Homotopy limits of model categories over inverse index categories (Q847676)

Let \(I\) be a direct category and \(\Phi: I^{op} \rightarrow \text{ModCat}\) a diagram of model categories (i.e. a pseudofunctor). This article provides the tools to deal with the homotopy limit of this diagram and to give an interpretation in terms of sections. The homotopy limit is constructed in two steps. First, take the Dwyer-Kan simplicial localization of each model category so as to get a diagram \(I^{op} \rightarrow \text{sCat}\). Second, construct a fibrant replacement of this diagram in the injective model structure. Its limit is a model for the homotopy limit of \(\Phi\) (the author takes care of set theoretic issues and rigidifies the pseudofunctor to a strict functor so these constructions do make sense). The category \( \text{PSect}(I, \Phi)\) of presections consists of collections of objects \(X_i\) of \(\Phi(i)\) together with compatible structure maps \(\Phi(f) X_i \rightarrow X_j\) for all morphisms \(f: j \rightarrow i\) in \(I\). A presection is said to be homotopy compatible if the maps \(\mathbb L \Phi(f) X_i \rightarrow X_j\) are isomorphisms in \( \text{Ho} \Phi(j)\). Here \(\mathbb L \Phi(f)\) is the total left derived functor of \(\Phi(f)\). The category of presections is equipped with an injective model category structure and one can thus consider its simplicial localization. The main result of this article is then that the full simplicial subcategory consisting of homotopy compatible systems is naturally isomorphic in \( \text{Ho sCat}\) to the homotopy limit of \(\Phi\).