Journal of Noncommutative Geometry· 2026Q1
Lie groupoids determined by their orbit spaces
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- Q1SCImago
- 2026year
Short summary
Lift-complete Lie groupoids are uniquely determined by their orbit spaces, establishing an equivalence between categories of Lie groupoids and diffeological spaces.
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Key points
- Introduces lift-complete Lie groupoids.
- Establishes an equivalence between lift-complete Lie groupoids and quasi-étale diffeological spaces.
- Shows that the Morita equivalence class of a lift-complete Lie groupoid is determined by its diffeological orbit space.
- Provides examples including quasifold groupoids and étale holonomy groupoids.
AI-generated from the title and abstract; the full text is not read.
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 submersive bibundles as arrows, and of quasi-étale diffeological spaces with plotwise submersions 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.
The authors' abstract, as published at the source. Journal of Noncommutative Geometry, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics