Localization of algebras over coloured operads

Details Localization of algebras over coloured operads Localization of algebras over coloured operads

2008-06-24

arxiv.org/abs/0806.3983v1

Proceedings of the London Mathematical Society (3) 101 (2010), no. 1, 105-136

Mathematics

Algebraic Topology

34 pages

Scientific paper

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.

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