Adjoint functor theorems for homotopically enriched categories

TL;DR

This paper develops an adjoint functor theorem for categories enriched in monoidal model categories, extending classical results to homotopical contexts and providing new tools for $ ext{infty}$-categories.

Contribution

It introduces a homotopical adjoint functor theorem for enriched categories, generalizing classical theorems to the setting of model categories and $ ext{infty}$-categories.

Findings

Established an adjoint functor theorem for $ ext{V}$-enriched categories with model structures.

Recaptured the classical enriched Freyd's theorem in trivial model structures.

Derived new homotopical adjoint theorems for categories of simplicial sets and $ ext{infty}$-cosmoi.

Abstract

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour - in particular, when $\mathcal V$ is the category of simplical sets we obtain a homotopical adjoint functor theorem appropriate to the $\infty$-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched setting, in particular obtaining homotopical cocompleteness results for accessible $\infty$-cosmoi.