The Homotopy Theory of Coalgebras over a Comonad

Let K be a comonad on a model category M. We provide conditions under which the associated category MK of K-coalgebras admits a model category structure such that the forgetful functor MK →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 (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 MA of right A-modules satisfying the conditions of our existence theorem with respect to the comonad −⊗A V and conclude that the category MA of V -comodules in MA admits a model category structure of the desired type. Finally, under extra conditions on R, A, and V , we describe fibrant replacements in MA in terms of a generalized cobar construction.