Classifying topos of topological categories

I gave a two CARGO talks on classifying topos of topological categories on 14 and 21 November 2017. Here is the abstract:

Abstract: 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]. 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

*Warning**: Notes of this talk will be up here very soon.