Homotopic Descent over Monoidal Model Categories

The starting point for this project is the article of Kathryn Hess [11]. In this article, a homotopic version of monadic descent is developed. In the classical setting, one constructs a category D(𝕋) of coalgebras in the Eilenberg-Moore category of algebras D𝕋 for a given monad 𝕋 on a category D. There is a canonical functor Can𝕃𝕋 from D to D(𝕋), and if Can𝕃𝕋 is fully faithful, then 𝕋 satisfies descent, while if Can𝕃𝕋 is an equivalence of categories, then 𝕋 satisfies effective descent [19]. In [11], these two conditions are replaced by a weaker one, that these hold only up to homotopy. This is achieved by working with model categories that are enriched over simplicial sets. Homotopic descent is then defined by demanding that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence of simplicial sets. A similar but stronger condition involving the path components in D(𝕋) expresses effective homotopic descent. The first goal of this project is to develop a framework of homotopic descent for model categories that are enriched over model categories other than simplicial sets. The most important examples we have in mind are chain complexes and spectra. In order to achieve this goal, we tried to determine the most general conditions that are sufficient and necessary to make the theory work. To ease the formulation, let us say that we are working with a model category D that is enriched over a monoidal model category V. The crucial constructions we need are realization, respectively totalization, of (co)simplicial objects in D. These functors have to be Quillen functors to ensure that they have the correct homotopical behaviour. This implies that there must exist a Quillen adjunction between V and simplicial sets. Furthermore, we need to be able to transfer the enrichment and (co)tensoring over V to an enrichment and (co)tensoring over simplicial sets. This forces the Quillen adjunction to be monoidal. Another main point that has to be adressed is the question, of whether the enrichment of D carries over to an enrichment of D𝕋 and D(𝕋) and how this enrichment behaves. It turns out that this works well under mild assumptions on V. This leads then to the definition of homotopic descent by requiring that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence in M and similarly for effective homotopic descent. Using this definition, the theorems in [11] carry over to this more general context. Although the conditions on V are rather constraining regarding the relation with simplicial sets, the cases of chain complexes and spectra are included. For the time being we do not see how the constraints on V could be weakened. The second goal of this project is