Topological groups, Lie groups Definition

Morita equivalence of Lie groupoids

Builds on Lie groupoid Submersion Smooth manifold

Definition

For a Lie groupoid $G$ , 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.

The map $G_1 \times_{s, G_0, \phi_0} K_0 \to G_0$ , $(g, y) \mapsto t(g)$ , is a surjective submersion.
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. 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.

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Lie groupoid concept
Smooth manifold concept
Submersion concept
Morita equivalence of Lie groupoids definition

Tag	`TY60P5RE56`
URI	`m2://TY60P5RE56`
Kind	definition
Writing	W0 stub
MSC	22A22, 58H05
Defines	Morita equivalence of Lie groupoids
Uses	Lie groupoid, Submersion, Smooth manifold

Morita equivalence of Lie groupoids

Definition

Remarks

Reading path

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