Bousfield Localization and Eilenberg-Moore Categories

TL;DR

This paper establishes conditions under which Bousfield localization interacts well with algebraic structures in model categories, providing a unified framework with applications to operads, spaces, spectra, and chain complexes.

Contribution

It proves equivalences of hypotheses for localizing algebras over monads and identifies conditions for model structures on local algebras and their preservation under localization.

Findings

Conditions for model structures on local algebras

Localization preserves algebraic structures under certain hypotheses

Applications to operads, spaces, spectra, and chain complexes

Abstract

We prove the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. We find conditions so that there is a model structure for local algebras, so that localization preserves algebras, and so that localization lifts to the level of algebras. We include examples coming from the theory of colored operads, and applications to spaces, spectra, and chain complexes.