Classifying toposes and foliations
Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 189-209.

On démontre que pour tout groupoïde topologique étale G (par exemple, le groupoïde d’holonomie d’un feuilletage) l’espace classifiant est du même type d’homotopie que le topos classifiant. On déduit que le groupe fondamental de l’espace des feuilles au sens de Haefliger est isomorphe à celui de Van Est. En plus, on donne une nouvelle démonstration du théorème de Segal concernant l’espace classifiant BΓq de Haefliger.

For any etale 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 BΓq.

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Moerdijk, Ieke. Classifying toposes and foliations. Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 189-209. doi : 10.5802/aif.1254. https://numdam.org/articles/10.5802/aif.1254/

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