#### Volume 13, issue 2 (2013)

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An étalé space construction for stacks

### David Carchedi

Algebraic & Geometric Topology 13 (2013) 831–903
##### Abstract

We generalize the notion of a sheaf of sets over a space to define the notion of a small stack of groupoids over an étale stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of $2$–categories between small stacks over an étale stack and local homeomorphisms over it. These results hold for a wide variety of types of spaces, for example, topological spaces, locales, various types of manifolds, and schemes over a fixed base (where stacks are taken with respect to the Zariski topology). Along the way, we also prove that the $2$–category of topoi is fully reflective in the $2$–category of localic stacks.

##### Keywords
étalé space, étale stack, groupoid, topological stack, differentiable stack, action groupoid, topos, topoi
##### Mathematical Subject Classification 2010
Primary: 22A22, 58H05, 53C08
Secondary: 18B25, 14A20, 18F20
##### Publication
Received: 24 April 2012
Revised: 16 October 2012
Accepted: 28 October 2012
Published: 30 March 2013
##### Authors
 David Carchedi Max Planck Institute for Mathematics Vivatsgasse 7 D-53113 Bonn Germany http://people.mpim-bonn.mpg.de/carchedi/