Accessible $\mathrm{\infty }$-cosmoi

We introduce the notion of an accessible $\mathrm{\infty }$-cosmos and prove that these include the basic examples of $\mathrm{\infty }$-cosmoi and are stable under the main constructions. A consequence is that the vast majority of known examples of $\mathrm{\infty }$-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 $\mathrm{\infty }$-cosmoi have flexibly weighted homotopy colimits.