Oberseminar Deformationsquantisierung und Geometrie: David Miyamoto (Max Planck Institut für Mathematik, Bonn)

Lie groupoids determined by their orbit spaces

Abstract: The orbit space of a Lie groupoid inherits a natural quotient diffeology. More generally, we have a quotient functor from the Hilsum-Skandalis category of Lie groupoids to the category of diffeological spaces. We introduce the notion of a lift-complete Lie groupoid, and show that the quotient functor restricts to an equivalence of the categories: of lift-complete Lie groupoids with isomorphism classes of submersive bibundles as arrows, and of quasi-étale diffeological spaces with local subductions as arrows. In particular, the Morita equivalence class of a lift-complete Lie groupoid is determined by its diffeological orbit space. Examples of lift-complete Lie groupoids include quasifold groupoids and étale holonomy groupoids of Riemannian foliations.

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