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

TL;DR

This paper develops adjoint functor theorems and Brown representability results for weakly (co)complete n-categories, extending classical concepts to higher categorical contexts including homotopy n-categories of presentable and stable ∞-categories.

Contribution

It introduces Brown representability for (homotopy) n-categories and proves a new Brown representability theorem for localizations of compactly generated n-categories.

Findings

Established adjoint functor theorems for weakly (co)complete n-categories.

Proved Brown representability theorem for localizations of compactly generated n-categories.

Extended classical higher category theory results to the setting of homotopy n-categories.

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