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Classifying toposes and foliations. (English) Zbl 0727.57029
For any étale topological groupoid G (for example, the holonomy groupoid of a foliation), it is shown that its classifying topos is homotopy equivalent to its classifying space. As an application, we prove that the fundamental group of Haefliger for the (leaf space of) a foliation agrees with the one introduced by Van Est. We also give a new proof of Segal’s theorem on Haefliger’s classifying space \(B\Gamma^ q\).
Reviewer: I.Moerdijk

MSC:
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
18F10 Grothendieck topologies and Grothendieck topoi
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