctions are Frobenius", "authors": "Christopher Townsend", "keywords": ["topos", "locale", "geometric morphism", "Frobenius reciprocity", "power monad"], "abstract": "We give a direct proof that between two toposes, F and E, bounded over a base topos S, adjunctions  L -| R: Loc_F -> Loc_E over Loc_S are Frobenius if and only if R commutes with the double power locale monad and finite coproducts. The proof uses only certain categorical properties of the category of locales, Loc. This implies that between categories axiomatized to behave like categories of locales, it does not make a difference whether maps are defined as structure preserving adjunctions (i.e. those that commute with the double power monads) or Frobenius adjunctions."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/16/33-16abs.html", "title": "Lax pullback complements and pullbacks of spans", "authors": "Seyed Naser Hosseini, Walter Tholen, and Leila Yeganeh", "keywords": ["lax pullback complement", "exponentiable morphism", "partial product", "adhesive category", "span category", "total spans", "cototal spans", "partial morphisms", "categories of presheaves", "graphs", "topological spaces"], "abstract": "The formation of the \"strict\" span category Span(C) of a category C with pullbacks is a standard organizational tool of category theory. Unfortunately, limits or colimits in Span(C) are not easily computed in terms of constructions in C. This paper shows how to form the pullback in Span(C) for many, but not all, pairs of spans, given the existence of some specific so-called lax pullback complements in C of the \"left legs\" of at least one of the two given spans. For some types of spans we require the ambient category to be adhesive to be able to form at least a weak pullback in Span(C). The existence of all lax pullback complements in C along a given morphism is equivalent to the exponentiability of that morphism. Since exponentiability is a rather restrictive property of a morphism, the paper first develops a comprehensive framework of rules for individual lax pullback complement diagrams, which resembles the set of pasting and cancellation rules for pullback diagrams, including their behaviour under pullback. We also present examples of lax pullback complements along non-exponentiable morphisms, obtained via lifting along a fibration."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/15/33-15abs.html", "title": "Pseudo-Kan extensions and descent theory", "authors": "Fernando Lucatelli Nunes", "keywords": ["descent objects", "descent category", "Kan extensions", "pseudomonads", "biadjunctions", "(effective) descent morphism", "weighted bilimits", "Benabou-Roubaud Theorem", "Galois Theory", "commutativity of bilimits"], "abstract": "There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits. We give a formal approach to descent theory, employing formal consequences of commuting properties of bilimits to prove classical and new theorems in the context of Janelidze-Tholen ``Facets of Descent II'', such as Benabou-Roubaud Theorems, a Galois Theorem, embedding results and formal ways of getting effective descent morphisms. In order to do this, we develop the formal part of the theory on commuting bilimits via pseudomonad theory, studying idempotent pseudomonads and proving a 2-dimensional version of the adjoint triangle theorem. Also, we work out the concept of pointwise pseudo-Kan extension, used as a framework to talk about bilimits, commutativity and the descent object. As a subproduct, this formal approach can be an alternative perspective/guiding template for the development of higher descent theory."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/14/33-14abs.html", "title": "Revisiting the canonicity of canonical triangulations", "authors": "Moritz Groth", "keywords": ["stable derivators", "triangulated categories", "exactness properties"], "abstract": "Stable derivators provide an enhancement of triangulated categories as is indicated by the existence of canonical triangulations. In this paper we show that exact morphisms of stable derivators induce exact functors of canonical triangulations, and similarly for arbitrary natural transformations. This 2-categorical refinement also provides a uniqueness statement concerning canonical triangulations.\nThese results rely on a more careful study of morphisms of derivators and this study is of independent interest. We analyze the interaction of morphisms of derivators with limits, colimits, and Kan extensions, including a discussion of invariance and closure properties of the class of Kan extensions preserved by a fixed morphism."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/13/33-13abs.html", "title": "On the normally ordered tensor product and duality for Tate objects", "authors": "O. Braunling, M. Groechenig, A. Heleodoro, J. Wolfson", "keywords": ["Tate vector space", "Tate object", "normally ordered product", "higher adeles", "higher local fields"], "abstract": "This paper generalizes the normally ordered tensor product from Tate vector spaces to Tate objects over arbitrary exact categories. We show how to lift bi-right exact monoidal structures, duality functors, and construct external Homs. We list some applications: (1) Adeles of a flag can be written as ordered tensor products; (2) Intersection numbers can be interpreted via these tensor products; (3) Pontryagin duality uniquely extends to n-Tate objects in locally compact abelian groups."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/11/33-11abs.html", "title": "A characterization of final functors between internal groupoids in exact categories", "authors": "Alan S. Cigoli", "keywords": ["exact category", "internal groupoid", "final functor", "comprehensive  factorization"], "abstract": "This paper provides three characterizations of final functors between internal groupoids in an exact category (in the sense of Barr). In particular, it is proved that a functor between internal groupoids is final if and only if it is internally full and essentially surjective."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/10/33-10abs.html", "title": "Nearly locally presentable categories", "authors": "L. Positselski and J. Rosicky", "keywords": ["locally presentable categories", "special directed colimits", "nearly presentable objects", "regular factorizations", "abelian categories", "complete partial orders"], "abstract": "We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/9/33-09abs.html", "title": "A Brauer-Clifford-Long group for the category of dyslectic Hopf Yetter-Drinfel'd (S,H)-module algebras", "authors": "Thomas Guedenon and Allen Herman", "keywords": ["Hopf algebras", "Yetter-Drinfel'd modules", "Braided monoidal categories", "Brauer groups"], "abstract": "Brauer-Clifford groups are equivariant Brauer groups for which a Hopf algebra acts or coacts nontrivially on the base ring.  Brauer-Clifford groups have been established previously in the category of modules for a skew group ring S#G, the category of modules for the smash product S#H over a cocommutative Hopf algebra H, and its dual category of (S,H)-Hopf modules over a commutative Hopf algebra H.  In this article the authors introduce a Brauer-Clifford group for the category of dyslectic Hopf Yetter-Drinfel'd (S,H)-modules for an H-commutative base ring S and quantum group H.  This is the first such example in a category of modules for a quantum group, and it gives a new example of an equivariant Brauer group in a braided monoidal category."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/12/33-12abs.html", "title": "Generating the algebraic theory of C(X): the case of partially ordered compact spaces", "authors": "Dirk Hofmann, Renato Neves, and Pedro Nora", "keywords": ["Ordered compact space", "quasivariety", "duality", "coalgebra", "Vietoris functor", "copresentable object", "metrisable"], "abstract": "It is known since the late 1960's that the dual of the category of compact Hausdorff spaces and continuous maps is a variety - not finitary, but bounded by $\\aleph_1$. In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is an $\\aleph_1$-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms on ordered compact spaces. We also characterise the $\\aleph_1$-copresentable partially ordered compact spaces."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/8/33-08abs.html", "title": "A construction of certain weak colimits and an exactness property of the 2-category of categories", "authors": "Descotte M.E., Dubuc E.J., Szyld M.", "keywords": ["weak colimit", "filtered", "2-category", "exactness property"], "abstract": "Given a 2-category $A$, a 2-functor $F : A \\to Cat$ and a distinguished 1-subcategory $\\Sigma \\subset A$ containing all the objects, a $\\sigma$-cone for $F$ (with respect to $\\Sigma$) is a lax cone such that the structural 2-cells corresponding to the arrows of $\\Sigma$ are invertible.  The conical $\\sigma$-limit} is the universal (up to isomorphism) $\\sigma$-cone. The notion of $\\sigma$-limit generalizes the well known notions of pseudo and lax limit. We consider the fundamental notion of $\\sigma$-filtered pair $(A, \\Sigma)$ which generalizes the notion of 2-filtered 2-category. We give an explicit construction of $\\sigma$-filtered $\\sigma$-colimits of categories, a construction which allows computations with these colimits.  