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arXiv · 2310.11968

Lie groupoids determined by their orbit spaces

David Miyamoto
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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-\'{e}tale 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 \'{e}tale holonomy groupoids of Riemannian foliations.

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David Miyamoto. 2023-10-18. Lie groupoids determined by their orbit spaces. https://arxiv.org/abs/2310.11968

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