Lie groupoids determined by their orbit spaces

Abstract

Given a Lie groupoid, we can form its orbit space, which carries a natural 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 surjective submersive bibundles as arrows, and of quasi-étale diffeological spaces with surjective local subductions as arrows. In particular, the Morita equivalence class of a lift-complete Lie groupoid, alternatively a lift-complete differentiable stack, is determined by its diffeological orbit space. Examples of lift-complete Lie groupoids include quasifold groupoids and étale holonomy groupoids of Riemannian foliations.

34 pages. Grouped several lemmas in a new subsection 3.1. Added subsection 3.2, where we show how certain PDEs generate examples of lift-complete pseudogroups. Section 5 is slightly re-arranged in response to the addition of subsection 3.2. In particular the proof of Proposition 5.4 in v1 has been re-done as Proposition 5.3