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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 3

### Proceedings of a Conference in Honor of Douglas C. Ravenel and W. Stephen Wilson

### Diagrams indexed by Grothendieck constructions

Pages: 193 – 221

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n3.a10

#### Author

#### Abstract

Let $I$ be a small indexing category, $G\colon I^{op} \to {\mathcal Cat}$ be a functor and $BG \in {\mathcal Cat}$ 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 $\mathcal M$.

#### Keywords

Grothendieck construction; stacks; homotopy theory of diagrams

#### 2010 Mathematics Subject Classification

18G55, 55Pxx

Published 1 January 2008