We then state and prove a basic exactness property of the 2-category of categories, namely, that $\\sigma$-filtered $\\sigma$-colimits commute with finite weighted pseudo (or bi) limits. An important corollary of this result is that a $\\sigma$-filtered $\\sigma$-colimit of exact category valued 2-functors is exact. This corollary is essential in the 2-dimensional theory of flat and pro-representable 2-functors, that we develop elsewhere."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/7/33-07abs.html", "title": "Regular patterns, substitudes, Feynman categories and operads", "authors": "Michael Batanin, Joachim Kock, and Mark Weber", "keywords": ["operads", "symmetric monoidal categories"], "abstract": "We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras.  There are three main ingredients in establishing these biequivalences.  The first is a strictification theorem (exploiting Power's General Coherence Result) which allows to reduce to the case where the structure maps are identity-on-objects functors and strict monoidal.  Second, we subsume the Getzler and Kaufmann-Ward hereditary axioms into the notion of Guitart exactness, a general condition ensuring compatibility between certain left Kan extensions and a given monad, in this case the  free-symmetric-monoidal-category monad.  Finally we set up a biadjunction between substitudes and what we call pinned symmetric monoidal categories, from which the results follow as a consequence of the fact that the hereditary map is precisely the counit of this biadjunction."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/6/33-06abs.html", "title": "Spans of cospans", "authors": "Daniel Cicala", "keywords": ["spans", "cospans", "bicategory", "graph rewriting", "adhesive category", "network theory"], "abstract": "We study spans of cospans in a category C and explain how to horizontally and vertically compose these. When C is a topos and the legs of the spans are monic, these two forms of composition satisfy the interchange law. In this case there is a bicategory of objects, cospans, and `monic-legged' spans of cospans in C. One motivation for this construction is an application to graph rewriting."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/5/33-05abs.html", "title": "Contravariance through enrichment", "authors": "Michael Shulman", "keywords": ["opposite category", "contravariant functor", "generalized multicategory", "enriched category", "coherence theorem"], "abstract": "We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution.  For this purpose we introduce \"2-categories with contravariance\", a sort of enhanced 2-category with a basic notion of \"contravariant morphism\", which can be regarded either as generalized multicategories or as enriched categories.  This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/4/33-04abs.html", "title": "Dualizing cartesian and cocartesian fibrations", "authors": "Clark Barwick, Saul Glasman and Denis Nardin", "keywords": ["cocartesian fibrations", "cartesian fibrations", "quasicategories"], "abstract": "In this technical note, we proffer a very explicit construction of the dual cocartesian fibration of a cartesian fibration, and we show they are classified by the same functor to the $\\infty$-category of $\\infty$-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/3/33-03abs.html", "title": "On fibrant objects in model categories", "authors": "Valery Isaev", "keywords": ["Quillen model structures", "fibrant objects"], "abstract": "In this paper, we study properties of maps between fibrant objects in model categories. We give a characterization of weak equivalences between fibrant objects. If every object of a model category is fibrant, then we give a simple description of a set of generating cofibrations. We show that to construct such a model structure it is enough to check some relatively simple conditions."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/2/33-02abs.html", "title": "Actor of a crossed module of Leibniz algebras", "authors": "José Manuel Casas, Rafael Fernández-Casado, Xabier García-Martínez, Emzar Khmaladze", "keywords": ["Leibniz algebra", "crossed module", "representation", "actor"], "abstract": "We extend to the category of crossed modules of Leibniz algebras the notion of biderivation via the action of a Leibniz algebra. This results into a pair of Leibniz algebras which allow us to construct an object which is the actor under certain circumstances. Additionally, we give a description of an action in the category of crossed modules of Leibniz algebras in terms of equations. Finally, we check that, under the aforementioned conditions, the kernel of the canonical map from a crossed module to its actor coincides with the center and we introduce the notions of crossed module of inner and outer biderivations."},
{"url": "http://www.tac.mta.ca/tac/volumes/33/1/33-01abs.html", "title": "Spans of cospans in a topos", "authors": "Daniel Cicala and Kenny Courser", "keywords": ["bicategory", "graph rewrite", "network", "span", "symmetric monoidal", "topos"], "abstract": "For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph))."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/46/34-46abs.html", "title": "Categories of weak fractions", "authors": "Pierre-Alain Jacqmin", "keywords": ["split monomorphism", "locally posetal 2-category", "category of fractions", "weak reflection", "weak adjoint", "injective subcategory problem"], "abstract": "Given a set $\\Sigma$ of morphisms in a category C, we construct a functor F which sends elements of $\\Sigma$ to split monomorphisms. Moreover, we prove that F is weakly universal with that property when considered in the world of locally posetal 2-categories. Besides, we also use locally posetal 2-categories in order to construct weak left adjoints to those functors for which any object in the codomain admits a weak reflection. We then apply these two results in order to restate the Injective Subcategory Problem for $\\Sigma$ into the existence of some kind of weak right adjoint for F."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/45/34-45abs.html", "title": "Models of Linear Logic based on the Schwartz $\\varepsilon$-product", "authors": "Yoann Dabrowski and Marie Kerjean", "keywords": ["Topological vector spaces", "$\\ast$-autonomous and dialogue categories", "differential linear logic"], "abstract": "From the interpretation of Linear Logic multiplicative disjunction as the epsilon product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on the usual mathematical notions of smooth maps. This improves on previous results by Blute, Ehrhard and Tasson based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion which is stable under duality called k-reflexivity, allowing for a star-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the epsilon-product. We adapt Meise's definition of smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same star-autonomous categories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the epsilon-product (without reflexivization)."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/44/34-44abs.html", "title": "The dual of compact ordered spaces is a variety", "authors": "Marco Abbadini", "keywords": ["compact ordered space", "variety", "duality", "axiomatizability"], "abstract": "In a recent paper (2018), D. Hofmann, R. Neves and P.  Nora proved that the dual of the category of compact ordered spaces and monotone continuous maps is a quasi-variety -not finitary, but bounded by $\\aleph_1$. An open question was: is it also a variety? We show that the answer is affirmative. We describe the variety by means of a set of finitary operations, together with an operation of countably infinite arity, and equational axioms. The dual equivalence is induced by the dualizing object [0,1]."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/43/34-43abs.html", "title": "Products and coequalizers in pointed categories", "authors": "Michael Hoefnagel", "keywords": ["product preserves coequalizers", "stability of coequalizers under product", "product commutes with coequalizers", "normal projections", "shifting lemma"], "abstract": "In this paper, we investigate the property (P) that binary products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well as any pointed factor permutable category with coequalizers. We establish a Mal'tsev term condition characterizing pointed varieties of universal algebras satisfying (P). We then consider categories satisfying (P) locally, i.e., those categories for which every fibre of the fibration of points satisfies (P). Examples include any regular Mal'tsev or majority category with coequalizers, as well as any regular Gumm category with coequalizers. Varieties satisfying (P) locally are also characterized by a Mal'tsev term condition, which turns out to be equivalent to a variant of Gumm's shifting lemma. Furthermore, we show that the varieties satisfying (P) locally are precisely the varieties with normal local projections in the sense of Z. Janelidze."