2-Representations and associated coalgebra 1-morphisms for locally wide finitary 2-categories

We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over the coalgebra 1-morphism. We finish the paper by examining two classes of examples of locally wide weakly fiat 2-categories: 2-categories associated to certain classes of infinite quivers, and singular Soergel bimodules associated to Coxeter groups with finitely many simple reflections.     

On Certain 2-Categories Admitting Localisation by Bicategories of Fractions

Pronk’s theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved. The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness. This is the published version of arXiv:1402.7108.     

Multifusion categories and finite semisimple 2-categories

We give a 3-universal property for the Karoubi envelope of a 2-category. Using this, we show that the 3-categories of finite semisimple 2-categories (as introduced in [1]) and of multifusion categories are equivalent.     

A categorical characterization of strong Steiner ω-categories

Strong Steiner ω-categories are a class of ω-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner ω-categories are traditionally expressed in terms of the associated chain complex, making them somewhat disconnected from the ω-categorical intuition. The purpose of this paper is to characterize this class as the class of ω-categories generated by polygraphs that satisfy a loop-freeness condition that does not make explicit use of the associated chain complex and instead relies on the categorical features of ω-categories.     

Accessible aspects of 2-category theory

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the structure of monoidal, but not strict monoidal, categories) then the 2-category in question is accessible. Furthermore, we explore the flexible limits that such 2-categories possess and their interaction with filtered colimits.     

Pseudo-commutativity of KZ 2-monads

In this paper we prove that KZ 2-monads (also known as lax-idempotent 2-monads) are pseudo-commutative. The main examples of KZ 2-monads for us will be 2-monads whose algebras are V-categories with chosen colimits of a given class; this provides a large family of examples of pseudo-commutative 2-monads. In order to achieve this we characterise pseudo-commutativities on a 2-monad in terms of extra structure on its 2-category of algebras and pseudomorphisms. We also consider tensor products associated to pseudo-closed structures and show some results on preservation of colimits. To cover the general case of V-enriched categories and not only ordinary categories we are led to consider monads enriched in a 2-category, and some of the associated two-dimensional monad theory.     

The 2-dimensional stable homotopy hypothesis

We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types.     

Approximation in quantale-enriched categories

Our work is a foundational study of the notion of approximation in Q-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Q- and (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (resp. (U,Q)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale Q and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory.     

Categorified quantum sl2 is an inverse limit of flag 2-categories

We prove that categorified quantum sl₂ is an inverse limit of Flag 2-categories defined using cohomology rings of iterated flag varieties. This inverse limit is an instance of a 2-limit in a bicategory giving rise to a universal property that characterizes the categorification of quantum sl₂ uniquely up to equivalence. As an application we characterize all bimodule homomorphisms in the Flag 2-category and prove that on the homological level the categorified quantum Casimir of sl₂ acts appropriately on these 2-representations.     

On the Geometry of 2-Categories and their Classifying Spaces

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen's Theorem A.     

A Quillen Model Structure for 2-Categories

We describe a cofibrantly generated Quillen model structure on the locally finitely presentable category 2-Cat of (small) 2-categories and 2-functors; the weak equivalences are the biequivalences, and the homotopy relation on 2-functors is just pseudonatural equivalence. The model structure is proper, and is compatible with the monoidal structure given by the Gray tensor product. It is not compatible with the Cartesian closed structure, in which the tensor product is the product.The model structure restricts to a model structure on the full subcategory PsGpd of 2-Cat, consisting of those 2-categories in which every arrow is an equivalence and every 2-cell is invertible. The model structure on PsGpd is once again proper, and compatible with the monoidal structure given by the Gray tensor product.     

Equivaleces between D-categories of Inverse Monoids

There exist two equivalent small categories associated to an inverse monoid which reflect its divisorial structure, called its D-categories. It is well known the relationship between the Green's relation D and the structure, of certain subsemigroups, of an inverse monoid. In this work, an analogous result is established between the Green's relation J and the D-categories of those subsemigroups. They are also given equivalent conditions for the equivalence of the D-categories of two inverse monoids and likewise equivalent conditions for the isomorphism of two inverse monoids in terms of its D-categories. It is proved that for many important classes of inverse monoids the multiplicative structure is determined by the associated category. On the contrary, a sufficient condition to obtain families of counterexamples to the above is provided and three examples are explicitely exhibited.     

Formal aspects of Gray’s tensor products of 2-categories

The category of small 2-categories has two monoidal structures due to John Gray: one biclosed and one closed. We propose a formalisation of the construction of the right internal and internal homs of these monoidal structures.     

Homotopy fiber sequences induced by 2-functors

This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors.     

An Equivalent of Algebraic ?-Categories

In this paper, based on ?-categories, some properties of(continuous) I-cocomplete ?-categories are studied. Then, we introduce the concepts of bicomplete ?-category and approximable bimodule, discuss their properties and we also show any I-cocomplete ?-category is a bicomplete ?-category. Finally, it is proved that the category of algebraic ?-categories is equivalent to the category of bicomplete ?-categories.     

C*-Categories

The purpose of this paper is to give a detailed study of thebasic theory of C^*-categories. The study includes some examplesof C^*-categories that occur naturally in geometric applications,such as groupoid C^*-categories, and C^*-categories associatedto structures in coarse geometry. We conclude the paper witha brief survey of Hilbert modules over C^*-categories. 2000 MathematicalSubject Classification: 18D99, 46L05, 46L08.     

Coherence, Homotopy and 2-Theories

2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a Quillen model category structure on the category of 2-theories and 2-theory-morphisms where the weak equivalences are biequivalences of 2-theories. A biequivalence of 2-theories induces a biequivalence of 2-categories of algebras. This model category structure allows one to talk of the homotopy of 2-theories and discuss the universal properties of coherence.     

The 2-category theory of quasi-categories

In this paper we re-develop the foundations of the category theory of quasi-categories (also called ∞-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples.     

Fibred 2-categories and bicategories

We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.