Related papers: Accessible aspects of 2-category theory

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then we give…

We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$,…

In this paper we obtain several model structures on {\bf DblCat}, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double…

In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have…

In his paper "Th\'eories homotopiques des 2-cat\'egories", Jonathan Chiche studies homotopy theories on 2-Cat, the category of small strict 2-categories, given by classes of weak equivalences which he calls basic localizers of 2-Cat. These…

We study lax epimorphisms in 2-categories, with special attention to $\mathsf{Cat}$ and $\mathcal{V}$-$\mathsf{Cat}$. We show that any 2-category with convenient colimits has an orthogonal $LaxEpi$-factorization system, and we give a…

Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…

Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched…

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…

We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…

In this paper we present $2$-category theory from the perspective of Gray-categories using the graphical calculus of separated surface diagrams. As an extended example we consider cones and limits of $2$-functors. Then we use the canonical…

We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The…

This paper considers the possible underlying multicategories for a symmetric monoidal category, and shows that, up to canonical and coherent isomorphism, there really is only one. As a result, there is a well-defined forgetful functor from…

Univalent categories constitute a well-behaved and useful notion of category in univalent foundations. The notion of univalence has subsequently been generalized to bicategories and other structures in (higher) category theory. Here, we…

For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack…

In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. These are…

We prove that categories enriched in the Thomason model structure admit a model structure that is Quillen equivalent to the Bergner model structure on simplicial categories, providing a new model for (infinity,1)-categories. Along the way,…

We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained "by abstract nonsense" via the adjoint functor theorem. This…

In this paper we study compact closed categories within the context of homotopical algebra. We construct two new model category structures by localizing two (Quillen equivalent) model categories of symmetric monoidal categories with the…