Accessible ∞-cosmoi

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