Accessible aspects of 2-category theory

Abstract

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.

Adjoint functor theorems for homotopically enriched categories

“…The difference between the two cases is that the definition of strict monoidal category involves equations between objects whereas that of monoidal category does not -in terms of the associated 2-monads this corresponds to the fact that the 2-monad for strict monoidal categories is not cofibrant (=flexible) whereas that for monoidal categories is. One result generalising this is Corollary 7.3 of [8], which asserts that if T is a finitary flexible 2-monad on Cat then the 2-category T-Alg p of strict algebras and pseudomorphisms is accessible with flexible limits and filtered colimits, and these are preserved by the forgetful functor to Cat. It now follows from our theorem that any such 2-category admits E-weak colimits, and in particular bicolimits, and also that if f : S → T is a morphism of such 2-monads, then the induced map T-Alg p → S-Alg p has an E-weak left adjoint, and so a left biadjoint.…”

“…The utility of these results lies in the fact that many, though not all, 2categories of pseudomorphisms are in fact accessible with flexible limits. For instance, the 2-category of monoidal categories and strong monoidal functors is accessible (although, as recalled in Section 5, the 2-category of strict monoidal categories and strong monoidal functors is not [8,Section 6.2]). The difference between the two cases is that the definition of strict monoidal category involves equations between objects whereas that of monoidal category does not -in terms of the associated 2-monads this corresponds to the fact that the 2-monad for strict monoidal categories is not cofibrant (=flexible) whereas that for monoidal categories is.…”

“…For (not necessarily accessible) ∞-cosmoi this was done in Chapter 6 of [35], and it remains to add accessibility to the mix. (The corresponding stability results for the appropriate class of accessible 2-categories have been established in [8]. )…”

On coslices and commas of locally finitely presentable categories

“…The intuition behind this claim is the following: from Gabriel-Ulmer duality, we know that the 2-category LFP of locally finitely presentable categories is equivalent to the opposite 2category Lex op of small lex categories with lex functors between them. But ongoing investigation on 2-categorical model theory, as well as the work of [4], tell us that Lex is a locally finitely bipresentable 2-category -or at least a locally presentable 2-category in [4] -which indicate its closure under both small pseudolimits and pseudocolimits. Hence if Lex is closed under pseudocolimit, LFP must be closed under pseudolimits, in particular under comma objects.…”