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 $\infty$-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 $\infty$-categories if $n\ge 2$, and the homotopy $n$-categories of presentable stable $\infty$-categories for any $n\ge 1$.

n-GAFT is shown also for n=2; corrected some errors around the proof of Lemma 3.2.9 (Criterion B) and the corresponding examples; other minor improvements in the exposition and some more details. To appear in Documenta Mathematica