Adjoint functor theorems for homotopically enriched categories

Abstract

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category V admitting certain limits. When V is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour — in particular, when V is the category of simplicial sets we obtain a homotopical adjoint functor theorem appropriate to the ∞-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched setting, in particular obtaining homotopical cocompleteness results for accessible ∞-cosmoi.