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The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program of cartesian bicategories, the precise relationship between the double-categorical and bicategorical approaches has so far remained mysterious, except in special cases. We provide a formal connection by showing that every double category with iso-strong finite products, and in particular every cartesian equipment, has an underlying cartesian bicategory. To do so, we develop broadly applicable techniques for transposing natural transformations and adjunctions between double categories, extending a line of previous work rooted in the concepts of companions and conjoints., Comment: Improvements to the Introduction

In this paper, we exhibit a "Yoneda"-style embedding of any virtual equipment into the virtual equipment of categories enriched in it. We show that this embedding preserves composition, is full on 2-cells and arrows, and coreflective on proarrows., Comment: 21 pages. arXiv admin note: text overlap with arXiv:1612.02762

In 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. Any 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. An 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., Comment: This article comprises a streamlined and expanded version of Sections 1, 2 and 3 of arXiv:1511.04070. v2 contains several improvements following the referee's suggestions. This is the final version as published in TAC. v3: minor corrections in 1.6, 1.9, 2.8, 7.1, 7.7, 9.4, 9.6

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., Comment: 18 pages

The scientific and practical needs of the twenty-first century lead humankind to convergence of the specialized and diverse branches of science and technology. This convergence reveals the need for new mathematical theories capable of providing common languages and frameworks to be utilized by professionals from different fileds in solving interdisciplinary and challenging problems. The present thesis is done in the same direction. Here, we develop a new formalism with the central idea of "unification of various mathematical branches". For this purpose, we utilize three major tools from today's mathematics, each of which possessing a unifying nature itself: category theory and especially the theory of "double cateogries", the theory of "universal dialgebra", and the "Chu construction". With the aid of these tools, we define and study a double category that subsumes a significant portion of the formalisms usual within the body of mathematics and theoretical computer science. We show that this double category possesses the properties of "horizontal self-duality" and "vertical self-duality". Also, we perform a primary investigation about existence of binary horizontal products and coproducts in this category. Finally, we give some suggestions for future work., Comment: 141 pages

This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful., Comment: v1: 20 pages, many diagrams. v2: updated after referee comments, 32 pages. v3: one minor terminology change, one proof streamlined after additional feedback, 32 pages; v4: minor corrections, final version to appear in journal

We establish the universal properties of the bicategory of polynomials, considering both cartesian and general morphisms between these polynomials. A direct proof of these universal properties would be impractical due to the complicated coherence conditions arising from polynomial composition; however, in this paper we avoid most of these coherence conditions using the properties of generic bicategories. In addition, we give a new proof of the universal properties of the bicategory of spans, and also establish the universal properties of the bicategory of spans with invertible 2-cells; showing how these properties may be used to describe the universal properties of polynomials., Comment: 48 pages; final author version incorporating the suggestions of the anonymous referee; to appear in the Journal of Pure and Applied Algebra

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. We 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. Then, 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., Comment: 82 pages. v2: revised following round of review; changed terminology (divisible->universal). v3: minor changes, published version

A short introduction to Grothendieck weak omega-groupoids is given. Our aim is to give evidence that, in certain contexts, this simple language is a convenient one for constructing globular weak omega-groupoids. To this end, we give a short reworking of van den Berg and Garner's construction of a Batanin weak omega-groupoid from a type using the language of Grothendieck weak omega-groupoids., Comment: 10 pages. Slightly expanded final version. Cahiers de Topologie et Geometrie Differentielle Categoriques Volume LVII (2016)

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., Comment: 25 pages. Version 2 has updated reference in bibliography. Paper has been published

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., Comment: 49 pages

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. We 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., Comment: Fixed typo, published to TAC

The behaviour of limits of weak morphisms in 2-dimensional universal algebra is not 2-categorical in that, to fully express the behaviour that occurs, one needs to be able to quantify over strict morphisms amongst the weaker kinds. F-categories were introduced to express this interplay between strict and weak morphisms. We express doctrinal adjunction as an F-categorical lifting property and use this to give monadicity theorems, expressed using the language of F-categories, that cover each weaker kind of morphism., Comment: v2: final journal version, some technical improvements to Section 4 which enabled the removal of unnecessary hypotheses (concerning cotensors with 2 and pullbacks) from the main results

Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in \'etale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more generally the gluing of finitely many pseudo functors. Given pseudo functors $F_i\colon \mathcal{A}_i\to \mathcal{D}$ defined on sub-$2$-categories $\mathcal{A}_i$ of a $2$-category $\mathcal{C}$, we are concerned with the problem of finding pseudo functors $\mathcal{C}\to \mathcal{D}$ extending $F_i$ up to pseudo natural equivalences. With the help of $n$-fold categories, we organize gluing data for $n$ pseudo functors into $2$-categories. We establish general criteria for equivalence between such $2$-categories for $n$ pseudo functors and for $n-1$ pseudo functors, which can be applied inductively to the gluing problem. Results of this article are used in arXiv:1006.3810 to construct extraordinary direct image functors in \'etale cohomology of Deligne-Mumford stacks., Comment: 61 pages. This is an updated version of the appendix to arXiv:1006.3810v1. v6: DOI; v5: minor changes; v4: various improvements including change of notation and terminology suggested by the referee; v2: appendix in v1 removed

