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Classifying topos of a topological category

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I will begin by reviewing the ideas of points of toposes and homotopy between maps of toposes ([1],[2]). I then define for a discrete group G, the geometric theory of G-torsors whose models are G-torsors in toposes (aka principal bundles), and construct a classifying topos for this theory. This was first introduced by Grothendieck and Verdier in SGA4 . One will see that the situation is a generalization of construction of classifying spaces in algebraic topology. Next, I will show the construction of classifying toposes of topological categories (i.e. categories internal to Top, see [3]) and Deligne classifying toposes of topological categories (see [4]). I will show that for source-étale topological categories, the classifying topos and Deligne classifying topos are weakly homotopy equivalent. The proof of this appears in [5].

[1] S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic

[2] Peter Johnstone, Topos theory

[3] Graeme Segal, Classifying spaces and spectral sequences

[4] P. Deligne, Théorie de Hodge III

[5] I.Moerdijk, Classifying spaces and classifying topoi

This talk is part of the Cargo series.

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