---
tag: TY60P5RE56
kind: definition
name: Morita equivalence of Lie groupoids
msc: [22A22, 58H05]
defines: FYXTVSVHT2
uses: [HNVW0E732J, BMY55EW3RF, F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_groupoid
---

## Definition

For a [[Lie groupoid|HNVW0E732J]] $G$, write $G_0$ for its manifold of objects and $G_1$ for its manifold of arrows.
Write $s, t \colon G_1 \to G_0$ for the source and target maps.
Let $\phi = (\phi_1, \phi_0) \colon K \to G$ be a morphism of Lie groupoids.
The morphism $\phi$ is a *weak equivalence* if both of the following hold.

1. The map $G_1 \times_{s, G_0, \phi_0} K_0 \to G_0$, $(g, y) \mapsto t(g)$, is a surjective [[submersion|BMY55EW3RF]].
2. The fibre product $(K_0 \times K_0) \times_{G_0 \times G_0} G_1$, formed with $\phi_0 \times \phi_0$ and $(t, s)$, is a [[smooth manifold|F45VQ47GPV]].
The map $k \mapsto ((t(k), s(k)), \phi_1(k))$ from $K_1$ to it is a diffeomorphism.

Two Lie groupoids $G$ and $H$ are *Morita equivalent* if there are a Lie groupoid $K$ and weak equivalences $K \to G$ and $K \to H$.

## Remarks

Clause 1 says that every object of $G$ is isomorphic to an object in the image of $\phi_0$, in a smooth way.
Clause 2 says that the arrows of $K$ between two objects are exactly the arrows of $G$ between their images.
Morita equivalence is an equivalence relation.
An equivalent definition uses a principal bibundle: a manifold $P$ with commuting actions of $G$ and $H$ and surjective submersions as moment maps.
Each action must be free and transitive on the fibres of the moment map of the other action.

Example: let $\pi \colon U \to M$ be a surjective submersion, and let $K$ be the groupoid with $K_0 = U$ and $K_1 = U \times_M U$, where $(u, v)$ is an arrow from $v$ to $u$.
The map $\pi$ gives a weak equivalence from $K$ to the groupoid $M$ that has only identity arrows.
Clause 1 holds because the map in question is $\pi$ itself.
Clause 2 holds because $K_1$ is exactly the set of pairs of points of $U$ with the same image in $M$.
So $K$ and $M$ are Morita equivalent; taking $U$ to be the disjoint union of the sets of an open cover gives the usual Čech groupoid.

Non-example: a nontrivial Lie group $G$, viewed as a groupoid with one object, is not Morita equivalent to the one-point groupoid with only the identity arrow.
Weak equivalences induce isomorphisms of isotropy groups, and the isotropy groups here are $G$ and the trivial group.

## Explanations

### L1 human

Two Lie groupoids are Morita equivalent when one can pass between them by refining or coarsening the space of objects without changing the symmetries between objects.