---
tag: FYXTVSVHT2
kind: concept
name: Morita equivalence of Lie groupoids
msc: [22A22, 58H05]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_groupoid
---

## Summary

Two [[Lie groupoids|HNVW0E732J]] are Morita equivalent when they describe the same transverse geometry, even if their spaces of objects differ.
Formally, they are joined by a third Lie groupoid that maps to both by weak equivalences, or equivalently by a principal bibundle.
Morita equivalent groupoids have homeomorphic orbit spaces, isomorphic isotropy groups along corresponding orbits and the same invariants such as cohomology and K-theory.
This notion makes Lie groupoids a model for singular spaces such as orbifolds, leaf spaces of foliations and quotients by group actions.

## Explanations

### L1 human

A groupoid can be thought of as a space together with rules saying which points count as the same and in how many ways.
Two groupoids can present the same "quotient" in different ways, for example by using a different set of charts.
Morita equivalence says exactly when two groupoids present the same quotient, with the same symmetries at each point.
For example, let a Lie group act freely and properly on a manifold.
Then the action groupoid is Morita equivalent to the quotient manifold, viewed as a groupoid with only identity arrows.
The idea parallels [[Morita equivalence|RD5WKA1VK7]] of rings, where two rings count as equivalent when their modules behave the same.