Comparing localizations across adjunctions

TL;DR

This paper explores how comparison maps from adjoint functors relate various localizations in homotopy theory, enabling functorial liftings to algebra categories, with applications to module spectra and operad algebras.

Contribution

It demonstrates that localization formulas are induced by comparison maps from adjunctions, facilitating their liftings to categories of algebraic structures in homotopy theory.

Findings

Comparison maps induce localization formulas

Functorial liftings of localizations to algebra categories

Localization equivalences for homotopy T-algebras

Abstract

We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if $T$ is the reduced monad associated with a simplicial operad and $f$ is any map of pointed simplicial sets, then $f$-localization coincides with $Tf$-localization on spaces underlying homotopy $T$-algebras, and similarly for cellularizations.