The homotopy theory of coalgebras over a comonad

TL;DR

This paper establishes conditions for the existence of model category structures on categories of coalgebras over a comonad in a model category, with applications to descent theory and Hopf-Galois extensions.

Contribution

It provides a general framework and concrete examples for when categories of K-coalgebras admit model structures, extending the theory to comodules over corings in algebraic contexts.

Findings

Model category structures exist for categories of K-coalgebras under certain conditions.

Concrete examples include comodules over corings over a semihereditary commutative ring.

Fibrant replacements can be described via a generalized cobar construction.

Abstract

Let K be a comonad on a model category M. We provide conditions under which the associated category of K-coalgebras admits a model category structure such that the forgetful functor to M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are specific instances of the following categories of comodules over a coring. For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that is semifree as a left A-module on a degreewise R-free, homologically simply connected graded module of finite type. We show that there is a model category structure on the category of right A-modules satisfying the conditions of our existence theorem with respect to the comonad given by tensoring over A with V and conclude that the category of V-comodules in the category of right A-modules admits a model category structure of the desired type. Finally, under extra conditions on R, A, and V, we describe fibrant replacements in this category of comodules in terms of a generalized cobar construction.