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

Sharon Hollander (Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Lisboa, Portugal)

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

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Published 1 January 2008