In category theory, one often encounters weak colimits as well as ordinary colimits. Lack and Rosicky [1] realised that can be viewed as instances of a common concept, if we work with categories enriched over a base V equipped with an appropriate class of morphisms E.

Dealing with flavours of weak colimit becomes more important in enriched approaches to higher categories. Here one typically enriches over a monoidal (Quillen) model category V. For instance, the infinity-cosmoi of Riehl and Verity correspond to enrichment over simplicial sets with the Joyal model structure.

In such a setting, there is a canonical choice of morphisms E in V, called the shrinking morphisms, and the theory of weak adjoints and weak colimits relative to this class E develops very nicely. I will talk about some aspects of this story, and describe applications to several frameworks including the cases of ordinary categories, 2-categories and infinity-cosmoi. This will draw on the papers [2,3,4].

References:

[1] Stephen Lack and Jirı Rosicky. Enriched weakness. J. Pure Appl. Algebra, 2012.

[2] John Bourke. Accessible aspects of 2-category theory, Journal of Pure and Applied Algebra 2021.

[3] John Bourke, Stephen Lack, and Lukas Vokrınek. Adjoint functor theorems for homotopically enriched categories, arXiv:2006.07843v1, 2020.

[4] John Bourke and Stephen Lack. Accessible infinity-cosmoi. arxiv.org:2111.00147, 2021.