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 n -categories 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.