Rigidification of connective comodules

Maximilien Péroux

Published 2020-06-16Version 1

We show that we can rigidify homotopy coherent comodules in connective modules over the Eilenberg-Mac Lane spectrum of a field, or more generally of a finite product of fields $\mathbb{k}$. That is, the $\infty$-category of homotopy coherent comodules is represented by a model category of strict comodules in non-negative chain complexes over $\mathbb{k}$ or in simplicial $\mathbb{k}$-modules. These comodules are over a coalgebra that is strict and simply connected. The rigidification result allows us to derive the notion of cotensor product of comodules and endows the $\infty$-category of comodules with a symmetric monoidal structure. We lift the usual Dold-Kan correspondance to these model categories of comodules. We also show a rigidification result for comodules in the non-connective case when the coalgebra is dualizable. To prove these results, we introduce Postnikov towers of comodules that dualize the cellularity of combinatorial model categories. Moreover, we define the notion of symmetric comonoidal Quillen model categories and weak comonoidal Quillen equivalences in order to describe the homotopical behavior of our derived cotensor product of comodules.