Accessible $\infty$-cosmoi

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.