"Cartan Stacks" and Reduction of Lie Pseudogroups

A Lie pseudogroup is a geometric object that encodes local symmetries of geometric structures. An example is the set of locally defined Poisson diffeomorphisms of a Poisson manifold. In his work on the structure theory of Lie pseudogroups, Élie Cartan introduced a Morita-like notion of equivalence of Lie pseudogroups that aims to remember the abstract group-like structure underlying a Lie pseudogroup by removing, as much as possible, the dependence on the manifold that is acted upon. One can think of a Lie pseudogroup modulo this equivalence as a "Cartan stack". In this talk, I will present an intrinsic reduction procedure of Lie pseudogroups that preserves the equivalence class, thus allowing us to find a "good" representative of a given "Cartan stack". This procedure builds upon Cartan's classical structure theory, which we will briefly present, and uses modern notions such as Lie groupoid/algebroid actions and Lie-Pfaffian groupoids. This is joint work with Marius Crainic.