Adjoint functor theorems for $\infty$-categories

TL;DR

This paper extends the classical adjoint functor theorems to the setting of $infty$-categories, providing new conditions and generalizations, including applications to presentable $infty$-categories and Brown representability.

Contribution

It proves a general $infty$-categorical adjoint functor theorem, generalizes Freyd's theorem, and discusses the relation between $infty$-category adjunctions and homotopy adjunctions.

Findings

Established an $infty$-categorical version of Freyd's General Adjoint Functor Theorem.

Recovered Lurie's adjoint functor theorems for presentable $infty$-categories.

Provided a categorical approach to Brown representability in $infty$-categories.

Abstract

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 $\infty$-categories. One of our main results is an $\infty$-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 $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-categories based on Heller's purely categorical formulation of the classical Brown representability theorem.