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., Comment: 20 pages. Revised version has an extra figure

This paper introduces the notion of a categorical pair, a pair of categories (C,C') such that every morphism in C is an object in C'. Categorical pairs are precursors to 2-categories. Arrows in C' can express relationships among the morphisms of C. In particular we show that by using a model of the linguistic process of naming, we can ensure that every morphism in C has an indirect self-reference of the form a -----> Fa where this arrow occurs in the category C'. This result is shown to generalize and clarify known fixed point theorems in logic and categories, and is applied to Goedel's Incompleteness Theorem, the Cantor Diagonal Process and the Lawvere Fixed Point Theorem. In particular we show that the indirect self-reference that is central to Goedel's Theorem is an instance of a general pattern here called the indicative shift., Comment: LaTeX document, 34 pages, two figures

The nerve of a strict omega-category is a simplicial set with additional structure, making it into a so-called complicial set, and strict omega-categories are in fact equivalent to complicial sets. The nerve functor is represented by a sequence of strict omega-categories, called orientals, which are associated to simplexes. In this paper we give a detailed algebraic description of the morphisms between orientals. The aim is to describe complicial sets algebraically, by operators and equational axioms., Comment: 25 pages. As to appear in Journal of Pure and Applied Algebra. Minor corrections, addtional explanations, some propositions reclassified as lemmas

We present a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories that satisfy a lifting condition. Such symmetric monoidal double categories frequently occur in nature, so the method is widely applicable, though not universally so., Comment: 21 pages

This note informally describes a way to build certain cubical n-categories by iterating a process of taking models of certain finite limits theories. We base this discussion on a construction of "double bicategories" as bicategories internal to Bicat, and see how to extend this to n-tuple bicategories (and similarly for tricategories etc.) We briefly consider how to reproduce "simpler" definitions of weak cubical n-category from these., Comment: 14 pages; submitted to proceedings of MakkaiFest 2009; expository

We offer two proofs 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., Comment: 12 pages, Comments welcome

We define weak units in a semi-monoidal 2-category $\CC$ as cancellable pseudo-idempotents: they are pairs $(I,\alpha)$ where $I$ is an object such that tensoring with $I$ from either side constitutes a biequivalence of $\CC$, and $\alpha: I \tensor I \to I$ is an equivalence in $\CC$. We show that this notion of weak unit has coherence built in: Theorem A: $\alpha$ has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: $\alpha$ alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2-cells (one for each pair of objects), satisfying the relevant coherence axioms., Comment: 37 pages, LaTeX; does not compile correctly with pdflatex due to some ps rotations. Minor typographical imperfections

The Grothendieck construction is a process to form a single category from a diagram of small categories. In this paper, we extend the definition of the Grothendieck construction to diagrams of small categories enriched over a symmetric monoidal category satisfying certain conditions. Symmetric monoidal categories satisfying the conditions in this paper include the category of $k$-modules over a commutative ring $k$, the category of chain complexes, the category of simplicial sets, the category of topological spaces, and the category of modern spectra. In particular, we obtain a generalization of the orbit category construction in [math/0312214]. We also extend the notion of graded categories and show that the Grothendieck construction takes values in the category of graded categories. Our definition of graded category does not require any coproduct decompositions and generalizes $k$-linear graded categories indexed by small categories defined by Lowen. There are two popular ways to construct functors from the category of graded categories to the category of oplax functors. One of them is the smash product construction defined and studied in [math/0312214,0807.4706,0905.3884] for $k$-linear categories and the other one is the fiber functor. We construct extensions of these functors for enriched categories and show that they are ``right adjoint'' to the Grothendieck construction in suitable senses. As a byproduct, we obtain a new short description of small enriched categories., Comment: 65 pages

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., Comment: 34 pages

In the efforts to define a 2-categorical analog of an abelian category, two (or three) notions of "abelian 2-categories" are defined. One is the relatively exact 2-category, and the other(s) is the (2-)abelian Gpd-category. We compare these notions. Since they proceed independently in their own way, in different settings and terminologies, it will be worth while to collect and unify them. In this paper, by comparing their definitions and arguments, we show the relationship among these classes of 2-categories., Comment: 8 pages

This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate `nerves of C' are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major result proves a direct extension of Thomason's `Homotopy Colimit Theorem' to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the `Grothendieck construction on the diagram'. Our results provide coherence for all reasonable extensions to bicategories of Quillen's definition of the `classifying space' of a category as the geometric realization of the category's Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental `delooping' construction., Comment: 42 pages