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/40/34-40abs.html", "title": "A unified framework for notions of algebraic theory", "authors": "Soichiro Fujii", "keywords": ["Algebraic theories", "clones", "operads", "double limits"], "abstract": "Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called {notions of algebraic theory}. In this paper, we develop a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called {metamodel}, which formalises a definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structure-semantics adjointness result and a double categorical universal property of categories of models are also shown."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/42/34-42abs.html", "title": "Free globularly generated double categories I", "authors": "Juan Orendain", "keywords": ["Category", "double category", "bicategory", "von Neumann algebra", "representation", "fusion", "tensor category", "length"], "abstract": "This is the first part of a two paper series studying free globularly generated double categories. In this first installment we introduce the free globularly generated double category construction. The free globularly generated double category construction canonically associates to every bicategory together with a possible category of vertical morphisms, a double category fixing this set of initial data in a free and minimal way. We use the free globularly generated double category to study length, free products, and problems of internalization. We use the free globularly generated double category construction to provide formal functorial extensions of the Haagerup standard form construction and the Connes fusion operation to inclusions of factors of not-necessarily finite Jones index."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/41/34-41abs.html", "title": "Distributive laws between the Three Graces", "authors": "Murray Bremner and Martin Markl", "keywords": ["Algebraic operads", "distributive laws", "Koszul duality", "associative algebras", "commutative associative algebras", "Lie algebras", "Poisson algebras", "linear algebra over polynomial rings", "Gr\\\"obner bases for polynomial ideals", "computer algebra"], "abstract": "By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and  Groebner."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/38/34-38abs.html", "title": "The Zassenhaus lemma in star-regular categories", "authors": "Olivette Ngaha Ngaha and Florence Sterck", "keywords": ["Factorisation systems", "ideal of morphisms", "normal category", "star-regular category", "ideal determined category", "good theory of ideals", "isomorphism theorems", "Zassenhaus lemma", "cocommutative Hopf algebra"], "abstract": "The Noether Isomorphism Theorems and the Zassenhaus Lemma from group theory have a non-pointed version in a suitable categorical context first considered by W. Tholen in his PhD thesis. This article leads to a unification of these results with the ones in the pointed categorical context previously considered by O.~Wyler, by working in the framework of \\emph{star-regular} categories introduced by M.~Gran, Z.~Janelidze and A.~Ursini. Some concrete examples of categories where these results hold are examined in detail."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/37/34-37abs.html", "title": "Finitely Presentable Algebras For Finitary Monads", "authors": "J. Adamek, S. Milius, L. Sousa and T. Wissmann", "keywords": ["Finitely presentable object", "finitely generated object", "finitary functor", "regular monad"], "abstract": "For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by finitely many generators and finitely many relations."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/39/34-39abs.html", "title": "A model structure on prederivators for (∞,1)-categories", "authors": "D. Fuentes-Keuthan, M. Kedziorek and M. Rovelli", "keywords": ["rederivator", "model structure", "(∞,1)-category", "quasi-category"], "abstract": "By theorems of Carlson and Renaudin, the theory of (∞,1)-categories embeds in that of prederivators. The purpose of this paper is to give a two-fold answer to the inverse problem: understanding which prederivators model (∞,1)-categories, either strictly or in a homotopical sense. First, we characterize which prederivators arise on the nose as prederivators associated to quasicategories. Next, we put a model structure on the category of prederivators and strict natural transformations, and prove a Quillen equivalence with the Joyal model structure for quasicategories."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/36/34-36abs.html", "title": "A note on internal object action representability of 1-cat groups and  crossed modules", "authors": "Pako Ramasu", "keywords": ["crossed module", "action of an object", "$ 1 $-cat group", "internal automorphism", "actor"], "abstract": "The category of 1-cat groups, which is equivalent to the category of crossed modules, has internal object actions which are representable (by internal automorphism groups). Moreover, it is known that the crossed module, corresponding to the representing object  [X] = Aut(X) $ associated with a 1-cat group X, must be isomorphic to the Norrie actor of the crossed module corresponding to X. We recall the description of Aut(X)  from the author's PhD thesis, and construct that isomorphism explicitly."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/35/34-35abs.html", "title": "On Finitary Functors", "authors": "Jiri Adamek, Stefan Milius, Lurdes Sousa and Thorsten Wissmann", "keywords": ["Finitely presentable object", "finitely generatd object", "(strictly) locally finitely presentable category", "finitary functor", "finitely bounded functor"], "abstract": "A simple criterion for a functor to be finitary is presented: we call F finitely bounded if for all objects X every finitely generated subobject of FX factorizes through the F-image of a finitely generated subobject of X. This is equivalent to F being finitary for all functors between `reasonable' locally finitely presentable categories, provided that F preserves monomorphisms. We also discuss the question when that last assumption can be dropped.  The answer is affirmative for functors between categories such as Set, K-Vec (vector spaces), boolean algebras, and actions of any finite group either on Set or on K-Vec for fields K of characteristic 0.\nAll this generalizes to locally $\\lambda$-presentable categories, $\\lambda$-accessible functors and $\\lambda$-presentable algebras. As an application we obtain an easy proof that the Hausdorff functor on the category of complete metric spaces is $\\aleph_1$-accessible."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/34/34-34abs.html", "title": "Weak $\\infty$-categories via terminal coalgebras", "authors": "Eugenia Cheng and Tom Leinster", "keywords": ["$\\infty$-category", "$\\omega$-category", "n-category", "higher category", "terminal coalgebra", "final coalgebra"], "abstract": "Higher categorical structures are often defined by induction on dimension, which a priori produces only finite-dimensional structures.  In this paper we show how to extend such definitions to infinite dimensions using the theory of terminal coalgebras, and we apply this method to Trimble's notion of weak n-category.  Trimble's definition makes explicit the relationship between n-categories and topological spaces; our extended theory produces a definition of Trimble $\\infty$-category and a fundamental $\\infty$-groupoid construction.\nFurthermore, terminal coalgebras are often constructed as limits of a certain type.  We prove that the theory of Batanin - Leinster weak $\\infty$-categories arises as just such a limit, justifying our approach to Trimble $\\infty$-categories.  In fact we work at the level of monads for $\\infty$-categories, rather than $\\infty$-categories themselves; this requires more sophisticated technology but also provides a more complete theory of the structures in question."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/33/34-33abs.html", "title": "The tangent bundle of a model category", "authors": "Yonatan Harpaz, Joost Nuiten, Matan Prasma", "keywords": ["Tangent category", "model category", "model fibration", "spectrum"], "abstract": "This paper studies the homotopy theory of parametrized spectrum objects in a model category from a global point of view. More precisely, for a model category $M$ satisfying suitable conditions, we construct a map of model categories $TM \\to M$, called the tangent bundle, whose fiber over an object in $M$ is a model category for spectra in its over-category. We show that the tangent bundle is a relative model category and presents the $\\infty$-categorical tangent bundle, as constructed by Lurie. Moreover, the tangent bundle $TM$ inherits an enriched model structure from $M$. This additional structure is used in subsequent work to identify the tangent bundles of algebras over an operad and of enriched categories, but may be of independent interest."