Rigidification of connective comodules

TL;DR

This paper proves that homotopy coherent comodules over certain spectra can be replaced by strict comodules, enabling classical constructions like cotensor products and symmetric monoidal structures in a more concrete setting.

Contribution

It establishes a rigidification theorem for homotopy coherent comodules over connective modules, connecting $$infinity-categories to model categories of strict comodules.

Findings

Homotopy coherent comodules can be represented by strict comodules in chain complexes.

The rigidification enables defining cotensor products for comodules.

The $$category of comodules gains a symmetric monoidal structure via the cobar resolution.

Abstract

Let $\mathbb{k}$ be a commutative ring with global dimension zero. We show that we can rigidify homotopy coherent comodules in connective modules over the Eilenberg-Mac Lane spectrum of $\mathbb{k}$. That is, the $\infty$-category of homotopy coherent comodules is represented by a model category of strict comodules in non-negative chain complexes over $\mathbb{k}$. These comodules are over a coalgebra that is strictly coassociative 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 via the two-sided cobar resolution.