Adjoint functor theorems for $\mathrm{\infty }$-categories

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\mathrm{\infty }$-categories. One of our main results is an $\mathrm{\infty }$-categorical generalization of Freyd’s classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie’s adjoint functor theorems for presentable $\mathrm{\infty }$-categories. We also discuss the comparison between adjunctions of $\mathrm{\infty }$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\mathrm{\infty }$-categories based on Heller’s purely categorical formulation of the classical Brown representability theorem.