Admissibility and rectification of colored symmetric operads. Issue 3 (12th June 2018)

• Record Type: Journal Article
• Authors: Pavlov, Dmitri; Scholbach, Jakob
• Is Part Of: Journal of topology. Volume 11:Issue 3(2018)
• Page Start: 559
• Page End: 601
• Publication Date: 2018-06-12
• Subjects: 55P48 -- 18D50 -- 18G55 -- 55U35 (primary) -- 55P43 -- 18D20 (secondary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12008
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, that is, the category of algebras over any operad admits a model structure transferred from the original model category. We also give a necessary and sufficient criterion that ensures that a given weak equivalence of admissible operads admits rectification, that is, the corresponding Quillen adjunction between the categories of algebras is a Quillen equivalence. In addition, we show that Quillen equivalences of underlying symmetric monoidal model categories yield Quillen equivalences of model categories of algebras over operads. Applications of these results include enriched categories, colored operads, prefactorization algebras, and commutative symmetric ring spectra.