Related papers: 2-Dimensional Categories

We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing…

In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally…

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a…

This note is a contribution written for the second volume of the Encyclopedia of mathematical physics. We give an informal introduction to the notions of an $(\infty,n)$-category and $(\infty,n)$-functor, discussing some of the different…

We develop the theory of strong and commutative monads in the 2-dimensional setting of bicategories. This provides a framework for the analysis of effects in many recent models which form bicategories and not categories, such as those based…

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and…

We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we…

Using the theory of measurable categories developped by Yetter in work in preparation, we provide a notion of representations of 2-groups more well-suited to physically and geometrically interesting examples than that proposed in…

In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us…

We provide bicategorical analogs of several aspects of the notion of geometry in the sense of the theory of spectrum. We first introduce a notion of local right biadjoint, and prove it to be equivalent to a notion of bistable pseudofunctor,…

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.

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…

The study of abstraction and composition - the focus of category theory - naturally leads to sophisticated diagrams which can encode complex algebraic semantics. Consequently, these diagrams facilitate a clearer visual comprehension of…

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we…

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every…

This is an extended version of my earlier articel "Projective and injective objects in symmetric categorical groups. arXiv:1007.0121v1." Several new facts added, including the material on the derived 2-functors and the proof of the…

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. A number of precise definitions of this notion have…

This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently…

In an abelian category, the (bi)fibration of subobjects is isomorphic to the (bi)fibration of quotients. This property captures substantial information about the exactness structure of a category. Indeed, as it was shown by the second…

This work hopes to be an introduction to Deligne categories for someone familiar with classical representation theory and some category theory. In the first chapter, we motivate and define (symmetric) tensor categories, construct the…