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Diagrams indexed by Grothendieck constructions. (English) Zbl 1160.18007

The Grothendieck construction takes a diagram of small categories over a category \({\mathcal I}^{\text{op}}\) to a small category over \({\mathcal I}\). The first observation is that this functor is both a left and a right adjoint. By considering the model structure on the category of small categories with weak equivalences the equivalences of categories and with cofibrations the functors which are inclusions on objects, these two adjunctions become Quillen adjunctions. These two adjunctions are even Quillen equivalences if \({\mathcal I}\) is a groupoid. Under some additional conditions, these adjunctions generalize to the setting of simplicial sets by replacing \({\mathcal I}\) by its simplicial nerve. Finally, the preceding constructions are generalized for diagrams of simplicial sets indexed by Grothendieck constructions. The author gives several applications of these results. The main application is a Quillen equivalence between the categories of presheaves of simplicial sets (resp. groupoids) on a stack \({\mathcal M}\) and presheaves of simplicial sets (resp. groupoids) over \({\mathcal M}\).

MSC:

18G55 Nonabelian homotopical algebra (MSC2010)
55P99 Homotopy theory
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