Transfer of algebras over operads along Quillen adjunctions

Abstract

Let 𝒱 be a cofibrantly generated monoidal model category and ℳ be a monoidal 𝒱-model category. If C is any set, then we denote by ℳC the product category of copies of ℳ indexed by C. Given a cofibrant C-coloured operad 𝒫 in 𝒱, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in ℳC to preserve 𝒫-algebra structures. In particular, we show how 𝒫-algebra structures can be transferred along Quillen adjunctions of monoidal 𝒱-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on ℳ. As an application, we prove that in the category of symmetric spectra the n-connective cover functor preserves A∞ and E∞ module spectra over connective ring spectra, for every n∈ℤ.