Transfer of algebras over operads along derived Quillen adjunctions

TL;DR

This paper establishes conditions under which algebraic structures over operads can be transferred along derived Quillen adjunctions in monoidal model categories, with applications to spectra and module spectra.

Contribution

It provides a framework for transferring P-algebra structures along derived Quillen adjunctions in monoidal V-model categories, including applications to spectra.

Findings

P-algebra structures can be transferred along derived Quillen adjunctions.

The n-connective cover functor preserves A∞ and E∞ module spectra.

Conditions for fibrant and cofibrant replacements to preserve algebra structures.

Abstract

Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived 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_{\infty} and E_{\infty} module spectra over connective ring spectra, for every integer n.