We give a rather general construction of double categories and so, under further conditions, double groupoids, from a structure we call a `double module'. We also give a homotopical construction of a double groupoid from a triad consisting of a space, two subspaces, and a set of base points, under a condition which also implies that this double groupoid contains two second relative homotopy groups., Comment: 7 pages, 1 figure, QED.sty v.2 This version explains and proves the relation of the homotopy double groupoid with two second relative homotopy groups. Also extra references added

For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties., Comment: accepted for a publication in "Homology, Homotopy and Applications"

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian., Comment: Incorporates corrections to the published version, which appeared in a separate correction note

The 2-categories of strict 2-groups and crossed modules are introduced and their 2-equivalence is made explicit., Comment: 18 pages, xy-pic diagrams

We define Jones's planar algebra as a map of multicategories and constuct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We prove finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus., Comment: 24 pages, Version 2, some minor changes and typographical errors corrected in sections 2.2 and 4

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, generalizing the one introduced by Squier for word rewriting systems. We characterize 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.

The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined and it is proved that homology can be developed in them, including the existence of a long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups. The examples include, in addition to symmetric 2-groups, the 2-modules on a 2-ring, which form a 2-abelian groupoid enriched category. Moreover, internal groupoids, functors and natural transformations in an abelian category C (in particular, Baez-Crans 2-vector spaces) form a 2-abelian groupoid enriched category if and only if the axiom of choice holds in C., Comment: x + 268 pages. This is the English version of my PhD thesis; the original French version is available at http://edoc.bib.ucl.ac.be:81/ETD-db/collection/available/BelnUcetd-06112008-231800/

We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras., Comment: 28 pages

In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals., Comment: Improvement on the exposition. Approximately 60 pages, 2 figures

We give a bicategorical version of the main result of A. Masuoka ({Corings and invertible bimodules,} {\em Tsukuba J. Math.} \textbf{13} (1989), 353--362) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings $R \subseteq S$, the relative Picard group $Pic(S/R)$ is isomorphic to the Amitsur 1--cohomology group $H^1(S/R,U)$ with coefficients in the units functor $U$.

We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the 2-cells are the maps of bimodules satisfying two simple compatibilities. The operation of getting the "composed coring" from a given entwining, is promoted here to a canonical morphism of bicategories from a bicategory of entwinings to the Street's bicategory of corings., Comment: 12 pages

We establish a model structure on the category of strict omega-categories. The constructions leading to the model structure in question are expressed entirely within the scope of omega-categories, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while cofibrant objects are exactly the free ones. Our model structure transfers to n-categories along right-adjoints, for each n, thus recovering the known cases n = 1 and n = 2., Comment: 33 pages, expanded version of the original 17 pages synopsis, new sections added

In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel'd double. If $A$ is a map pseudomonoid in a monoidal bicategory \M, the analogue of the category of Hopf modules for $A$ is an Eilenberg-Moore construction for a certain monad in $\mathbf{Hom}(\M^{\mathrm{op}},\mathbf{Cat})$. We study the existence of the internalisation of this notion, called the Hopf module construction, by extending the completion under Eilenberg-Moore objects of a 2-category to a endo-homomorphism of tricategories on $\mathbf{Bicat}$. Our main result is the equivalence between the existence of a left dualization for $A$ ({\em i.e.}, $A$ is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case the Hopf module construction for $A$ always exists. We use these results to study the lax centre of a left autonomous map pseudomonoid. We show that the lax centre is the Eilenberg-Moore construction for a certain monad on $A$ (one existing if the other does). If $A$ is also right autonomous, then the lax centre equals the centre. We look at the examples of the bicategories of \V-modules and of comodules in \V, and obtain the Drinfel'd double of a coquasi-Hopf algebra $H$ as the centre of $H$., Comment: 53 pages. Observation 9.15 corrected. Several string diagrams grouped into figures

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.

We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between \bfseries{BrMonCat} and \bfseries{Tricat}. We show how to alter the natural structure of \bfseries{Tricat} in two different ways to provide a comparison, but show that only the more brutal alteration yields an equivalence. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure \bfseries{MonBicat}., Comment: 51 pages

In the context of Synthetic Differential Geometry, we describe a notion of higher connection with values in a cubical groupoid. We do this by exploiting a certain structure of cubical complex derived from the first neighbourhood of the diagonal of a manifold. This cubical complex consists of infinitesimal parallelelpipeda., Comment: 31 pages

We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon cohomology., Comment: 29 pages, we have added further details and examples; this paper will appear in the Canadian Journal of Mathematics

We define a notion of cofibration among n-categories and show that the cofibrant objects are exactly the free ones, that is those generated by polygraphs., Comment: 16 pages

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., Comment: 3 pages; v2: minor changes only, journal ref added

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps., Comment: 27 pages

This note presents the first known class of termination orders for 3-polygraphs, together with an application., Comment: 4 pages, 12 figures