Comonadic Coalgebras and Bousfield Localization

TL;DR

This paper investigates how coalgebra categories over comonads interact with Bousfield localization in model categories, providing conditions and constructions for preserving coalgebra structures under localization.

Contribution

It establishes that coalgebra categories commute with Bousfield localization and offers criteria for when localization preserves coalgebras over a comonad.

Findings

Coalgebra categories commute with Bousfield localization under certain conditions.

A general existence theorem for model structures on coalgebras in localized categories.

Multiple applications demonstrating preservation of coalgebras in various settings.

Abstract

For a model category, we prove that taking the category of coalgebras over a comonad commutes with left Bousfield localization in a suitable sense. Then we prove a general existence result for the left-induced model structure on the category of coalgebras over a comonad in a left Bousfield localization. Next we provide several equivalent characterizations of when a left Bousfield localization preserves coalgebras over a comonad. These results are illustrated with many applications in chain complexes, (localized) spectra, the stable module category, and simplicial settings.