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/32/34-32abs.html", "title": "Characterization of left coextensive varieties of universal algebras", "authors": "David Neal Broodryk", "keywords": ["Coextensivity", "Universal Algebra", "Syntactic Characterization"], "abstract": "An extensive category can be defined as a category C with finite coproducts such that for each pair X,Y of objects in C, the canonical functor $+ : C/X \\times C/Y \\to C/(X + Y)$ is an equivalence. We say that a category C with finite products is left coextensive if the dual canonical functor $\\times :  X/C \\times Y/C \\to (X \\times Y)/C$ is fully faithful. We then give a syntactical characterization of left coextensive varieties of universal algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/31/34-31abs.html", "title": "Adjunction up to automorphism", "authors": "D. Tambara", "keywords": ["distributor", "adjoint", "nearly representable"], "abstract": "We say a set-valued functor on a category is nearly representable if it is a quotient of a representable functor by a group of automorphisms. A distributor is a set-valued functor in two arguments, contravariant in one argument and covariant in the other. We say a distributor is slicewise nearly representable if it is nearly representable in either of the arguments whenever the other argument is fixed. We consider such a distributor a weak analogue of adjunction. Under a finiteness assumption on the domain categories, we show that every slicewise nearly representable functor is a composite of two distributors, each of which may be considered as a weak analogue of (co-)reflective adjunction."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/30/34-30abs.html", "title": "Biproducts and commutators for noetherian forms", "authors": "Francois Koch Van Niekerk", "keywords": ["Forms", "product", "commutators", "Semi-abelian categories"], "abstract": "Noetherian forms provide an abstract self-dual context in which one can establish homomorphism theorems (Noether isomorphism theorems and homological diagram lemmas) for groups, rings, modules and other group-like structures. In fact, any semi-abelian category in the sense of G. Janelidze, L. M\\arki and W. Tholen, as well as any exact category in the sense of M. Grandis (and hence, in particular, any abelian category), can be seen as an example of a noetherian form. In this paper we generalize the notion of a biproduct of objects in an abelian category to a noetherian form and apply it do develop commutator theory in noetherian forms. In the case of semi-abelian categories, biproducts give usual products of objects and our commutators coincide with the so-called Huq commutators (which in the case of groups are the usual commutators of subgroups). This paper thus shows that the structure of a noetherian form allows for a self-dual approach to products and commutators in semi-abelian categories, similarly as has been known for homomorphism theorems."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/29/34-29abs.html", "title": "Weak units, universal cells, and coherence via universality for bicategories", "authors": "Amar Hadzihasanovic", "keywords": ["bicategories", "polycategories", "multicategories", "merge-bicategories", "polygraphs", "coherence", "strictification", "weak units"], "abstract": "Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces coherent algebraic structure on the 2-graph of single-input, single-output 2-cells. A special case of this theory was used by Hermida to produce a proof of strictification for bicategories. No full strictification is possible for higher-dimensional categories, seemingly due to problems with 2-cells that have degenerate boundaries; it was conjectured by C. Simpson that semi-strictification excluding units may be possible.\nWe study poly-bicategories where 2-cells with degenerate boundaries are barred, and show that we can recover the structure of a bicategory through a different construction of weak units. We prove that the existence of these units is equivalent to the existence of 1-cells satisfying lower-dimensional universal properties, and study the relation between preservation of units and universal cells.\nThen, we introduce merge-bicategories, a variant of poly-bicategories with more composition operations, which admits a natural monoidal closed structure, giving access to higher morphisms. We derive equivalences between morphisms, transformations, and modifications of representable merge-bicategories and the corresponding notions for bicategories. Finally, we prove a semi-strictification theorem for representable merge-bicategories with a choice of composites and units."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/27/34-27abs.html", "title": "Homotopy theory with *-categories", "authors": "Ulrich Bunke", "keywords": ["*-categories", "model categories", "$\\infty$-categories", "limits and colimits"], "abstract": "We construct model category structures on various types of (marked) *-categories. These structures are used to present the infinity categories of (marked) *-categories obtained by inverting (marked) unitary equivalences. We use this presentation to explicitly calculate the $\\infty$-categorical G-fixed points and G-orbits for G-equivariant (marked) *-categories."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/26/34-26abs.html", "title": "The genuine operadic nerve", "authors": "Peter Bonventre", "keywords": ["infinity operads", "equivariant operads", "symmetric monoidal categories"], "abstract": "We construct a generalization of the operadic nerve, providing a translation between the equivariant simplicially enriched operadic world to the parametrized $\\infty$-categorical perspective. This naturally factors through genuine equivariant operads, a model for ``equivariant operads with norms up to homotopy''. We introduce the notion of an op-fibration of genuine equivariant operads, extending Grothendieck op-fibrations, and characterize fibrant operads as the image of genuine equivariant symmetric monoidal categories. Moreover, we show that under the operadic nerve, this image is sent to G-symmetric monoidal G-$\\infty$-categories. Finally, we produce a functor comparing the notion of algebra over an operad in each of these two contexts."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/25/34-25abs.html", "title": "Monic skeleta, Boundaries, Aufhebung, and the meaning of `one-dimensionality'", "authors": "Matias Menni", "keywords": ["Topos theory", "Axiomatic Cohesion"], "abstract": "Let E be a topos. If l is a level of E with monic skeleta then it makes sense to consider the objects in E that have  l-skeletal boundaries. In particular, if p : E \\to S is a pre-cohesive geometric morphism then its centre (that may be called level 0) has monic skeleta. Let level 1 be the Aufhebung of level 0. We show that if level 1 has monic skeleta then the quotients of 0-separated objects with 0-skeletal boundaries are 1-skeletal. We also prove that in several examples (such as the classifier of non-trivial Boolean algebras, simplicial sets and the classifier of strictly bipointed objects) every 1-skeletal object is of that form."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/24/34-24abs.html", "title": "A universal characterisation of codescent objects", "authors": "Alexander S. Corner", "keywords": ["Codescent object", "pseudo-extranatural", "Fubini"], "abstract": "In this work we define a 2-dimensional analogue of extranatural transformation and use these to characterise codescent objects. They will be seen as universal objects amongst pseudo-extranatural transformations in a similar manner in which coends are universal objects amongst extranatural transformations. Some composition lemmas concerning these transformations are introduced and a Fubini theorem for codescent objects is proven using the universal characterisation description."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/22/34-22abs.html", "title": "Strictification tensor product of 2-categories", "authors": "Branko Nikolic", "keywords": ["Lax functor", "strictification", "distributive law", "lax Gray product", "free  monoid"], "abstract": "Given 2-categories C and D, let Lax (C,D) denote the 2-category of lax functors, lax natural transformations and modifications, and [C,D]_lnt its full sub-2-category of (strict) 2-functors. We give two isomorphic constructions of a 2-category C \\boxtimes D satisfying Lax (C, Lax(D,E)) \\cong [C \\boxtimes D, E}_lnt, hence generalising the case of the free distributive law 1 \\boxtimes 1. We also discuss dual constructions."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/21/34-21abs.html", "title": "Left-invariant vector fields on a Lie 2-group", "authors": "Eugene Lerman", "keywords": ["Lie 2-group", "Lie 2-algebra", "invariant vector fields", "2 limit"], "abstract": "A Lie 2-group G is a category internal to the category of Lie groups.   Consequently it is a monoidal category and a Lie groupoid.  The Lie groupoid structure on G gives rise to the Lie 2-algebra X(G) of multiplicative vector fields.  The monoidal structure on G gives rise to a left action of the 2-group G on the Lie groupoid G, hence to an action of G on the Lie 2-algebra X(G).  As a result we get the Lie 2-algebra X(G)^G of left-invariant multiplicative vector fields.