Accessibility of enriched and higher-dimensional categories

In this talk, I will give an overview of joint research about accessibility and local presentability in higher dimensions, including 2-category theory and the infinity-cosmoi of Riehl and Verity.

The classical story is that accessible categories are those in which each object can built out of a set of generators in a natural way. They capture sets with infinitary first order structure. When they are complete, they are cocomplete, and vice-versa - in this setting, they are called locally presentable and capture algebraic structures of a general kind. One reason that such categories are important in practice is that they admit easily applicable adjoint functor theorems.

The aim of this project has been to study analogues of the above story in enriched category settings appropriate for higher category theory, and to obtain new useful results such as adjoint functor theorems of a homotopical flavour. It is much inspired by Australian 2-category theory [1, 2], Makkai’s work on sketches [7], Lack and Rosický’s work on enriched weakness [6] and Riehl and Verity’s programme on studying infinity-categorical structures using 2-category theory and simplically-enriched category theory [8].

I will talk about some of the main ideas and insights, mainly from the papers [3, 4, 5] below, and also point to some recent developments and perhaps even some open questions.