Homotopical Adjoint Lifting Theorem

TL;DR

This paper develops a homotopical version of the adjoint lifting theorem, enabling the transfer of Quillen equivalences to categories of algebras over colored operads, with broad applications including rectification and base change.

Contribution

It introduces a general homotopical adjoint lifting theorem applicable to colored operads without requiring $ ext{Sigma}$-cofibrancy, simplifying and unifying existing results.

Findings

Recovers known rectification results

Reduces the number of Quillen equivalences needed in specific cases

Extends lifting of Quillen equivalences to localized model categories

Abstract

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $\Sigma$-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter-Shipley about a zig-zag of Quillen equivalences between commutative $H\mathbb{Q}$-algebra spectra and commutative differential graded $\mathbb{Q}$-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization.