Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lie groupoids determined by their orbit spaces

David Miyamoto

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.

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.
Paper Structure (10 sections, 29 theorems, 65 equations, 1 table)

This paper contains 10 sections, 29 theorems, 65 equations, 1 table.

Table of Contents

  1. Introduction
  2. Diffeology and Lie groupoids
  3. Diffeology
  4. Lie Groupoids
  5. Lift-complete Lie groupoids
  6. Necessary lemmas
  7. Lift-complete pseudogroups and PDEs
  8. An equivalence of categories
  9. Examples and extensions
  10. Application to Riemannian foliations

Key Result

Theorem 1

The quotient functor $\mathbf{F}$ restricts to a functor from the bicategory of effective quasifold groupoids, locally invertible bibundles, and isomorphisms of bibundles, to the category of diffeological quasifolds and local diffeomorphisms. This restriction is: Furthermore, the stack represented by an effective quasifold groupoid is determined by its diffeological orbit space, and $\mathbf{F}$

Theorems & Definitions (85)

  • Theorem : KM22
  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • ...and 75 more