\nOn the other hand there is a well-known construction that associates a Lie 2-algebra g to a Lie 2-group G: apply the functor Lie :  LieGp -> LieAlg to the structure maps of the category G.  We show that the Lie 2-algebra g is isomorphic to the Lie 2-algebra X(G)^G of left invariant multiplicative vector fields."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/23/34-23abs.html", "title": "Coequalizers and free triples, II", "authors": "Michael Barr, John Kennison, and R. Raphael", "keywords": ["Free triples", "coequalizers", "T-horns", "ordinal sequences"], "abstract": "This paper studies a category X with an endofunctor T : X \\to X. A T-algebra is given by a morphism Tx \\to x in X.  We examine the related questions of when T freely generates a triple (or monad) on X; when an object x in X freely generates a T-algebra; and when the category of T-algebras has coequalizers and other colimits.  The paper defines a category of ``T-horns'' which effectively contains X as well as all T-algebras.  It is assume that  Xs  is cocomplete and has a factorization system (E,M) satisfying reasonable properties. An ordinal-indexed sequence of T-horns is then defined which provides successive approximations to a free T-algebra generated by an object x in X, as well as approximations to coequalizers and other colimits for the category of T-algebras.  Using the notions of an M-cone and a separated T-horn it is shown that if X is M-well-powered, then the ordinal sequence stabilizes at the desired free algebra or coequalizer or other colimit whenever they exist.  This paper is a successor to a paper written by the first author in 1970 that showed that T generates a free triple when every x in X generates a free T-algebra.  We also consider colimits in triple algebras and give some examples of functors T for which no x in X generates a free T-algebra."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/19/34-19abs.html", "title": "Fibred pseudo double categories for game semantics", "authors": "Clovis Eberhart, Tom Hirschowitz", "keywords": ["concurrent game semantics", "pseudo double categories", "factorisation systems"], "abstract": "We unify previous constructions from our work on concurrent game   semantics into a single categorical framework. From an operational   description of positions and moves in some game, called a   \\emph{signature}, we produce a pseudo double category, in which   objects are positions and vertical morphisms are plays. The   considered games are multi-player, so it makes sense to consider   embeddings of positions: these are the horizontal   morphisms. Finally, cells may be thought of as embeddings of plays   preserving initial and final positions. In order to be suitable for   game semantics, the obtained pseudo double category should enjoy a   certain fibredness property. Under suitable hypotheses, we show that   our construction actually produces such a \\emph{fibred} pseudo   double category, from which we can define relevant categories of   plays, and thus of strategies. We give a first necessary and   sufficient criterion for this to hold and then a sufficient   criterion that can be checked more easily."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/16/34-16abs.html", "title": "A note on the categorical congruence distributivity", "authors": "Dominique Bourn", "keywords": ["Suprema of equivalence relations", "congruence modularity", "congruence distributivity"], "abstract": "Having given a characterization of the categorical congruence modularity getting rid of the assumption that the ground category is regular, we give now a characterization of the categorical congruence distributivity. We have a look as well at the case where the congruence distributivity is only involved, in some sense, for a subclass $\\Gamma$ of equivalence relations."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/20/34-20abs.html", "title": "Quotient categories and phases", "authors": "Sean Tull", "keywords": ["Phased coproduct", "phased biproduct", "quotient category", "phase", "global  phase"], "abstract": "We study properties of a category after quotienting out a suitable chosen group of isomorphisms on each object. Coproducts in the original category are described in its quotient by our new weaker notion of a `phased coproduct'. We examine these and show that any suitable category with them arises as such a quotient of a category with coproducts. Motivation comes from projective geometry, and also quantum theory where they describe superpositions in the category of Hilbert spaces and continuous linear maps up to global phase. The quotients we consider also generalise those induced by categorical isotropy in the sense of Funk et al."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/18/34-18abs.html", "title": "Limits in dagger categories", "authors": "Chris Heunen and Martti Karvonen", "keywords": ["Dagger category", "limit", "adjoint functors"], "abstract": "We develop a notion of limit for dagger categories, that we show is suitable in the following ways:  it subsumes special cases known from the literature;  dagger limits are unique up to unitary isomorphism;  a wide class of dagger limits can be built from a small selection of them;   dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor;  dagger limits can be built from ordinary limits in the presence of polar decomposition;  dagger limits commute with dagger colimits in many cases."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/17/34-17abs.html", "title": "Colimits  of monoids", "authors": "Hans-E. Porst", "keywords": ["Monoids in monoidal categories", "(reflexive) coequalizers", "(regularly)   monadic functors", "monoidal functors"], "abstract": "If C is a monoidal category with reflexive coequalizers which are preserved by tensoring from both sides, then the category MonC of monoids over C has all coequalizers and these are regular epimorphisms in C. This implies that MonC has all colimits which exist in C, provided that C in addition has (regular epi, jointly monomorphic)-factorizations of discrete cones and admits arbitrary free monoids. A further application is a lifting theorem for adjunctions with a monoidal right adjoint whose left adjoint is not necessarily strong to adjunctions between the respective categories of monoids."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/14/34-14abs.html", "title": "Well-closed subschemes of noncommutative schemes", "authors": "D. Rogalski", "keywords": ["Grothendieck category", "noncommutative blowing up", "adjoint functors", "locally noetherian", "closed subcategory"], "abstract": "Van den Bergh has defined the blowup of a noncommutative surface at a point lying on a commutative divisor. We study one aspect of the construction, with an eventual aim of defining more general kinds of noncommutative blowups. Our basic object of study is a quasi-scheme X (a Grothendieck category). Given a closed subcategory Z, in order to define a blowup of X along Z one first needs to have a functor F_Z which is an analog of tensoring with the defining ideal of Z.  Following Van den Bergh, a closed subcategory Z which has such a functor is called well-closed.  We show that well-closedness can be characterized by the existence of certain projective effacements for each object of X, and that the needed functor F_Z has an explicit description in terms of such effacements.  As an application, we prove that closed points are well-closed in quite general quasi-schemes."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/15/34-15abs.html", "title": "Affine geometric spaces in tangent categories", "authors": "R. F. Blute, G. S. H. Cruttwell,  and R. B. B. Lucyshyn-Wright", "keywords": ["Tangent categories", "affine manifolds", "connections"], "abstract": "We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds.  In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free connection on its tangent bundle. This equivalence allows us to define a category of affine objects associated to a tangent category and we show that the resulting category is also a tangent category, as are several related categories.   As a consequence of some of these ideas we also give two new characterizations of flat torsion-free connections.  We also consider 2-categorical structure associated to the category of tangent categories and demonstrate that assignment of the tangent category of affine objects to a tangent category induces a 2-comonad."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/28/34-28abs.html", "title": "On spans with right fibred right adjoints", "authors": "J. R. A. Gray", "keywords": ["action representable", "locally algebraically cartesian closed", "semi-abelian", "split extension", "normalizer", "prefibration", "right fibred right adjoints", "regular span"], "abstract": "We introduce a new condition on an abstract span of categories which we refer to as having right fibred right adjoints, RFRA  for short. We show that:\n(a) the span of split extensions of a semi-abelian category C has RFRA if and only if C is action representable;   (b) the reversed span to the one considered in (a) has RFRA if and only if C is locally algebraically cartesian closed;   (c) the span of split extensions of the category of morphisms of C has RFRA if and only if C is action representable and has normalizers;   (d) the reversed span to the one considered in (c) has RFRA if and only if C is locally algebraically cartesian closed.