• Grothendieck topos
• Foliations
• First-order theory
• Étale homotopy
• Classifying topos
• Classifying space
• groupoid
• Simplicial sets

In this dissertation we examine the notion of classifying topos of a topological category, and its connections with the theory of groupoids. In the first chapter we introduce the notion of Grothendieck topos, and the general definition of the classifying topos BC of a topological category C. Then we make examples of the “classifying property”, starting with the classical theorem that connects the classifying space of a group with the principal G-bundles on a topological space, and then studying the classifying topos of a group Sh(G) (namely, the category of right G-sets), proving a theorem by Diaconescu that rephrases the classifying property of the classifying space in a categorical context.
Also, we explain the connections between topos cohomology and group cohomology.
Also, a brief overview of the more general statement of the “classifying property”, in term of first-order theories, is given. In the second chapter we examine more in detail the notion of homotopy for a topos, in order to establish a connection with the homotopy of the classifying space.
We present the theory of ́etale homotopy, starting with the example of ́etale coverings on a scheme and then generalising to the case of general Grothendieck topoi. We are then able to consider the homotopy progroups of a Grothendieck topos, and to state the the so-called toposophic Whitehead theorem, that connects isomorphisms in ( ́etale) homotopy with isomorphisms in (topos) cohomology.
We then deepen the context of simplicial objects and sheaves on these, in order to define the nerve of a topological category. This allows us to define and study the clas- sifying space of a topological category, in a way that extends the case of groups.
In the third chapter we prove the comparison theorem: for an s- ́etale topological category C, there is a weak homotopy equivalence between the topoi Sh(BC) and Sh(C). As an application, we restrict to the case of topological groupoids, and consider the Haefliger groupoid Γq. This groupoid “classifies foliations”, in the sense that the existence of certain foliations on an open manifold is equivalent to the existence of a lifting in a diagram involving the classifying space of Γq . We prove a theorem by Segal, following Moerdijk’s alternative proof, according to which Sh(Γq) can be replaced, up to homotoopy, by the classifying space of M(Rq), the monoid of smooth embeddings of Rq into itself.
In the fourth chapter we consider a sort of “inverse question”: given a Grothendieck topos E, is it true that it can be represented as the classifying topos of a groupoid G?
The answer is, in general, negative. It is positive when taking topoi with “enough points”. Also, every topos can be represented as the classifying topos of a “localic groupoid”. These results are due, respectively, to Butz and Moerdijk, and to Joyal and Tierney. We examine the proof of the first theorem and remark the use of a set-theoretic argument that fails in the proof of the second result. Finally, following another article by Moerdijk, we examine how localic groupoids (more precisely, localic groups) enter into the problem of Morita-equivalence of sites, underlying the differences with the standard case of discrete groups on one side (that have a simpler behaviour) and topological groups on the other (whose problems can more easily be solved in the localic context).