Localization of algebras over coloured operads

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.

34 pages

References in corpus (2)

• Rings, modules, and algebras in infinite loop space theory
• Rigidification of algebras over multi-sorted theories

Cited by in corpus (10)

• Encoding Equivariant Commutativity via Operads
• On homotopy invariance for algebras over colored PROPs
• A model structure for coloured operads in symmetric spectra
• The Homotopy Theory of Simplicially Enriched Multicategories
• An "almost" full embedding of the category of graphs into the category of groups
• An almost full embedding of the category of graphs into the category of abelian groups
• Bousfield Localization and Eilenberg-Moore Categories
• Comparing localizations across adjunctions
• Spaces of Operad Structures
• Transfer of algebras over operads along derived Quillen adjunctions