Accessible $\infty$-cosmoi

John Bourke, Stephen Lack

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.