Localization of algebras over coloured operads

TL;DR

This paper establishes conditions under which homotopical localization functors preserve algebraic structures over coloured operads in monoidal model categories, extending previous results and including new cases like ring spectra and modules.

Contribution

It provides a unified framework using coloured operads to show preservation of various algebraic structures under localization in homotopical settings.

Findings

Homotopical localizations preserve ring spectra and modules.

Preservation of algebraic structures extends to maps between them.

New results include preservation of algebras over commutative ring spectra.

Abstract

We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.