Accessible $\infty$-cosmoi

TL;DR

This paper introduces accessible $ abla$-cosmoi, demonstrating their stability under key constructions and showing most known examples are accessible, which implies they possess flexible homotopy colimits.

Contribution

The paper defines accessible $ abla$-cosmoi, proves their stability under main constructions, and connects this to the existence of flexible homotopy colimits via an adjoint functor theorem.

Findings

Most known $ abla$-cosmoi are accessible.

Accessible $ abla$-cosmoi are stable under main constructions.

All such $ abla$-cosmoi have flexible homotopy colimits.

Abstract

We introduce the notion of an accessible $\infty$-cosmos and prove that these include the basic examples of $\infty$-cosmoi and are stable under the main constructions. A consequence is that the vast majority of known examples of $\infty$-cosmoi are accessible. By the adjoint functor theorem for homotopically enriched categories which we proved in an earlier paper, joint with Lukas Vokrinek, it follows, for instance, that all such $\infty$-cosmoi have flexibly weighted homotopy colimits.