Homotopy theory of modules over diagrams of rings (Q2814365)

Given a functor \(\mathcal{M}:{\mathrm D}\to {\mathrm Cat}\), for every morphism \(a:s\to t\) in \({\mathrm D}\) we have a functor \(a_*:\mathcal{M}(s)\to \mathcal{M}(t)\). It is said that \(\mathcal{M}\) is a diagram of model categories (or \(\mathcal{M}\)-diagram) if each category \(\mathcal{M}(s)\) has a model structure, the functors \(a_*\) all have right adjoints and the adjoint pair \(a_*\vdash a^*\) of functors relating the model categories form a Quillen pair.NEWLINENEWLINEThe authors study the category of \(\mathcal{M}\)-diagrams, whose objects are \(\{\mathcal{M}(s)\}_{s\in \mathrm{D}}\), in order to put a model structure on it: The main theorem says that if \({\mathrm D}\) is a direct category there is a diagram projective model category structure on the category of \(\mathcal{M}\)-diagrams whose fibrations and weak equivalences are defined objectwise. Dually, when \({\mathrm D}\) is an inverse category there is a diagram injective model category structure on the category of \(\mathcal{M}\)-diagrams whose cofibrations and weak equivalences are defined objectwise.NEWLINENEWLINENext they show that for \(\mathrm{D}\) a finite inverse category with at most one morphism in each set \(\mathrm{D}(s,t)\) and \(R\) a \(\mathrm{D}\)-diagram of ring spectra with homotopy inverse limit \(\hat{R}\), then there is a Quillen equivalence between the category of \(\hat{R}\)-modules and the cellularization with respect to \(R\) of \(R\)-modules.NEWLINENEWLINEIn the last section they consider the inclusion of a subcategory \(i:\mathrm{D}\to \mathrm{E}\) and its induced restriction functor on diagram categories over \(\mathrm{D}\) and \(\mathrm{E}\). It is showed that a Quillen equivalence is induced after certain cellularizations (or colocalizations).