Right Bousfield Localization and Eilenberg-Moore Categories

TL;DR

This paper establishes the equivalence of recent approaches to right Bousfield localization and algebras over monads, and applies these results to derive new properties and conditions for various mathematical categories.

Contribution

It proves the equivalence of different methods for studying right Bousfield localization and applies this to obtain new results and conditions for localizations in multiple mathematical contexts.

Findings

Proves the equivalence of approaches to right Bousfield localization and monad algebras.

Provides conditions for lifting localization to algebra categories.

Derives new results for spectra, spaces, and other categories.

Abstract

We compare several recent approaches to studying right Bousfield localization and algebras over monads. We prove these approaches are equivalent, and we apply this equivalence to obtain several new results regarding right Bousfield localizations (some classical, some new) for spectra, spaces, equivariant spaces, chain complexes, simplicial abelian groups, and the stable module category. En route, we provide conditions so that right Bousfield localization lifts to categories of algebras, so that right Bousfield localization preserves algebras over monads, and so that right Bousfield localization forms a compactly generated model category.