\nWe also examine our condition for the span of monoid actions (of monoids in a monoidal category C on objects in a given category on which C acts), and for various other spans."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/13/34-13abs.html", "title": "The operadic nerve, relative nerve and the Grothendieck construction", "authors": "Jonathan Beardsley and Liang Ze Wong", "keywords": ["simplicial categories", "Grothendieck construction", "higher category theory", "operads"], "abstract": "We relate the relative nerve $N_f(D)$ of a diagram of simplicial sets $f : D \\to sSet$ with the Grothendieck construction $Gr F$ of a simplicial functor $F : D \\to sCat$ in the case where $f = N F$.  We further show that any strict monoidal simplicial category $C$ gives rise to a functor $C^\\bullet : \\Delta^\\op \\to sCat$, and that the relative nerve of $\\N C^\\bullet$ is the operadic nerve $\\N^\\otimes(C)$.  Finally, we show that all the above constructions commute with appropriately defined opposite functors."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/12/34-12abs.html", "title": "The formal theory of multimonoidal monads", "authors": "Gabriella Bohm", "keywords": ["monoidal 2-category", "monoidal double category", "pseudomonoid", "(op)monoidal monad", "Eilenberg-Moore construction", "lifting"], "abstract": "Certain aspects of Street's formal theory of monads in 2-categories are extended to multimonoidal monads in symmetric strict monoidal 2-categories. Namely, any symmetric strict monoidal 2-category M admits a symmetric strict monoidal 2-category of pseudomonoids, monoidal 1-cells and monoidal 2-cells in M. Dually, there is a symmetric strict monoidal 2-category of pseudomonoids, opmonoidal 1-cells and opmonoidal 2-cells in M. Extending a construction due to Aguiar and Mahajan for M = Cat, we may apply the first construction p-times and the second one q-times (in any order). It yields a 2-category M_{pq}. A 0-cell therein is an object A of M together with p+q compatible pseudomonoid structures;  it is termed a (p+q)-oidal object in M. A monad in M_{pq} is called a (p,q)-oidal monad in M; it is a monad t on A in M together with p monoidal, and q opmonoidal structures in a compatible way. If M has monoidal Eilenberg-Moore construction, and certain (Linton type) stable coequalizers exist, then a (p+q)-oidal structure on the Eilenberg-Moore object A^t of a (p,q)-oidal monad (A,t) is shown to arise via a symmetric strict monoidal double functor to Ehresmann's double category Sqr(M) of squares in M, from the double category of monads in Sqr(M) in the sense of Fiore, Gambino and Kock. While q ones of the pseudomonoid structures of A^t are lifted along the `forgetful' 1-cell A^t -> A, the other p ones are lifted along its left adjoint. In the particular example when M is an appropriate 2-subcategory of Cat, this yields a conceptually different proof of some recent results due to Aguiar, Haim and Lopez Franco."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/9/34-09abs.html", "title": "Topoi of parametrized objects", "authors": "Marc Hoyois", "keywords": ["Higher topos theory"], "abstract": "We give necessary and sufficient conditions on a presentable $\\infty$-category $C$ so that families of objects of $C$ form an $\\infty$-topos. In particular, we prove a conjecture of Joyal that this is the case whenever $C$ is stable."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/11/34-11abs.html", "title": "Categorification of pre-Lie Algebras and solutions of 2-graded classical Yang-Baxter equations", "authors": "Yunhe Sheng", "keywords": ["2-algebras", "pre-Lie$_\\infty$-algebras", "Lie 2-algebras", "O-operators", "2-graded classical Yang-Baxter equations"], "abstract": "In this paper, we introduce the notion of a pre-Lie 2-algebra, which is the categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\\infty$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology group of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. O-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/10/34-10abs.html", "title": "Majority categories", "authors": "Michael Anton Hoefnagel", "keywords": ["arithmetical category", "internal relation", "Mal'tsev category", "majority category", "majority algebra", "majority term", "lattice", "protoarithmetical category"], "abstract": "We introduce the notion of a majority category - the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that parallels how Mal'tsev categories capture properties of the category of groups. Among algebraic majority categories are the categories of lattices, Boolean algebras and Heyting algebras. Many geometric categories such as the category of topological spaces, metric spaces, ordered sets, any topos, etc., are comajority categories (i.e.~their duals are majority categories), and we show that, under mild assumptions, the only categories which are both majority and comajority, are the preorders. Mal'tsev majority categories provide an alternative generalization of arithmetical categories to protoarithmetical categories in the sense of Bourn. We show that every Mal'tsev majority category is protoarithmetical, provide a counter-example for the converse implication, and show that in the Barr-exact context, the converse implication also holds. We can then conclude that a category is arithmetical if and only if it is a Barr-exact Mal'tsev majority category, recovering in the varietal context a well known result of Pixley."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/7/34-07abs.html", "title": "A probability monad as the colimit of spaces of finite samples", "authors": "Tobias Fritz and Paolo Perrone", "keywords": ["Categorical probability", "Giry monad", "graded monad", "optimal transport", "Wasserstein spaces", "Kantorovich-Rubinstein distance", "monoidal Kan extension"], "abstract": "We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces.\nWe prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limits of finite samples. The proof relies on a criterion for when an ordinary left Kan extension of lax monoidal functors is a monoidal Kan extension. The colimit characterization allows the development of integration theory and the treatment of measures on spaces of measures, without measure theory.\nWe also show that the category of algebras of the Kantorovich monad is equivalent to the category of closed convex subsets of Banach spaces with short affine maps as morphisms."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/8/34-08abs.html", "title": "Cluster-tilting subcategories in extriangulated categories", "authors": "Panyue Zhou and Bin Zhu", "keywords": ["Extriangulated categories", "Cluster-tilting subcategories", "Rigid subcategories", "Maximal rigid subcategories", "Quotient categories"], "abstract": "Let $(C,E,s)$ be an extriangulated category. We show that certain quotient categories of extriangulated categories are equivalent to module categories by some restriction of functor $E$, and in some cases, they are abelian.  This result can be regarded as a simultaneous generalization of Koenig-Zhu and Demonet-Liu. In addition, we introduce the notion of maximal rigid subcategories in extriangulated categories.  Cluster tilting subcategories are obviously strongly functorially finite maximal rigid subcategories, we prove that the converse is true if the 2-Calabi-Yau extriangulated categories admit a cluster tilting subcategories, which generalizes a result of Buan-Iyama-Reiten-Scott and Zhou-Zhu."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/6/34-06abs.html", "title": "Six operations formalism for generalized operads", "authors": "Benjamin C. Ward", "keywords": ["Operads", "modular operads", "graph complexes", "six-operations formalism", "Koszul duality"], "abstract": "This paper shows that generalizations of operads equipped with their respective bar/cobar dualities are related by a six operations formalism analogous to that of classical contexts in algebraic geometry.  As a consequence of our constructions, we prove intertwining theorems which govern derived Koszul duality of push-forwards and pull-backs."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/5/34-05abs.html", "title": "Distributors and the comprehensive factorization system for internal groupoids", "authors": "Giuseppe Metere", "keywords": ["distributor", "profunctor", "factorization system", "internal groupoid"], "abstract": "In this note we prove that distributors between groupoids in a Barr-exact category E form the bicategory of relations relative to the comprehensive factorization system in Gpd(E). The case E = Set is of special interest."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/4/34-04abs.html", "title": "Lifting bicategories into double categories: The globularily generated condition", "authors": "Juan Orendain", "keywords": ["categories", "double categories", "bicateogries", "von Neumann algebras", "cobordisms", "algebras", "bimodules"], "abstract": "This is the first part of a series of papers studying the problem of existence of double categories for which horizontal bicategory and object category are given. We refer to this problem as the problem of existence of internalizations for decorated bicategories. Motivated by this we introduce the condition of a double category being globularily generated. We prove that the problem of existence of internalizations for a decorated bicategory admits a solution if and only if it admits a globularily generated solution, and we prove that the condition of a double category being globularily generated is precisely the condition of a solution to the problem of existence of internalizations for a decorated bicategory being minimal in a sense which we will make precise. The study of the condition of a double category being globularily generated will thus be pivotal in our study of the problem of existence of internalizations."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/3/34-03abs.html", "title": "On the relative projective space", "authors": "Matias Data and Juliana Osorio", "keywords": ["symmetric monoidal category", "algebra object", "line object", "relative scheme"], "abstract": "Let $(C,\\otimes,1)$ be an abelian symmetric monoidal category satisfying certain exactness conditions. In this paper we define a presheaf $Proj{C}$ on the category of commutative algebras in $C$ and we prove that this functor is a $C$-scheme in the sense of B. Toen and M. Vaquie. We give another definition and prove that they give isomorphic $C$-schemes. This construction gives us a context of non-associative relative algebraic geometry. The most important example of the construction is the octonionic projective space."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/1/34-01abs.html", "title": "Lifting PIE limits with strict projections", "authors": "Martin Szyld", "keywords": ["PIE limit", "2-monad", "2-algebra", "2-category"], "abstract": "We give a unified direct proof of the lifting of PIE limits to the 2-category of algebras and (pseudo) morphisms, which specifies precisely which of the projections of the lifted limit are strict and detect strictness. In the literature, these limits were lifted one by one, so as to keep track of these projections in each case. We work in the more general context of weak algebra morphisms, so as to include lax morphisms as well. PIE limits are also all simultaneously lifted in this case, provided some specified arrows of the diagram are pseudo morphisms. Again, this unifies the previously known lifting of many particular PIE limits, which were also treated separately."},
{"url": "http://www.tac.mta.ca/tac/volumes/34/2/34-02abs.html", "title": "Involutive categories, colored *-operads and quantum field theory", "authors": "Marco Benini, Alexander Schenkel and Lukas Woike", "keywords": ["involutive categories", "involutive monoidal categories", "*-monoids", "colored operads", "*-algebras", "algebraic quantum field theory"], "abstract": "Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored *-operads and *-algebras. Central to the definition of colored *-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential 2-adjunction whose right adjoint forms involutive functor categories. For *-algebras over *-operads we obtain involutive analogs of the usual change of color and operad adjunctions. As an application, we turn the colored operads for algebraic quantum field theory into colored *-operads. The simplest instance is the associative *-operad, whose *-algebras are unital and associative *-algebras."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/34/35-34abs.html", "title": "An intrinsic approach to the non-abelian tensor product via internal crossed squares", "authors": "Davide di Micco and Tim Van der Linden", "keywords": ["Semi-abelian category", "pair of compatible actions", "internal action", "crossed module", "crossed square", "commutator", "non-abelian tensor product"], "abstract": "We explain how, in the context of a semi-abelian category, the concept of an internal crossed square may be used to set up an intrinsic approach to the Brown-Loday non-abelian tensor product."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/33/35-33abs.html", "title": "Segal enriched categories and applications", "authors": "Hugo V. Bacard", "keywords": ["Segal categories", "enriched categories", "homotopy algebras", "higher categories"], "abstract": "In this paper we develop a theory of Segal  enriched categories. Our motivation was to generalize the notion of up-to-homotopy monoid in a monoidal category, introduced by Leinster. Our formalism generalizes the classical theory of Segal categories and extends the theory of categories enriched over a 2-category. We introduce Segal dg-categories which did not exist so far. We show that the homotopy transfer problem for algebras leads directly to a Leinster-Segal algebra."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/32/35-32abs.html", "title": "Exponentiability in Double Categories  and the Glueing Construction", "authors": "Susan Niefield", "keywords": ["double category", "exponentiability", "cartesian closed", "glueing"], "abstract": "We consider pre-exponentiable objects of a pre-cartesian double category D, i.e., objects Y such that the lax functor - x Y: D --> D has a right adjoint in the 2-category  LxDbl of double categories and lax functors. When  D has  2-glueing, we show that Y is pre-exponentiable in  D if and only if Y is exponentiable in  D_0 and - x Y is an oplax functor. Thus, such a D is pre-cartesian closed as a double category if and only if D_0 is a cartesian closed category. Applications include the double categories  cat, pos, spaces, loc, and topos, whose objects are small categories, posets, topological space, locales, and toposes, respectively."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/30/35-30abs.html", "title": "On the construction of limits and colimits in ∞-categories", "authors": "Emily Riehl and Dominic Verity", "keywords": ["infinity category", "limit", "colimit"], "abstract": "In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an ∞-category --- for instance, a quasi-category, a complete Segal space, or a Segal category --- is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly that the modules that express the universal properties of (co)limits of diagrams of these shapes are reconstructible as limits of the modules that express the universal properties of (co)limits of the restricted diagrams. We also prove that the Yoneda embedding preserves and reflects limits in a suitable sense, and deduce our main theorems as a consequence."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/29/35-29abs.html", "title": "Cohesive toposes of sheaves on monoids of continuous endofunctions of the unit interval", "authors": "Luis Jesús Turcio", "keywords": ["TAC", "Cohesion", "Topos theory"], "abstract": "We determine the largest submonoid of the monoid of continuous endomorphisms of the unit interval [0,1] on which the finite partitions form the basis of a Grothendieck topology, and thus determine a cohesive topos over sets. We analyze some of the sheaf theoretic aspects of this topos.    Furthermore, we adapt the constructions of Menni to include another model of axiomatic cohesion.  We conclude the paper with a proof of the fact that a sufficiently cohesive topos of presheaves does not satisfy the continuity axiom."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/31/35-31abs.html", "title": "Monoidal Grothendieck Construction", "authors": "Joe Moeller and Christina Vasilakopoulou", "keywords": ["Fibrations", "indexed categories", "Grothendieck construction", "monoidal 2-categories", "monoidal pseudofunctors"], "abstract": "We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely lax monoidal pseudofunctors to the 2-category of categories. Furthermore, we investigate the relation between this `global' monoidal version where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, we show how lax monoidal structures on a pseudofunctor to Cat bijectively correspond to lifts of the pseudofunctor to MonCat. Finally, we give some examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/27/35-27abs.html", "title": "Classification of Constructible Cosheaves", "authors": "Justin Curry and Amit Patel", "keywords": ["Constructible cosheaves", "entrance path category", "Reeb graphs", "Reeb spaces"], "abstract": "In this paper we prove an equivalence theorem originally observed by Robert MacPherson.  On one side of the equivalence is the category of cosheaves that are constructible with respect to a  locally cone-like stratification.  Our constructibility condition is new and only requires that certain inclusions of open sets are sent to isomorphisms. On the other side of the equivalence is the category of functors from the entrance path category,  which has points for objects and certain homotopy classes of paths for morphisms.  When our constructible cosheaves are valued in Set we prove an additional equivalence with  the category of stratified coverings."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/28/35-28abs.html", "title": "Higher Segal spaces via higher excision", "authors": "Tashi Walde", "keywords": ["Simplicial objects", "cyclic objects", "higher Segal objects", "excision", "manifold calculus"], "abstract": "We show that the various higher Segal conditions of Dyckerhoff and Kapranov   can all be characterized in purely categorical terms   by higher excision conditions   (in the spirit of Goodwillie--Weiss manifold calculus)   on the simplex category Δ and the cyclic category Λ."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/26/35-26abs.html", "title": "A recipe for black box functors", "authors": "Brendan Fong and Maru Sarazola", "keywords": ["decorated corelation", "Frobenius monoid", "hypergraph category", "black box functor", "well-supported compact closed category"], "abstract": "The task of constructing compositional semantics for network-style diagrammatic languages, such as electrical circuits or chemical reaction networks, has been dubbed the black boxing problem, as it gives semantics that describes the properties of each network that can be observed externally, through composition, while discarding the internal structure. One way to solve these problems is to formalise the diagrams and their semantics using hypergraph categories, with semantic interpretation a hypergraph functor, called the black box functor, between them.   Building on a previous method for constructing hypergraph categories and functors, known as decorated corelations, in this paper we construct a category of decorating data, and show that the decorated corelations method is itself functorial, with a universal property characterised by a left Kan extension. We then show that any hypergraph category can be presented in terms of decorating data, and hence argue that the category of decorating data is a good setting in which to construct any hypergraph functor. As an example, we give a new construction of Baez and Pollard's black box functor for reaction networks."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/25/35-25abs.html", "title": "Lax limits of model categories", "authors": "Yonatan Harpaz", "keywords": ["Lax limit", "model categories", "strictification"], "abstract": "For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying ∞-categories. Our approach can also allow for the indexing category to be simplicial, as long as the diagram factors through its homotopy category. Analogous results for the associated homotopy limit (and other intermediate limits) directly follow."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/24/35-24abs.html", "title": "Weak model categories in classical and constructive mathematics", "authors": "Simon Henry", "keywords": ["Model categories", "constructive mathematics", "simplicial sets", "semi-simplicial sets", "complicial sets"], "abstract": "We introduce a notion of \"weak model category\" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences, and most of the usual constructions of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category.\nThe main advantage of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems for them to form a weak model category. The theory is developed in a very weak constructive, even predicative, framework and we use it to give constructive proofs of the existence of  weak versions of various standard model categories, including the Kan-Quillen model structure, Lurie's variant of the Joyal model structure on marked simplicial sets, and the Verity model structure for weak complicial sets. We also construct semi-simplicial versions of all these."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/23/35-23abs.html", "title": "A localization of bicategories via homotopies", "authors": "Descotte M.E., Dubuc E.J., and Szyld M.", "keywords": ["localization", "bicategory", "homotopy"], "abstract": "Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C,W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows to C. We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to W, a notion defined in this article which is closely related to Quillen's notion of homotopy for model categories but depends only on a single family of arrows.   This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair (C_{fc},W) given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/22/35-22abs.html", "title": "Cocompletion of restriction categories", "authors": "Richard Garner and Daniel Lin", "keywords": ["restriction categories", "cocompletion", "presheaves", "locally small"], "abstract": "Restriction categories were introduced as a way of generalising the notion of partial map category. In this paper, we define a notion of  \tcocompleteness for restriction categories, and describe the free cocompletion of a small restriction category as a suitably defined category  \tof restriction presheaves. We also consider free cocompletions in the case where our restriction category is only locally small."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/21/35-21abs.html", "title": "The folk model category structure on strict ω-categories is monoidal", "authors": "Dimitri Ara and Maxime Lucas", "keywords": ["augmented directed complexes", "folk model category structure", "Gray   tensor product", "join", "locally biclosed monoidal categories", "monoidal model   categories", "oplax transformations", "strict ω-categories", "strict   ω-groupoids", "strict (m, n)-categories"], "abstract": "We prove that the folk model category structure on the category of strict   ω-categories, introduced by Lafont, Métayer and Worytkiewicz, is   monoidal, first, for the Gray tensor product and, second, for the join of   ω-categories, introduced by the first author and Maltsiniotis. We   moreover show that the Gray tensor product induces, by adjunction, a   tensor product of strict (m,n)-categories and that this tensor product   is also compatible with the folk model category structure. In particular,   we get a monoidal model category structure on the category of strict   ω-groupoids. We prove that this monoidal model category structure   satisfies the monoid axiom, so that the category of Gray monoids, studied   by the second author, bears a natural model category structure."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/4/35-04abs.html", "title": "The 2-Chu-Dialectica construction and the polycategory of multivariable  adjunctions", "authors": "Michael Shulman", "keywords": ["multivariable adjunction", "polycategory", "Dialectica construction", "Chu construction"], "abstract": "Cheng, Gurski, and Riehl constructed a cyclic double multicategory   of multivariable adjunctions.  We show that the same information is   carried by a poly double category, in which opposite categories are   polycategorical duals.  Moreover, this poly double category is a   full substructure of a double Chu construction, whose objects are a   sort of polarized category, and which is a natural home for   2-categorical dualities.\nWe obtain the double Chu construction using a general   \"Chu-Dialectica\" construction on polycategories, which includes   both the Chu construction and the categorical Dialectica   construction of de Paiva.  The Chu and Dialectica constructions each   impose additional hypotheses making the resulting polycategory   representable (hence *-autonomous), but for different reasons;   this leads to their apparent differences."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/20/35-20abs.html", "title": "Network Models", "authors": "John C. Baez, John Foley, Joe Moeller, and Blake S. Pollard", "keywords": ["Grothendieck construction", "graph", "monoidal category", "network", "operad"], "abstract": "Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce `network models' to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to Cat, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/3/35-03abs.html", "title": "Completely distributive enriched categories  are not always continuous", "authors": "Hongliang Lai and Dexue Zhang", "keywords": ["Enriched category", "continuous t-norm", "forward Cauchy weight", "distributive law", "completely distributive quantale-enriched category", "continuous quantale-enriched category"], "abstract": "In contrast to the   fact that every completely distributive lattice is necessarily continuous in the sense of  Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the  unit interval with a continuous t-norm does not imply its continuity in general. Necessary and sufficient conditions for the implication   are presented."},
{"url": "http://www.tac.mta.ca/tac/volumes/35/2/35-02abs.html", "title": "Braided skew monoidal categories", "authors": "John Bourke and Stephen Lack", "keywords": ["Braiding", "skew monoidal category", "bialgebra", "quasitriangular", "2-category"], "abstract": "We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two.  Examples are shown to arise from 2-category theory and from bialgebras.  In order to describe the 2-categorical examples, we take a multicategorical approach.  We explain how certain braided skew monoidal structures in the 2-categorical setting give rise to braided monoidal  bicategories. For the  bialgebraic examples, we show that, for a skew monoidal category arising from a bialgebra, braidings on the skew monoidal category are in bijection with cobraidings  (also known as coquasitriangular structures) on the bialgebra."}
]