The homotopy theory of coalgebras over a comonad

Abstract

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 (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 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 V A 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 V A in terms of a generalized cobar construction.

Left-Induced Model Structures and Diagram Categories

“…First, we allow the generators to be a class of maps; observe that any model category is (trivially) cofibrantly generated by its cofibrations and acyclic cofibrations. The reason for this convention is that past work has shown that even trivial Postnikov presentations can have nontrivial applications; see [11,Theorem 6.2].…”

“…In [11], the authors provide conditions under which the left-induced model structure exists for such an adjunction.…”

“…This corollary has been applied in [9] and [11] to prove the existence of explicit left-induced model structures. A very similar proof to that in [9] enables us to establish an existence result in the presence of the weaker hypothesis of fibrant generation as well.…”

“…For this reason, the terminology we introduce in Section 2.1 separates the lifting properties, the cellular presentation, and the smallness conditions that are normally unified by the adjective "cofibrantly generated" (see Remarks 2.5 and 2.8). Our motivation for persevering despite the technical difficulties presented by the theory of Postnikov presentations and fibrant generation is that left-induced model structures include interesting examples that were previously unknown, e.g., [9,Theorem 2.10] and [11,Theorem 6.2]). …”

Homotopy-theoretically enriched categories of noncommutative motives

“…[Tate, by the way, worked with a commutative noetherian local ring Remark. In fact under very general conditions ( [38], § 3.1, [39], § 6.12), the bar construction associates to a morphism ϕ : A → B of suitable monoid objects, a pullback functor …”