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Classifying topos of topological categories (II)

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I will continue the discussion of last week. In previous meeting, I reviewed Giraud’s theorem for Grothedieck toposes and gave two example of categories which we proved are indeed toposes by verifying that they satisfy Giraud’s axioms. I also recalled the notions of etale bundles and torsors and gave examples of each. I also constructed the classifying topos of G-torsors for a discrete group G. This week I will extend last week’s results to C-torsors where C is a topological category.

We will see that the situation is a generalization of construction of classifying spaces in algebraic topology. Next, I will illustrate 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]). For source-étale topological categories, the classifying topos and Deligne classifying topos are weakly homotopy equivalent. The proof of this appears in [5].

Special attention will be paid to classifying topos of the simplicial category. I will demonstrate that this topos classifies geometric theory of linear orders.

[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.

This talk is part of the Cargo series.

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