A model structure for coloured operads in symmetric spectra

TL;DR

This paper develops a model structure for coloured operads in symmetric spectra, enabling the study of module spectra as algebras over spectrum-valued operads, and provides conditions for homotopical localizations to preserve module structures.

Contribution

It introduces a new model structure for coloured operads in symmetric spectra, facilitating algebraic and homotopical analysis of module spectra.

Findings

Established a model structure where fibrations and weak equivalences are defined at the collection level.

Enabled treatment of R-module spectra as algebras over spectrum-valued operads.

Provided conditions under which homotopical localizations preserve module structures.

Abstract

We describe a model structure for coloured operads with values in the category of symmetric spectra (with the positive model structure), in which fibrations and weak equivalences are defined at the level of the underlying collections. This allows us to treat R-module spectra (where R is a cofibrant ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as its first term. Using this model structure, we give suficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures.