The homotopy theory of coalgebras over a comonad

Abstract

Let K be a comonad on a model category M. We provide conditions under which the associated category M K of K-coalgebras admits a model category structure such that the forgetful functor M K → 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 MA of right A-modules satisfying the conditions of our existence theorem with respect to the comonad -⊗A V and conclude that the category M V A 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 M V A in terms of a generalized cobar construction.

Locations

• Proceedings of the London Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• Infoscience (Ecole Polytechnique Fédérale de Lausanne) - View - PDF
• DataCite API - View