Topology Seminar: David Carchedi

We will introduce infinity-topoi as generalized topological spaces, and show how using this language unifies the notion of manifold with that of etale differentiable stacks (generalized orbifolds) and their higher-categorical analogues. We will then give a completely categorical description of etale stacks in terms of a representability theorem. This theorem gives a recipe for constructing moduli stacks of geometric structures, and we will explain some examples of how this produces moduli-stacks presented by Lie groupoids that have been well studied in the foliation theory literature. Finally, we will explain how a generalization of Segal's theorem follows which describes the homotopy type of certain classifying spaces, and will explain the connection to the classification of foliations with transverse structures.