A model structure for coloured operads in symmetric spectra

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.

💡 Research Summary

The paper establishes a model category structure for coloured operads whose values lie in the category of symmetric spectra equipped with the positive model structure. The authors begin by recalling the positive model structure on symmetric spectra, which excludes the zeroth level from the cofibrancy requirements and thereby yields a convenient setting for homotopical algebra. They then define coloured operads as collections indexed by a set of colours, each entry being a symmetric spectrum, and show that if the underlying collection carries a model structure (with weak equivalences and fibrations detected level‑wise), this structure can be transferred to the operad level via the usual free‑forgetful adjunction. A key technical result is that the transfer works provided the operad is cofibrant as a symmetric‑spectrum‑valued object; in particular, any cofibrant operad gives rise to a model structure on its algebras in which weak equivalences and fibrations are created by the forgetful functor.

With this machinery in place, the authors focus on the case of a cofibrant ring spectrum (R). They construct a cofibrant coloured operad (\mathcal{O}) whose first entry (\mathcal{O}(1)) is precisely (R) and whose higher entries encode the appropriate multilinear operations. Algebras over (\mathcal{O}) are then identified with (R)-module spectra, showing that the category of (R)-modules can be regarded as the category of algebras over a cofibrant operad. This perspective allows one to apply the operadic model structure to study homotopical properties of module spectra, such as homotopy limits, colimits, and derived mapping spaces.

The final part of the paper addresses homotopical localizations in symmetric spectra. The authors give sufficient conditions under which a Bousfield localization functor (L) preserves the structure of (\mathcal{O})-algebras, i.e., when the localized object (L M) of an (R)-module (M) naturally inherits an (R)-module structure compatible with the operadic action. The conditions require that (L) be a left Quillen functor with respect to the transferred operadic model structure and that it preserve cofibrant objects. Under these hypotheses, the localization of an (R)-module remains an (R)-module up to homotopy, and the induced map (M \to L M) is a map of (\mathcal{O})-algebras.

In summary, the paper provides a robust framework for handling coloured operads in the setting of symmetric spectra, demonstrates how module spectra over a cofibrant ring spectrum can be treated as algebras over a cofibrant operad, and establishes criteria ensuring that homotopical localizations respect these module structures. This work bridges operadic homotopy theory and stable homotopy theory, opening new avenues for the systematic study of structured ring and module spectra.