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

The homotopy theory of coalgebras over a comonad