Diagrams indexed by Grothendieck constructions

Abstract

Let $I$ be a small indexing category, $G: I^{op} \to \mathcal{C}at$ be a functor and $BG \in \mathcal{C}$ at denote the Grothendieck construction on $G$. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on $BG$ and the category of $I$-diagrams over $N(G)$ (resp. $G$). As an application we obtain 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 $\mathcal{M}$.