Higher weak (co)limits, adjoint functor theorems, and higher Brown representability

Abstract

We prove general adjoint functor theorems for weakly (co)complete n-categories. This class of n-categories includes the homotopy n-categories of (co)complete ∞-categories, so these ncategories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) n-categories and prove a Brown representability theorem for localizations of compactly generated n-categories. This class of n-categories includes the homotopy n-categories of presentable ∞-categories if n ≥ 2, and the homotopy n-categories of presentable stable ∞-categories for any n ≥ 1.