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 (,). 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 ) and Deligne classifying toposes of topological categories (see ). 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 . Special attention will be paid to classifying topos of the simplicial category. I will demonstrate that this topos classifies geometric theory of linear orders.
 S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic
 Peter Johnstone, Topos theory
 Graeme Segal, Classifying spaces and spectral sequences
 P. Deligne, Théorie de Hodge III
 I.Moerdijk, Classifying spaces and classifying topoi
*Warning**: Notes of this talk will be up here very soon.