User:Francesco Cattafi/sandbox

Morita Equivalence (of Lie groupoids) - work in progress
In differential geometry, Morita equivalence is an equivalence relation between Lie groupoids that preserves many properties. Two Lie groupoids are Morita equivalent if and only if there is a principal bibundles between them or, alternatively, if their categories of principal bundles are equivalent.

It was introduced by Ping Xu in 1991, inspired by the similar notions of Morita equivalence for rings in abstract algebra, and for C* algebras, and building up on Morita equivalence between topological groupoids.

Definition
There are several equivalent definitions.

Definition with bibundles
Two Lie groupoids are Morita equivalent if there is a manifold $$P$$ together with two surjective submersions $$P \to G_0$$ and $$P \to H_0$$ together with a left $$G$$-action and a right $$H$$-action,


 * $$P \to H_0$$ is a principal $$G_1$$-bundle;
 * $$P \to G_0$$ is a principal $$H_1$$-bundle;
 * the two actions commute.

One says also that $$P$$ is a principal bibundle.

Definition with groupoid morphisms
A Lie groupoid map between two Lie groupoids $$G_1 \rightrightarrows G_0$$ and $$H_1\rightrightarrows H_0$$ is called Morita if it is


 * fully faithful: blabla
 * essentially surjective: blabla

Sometimes called a weak equivalence or essential equivalence

Two Lie groupoids $$G_1\rightrightarrows G_0$$ and $$H_1\rightrightarrows H_0$$ are Morita equivalent if and only if there exists a third Lie groupoid $$K_1\rightrightarrows K_0$$ together with two Morita maps from G to K and from H to K.

Equivalence between the definitions
A generalised equivalence is blabla

Then there is a 1-1 correspondence between Morita maps and generalised equivalence?

Properties and objects preserved by Morita equivalence
The following properties are Morita invariant


 * being proper (i.e. the source map is proper)
 * being Hausdorff
 * being transitive (actually much more holds; see below)
 * being ?

Other properties, i.e. being étale (see below), are not Morita invariant.

Moreover, a Morita equivalence between $$G_1\rightrightarrows G_0$$ and $$H_1\rightrightarrows H_0$$ preserves their transverse geometry, i.e. it induces:


 * a homeomorphism between the orbit spaces $$G_0/G_1$$ and $$H_0/H_1$$, where the orbit at $$x\in G_0$$ corresponds to that at $$y\in H_0$$ if blabla;
 * an isomorphism $$G_x\cong H_y$$ between the isotropy groups at corresponding points $$x\in G_0$$ and $$y\in H_0$$;
 * an isomorphism $$\mathcal{N}_x\cong \mathcal{N}_y$$ between the normal representations of the isotropy groups at corresponding points $$x\in G_0$$ and $$y\in H_0$$.

Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic.

Examples

 * Isomorphic Lie groupoids are trivially Morita equivalent.
 * Two Lie groups are Morita equivalent if and only if they are isomorphic as Lie groups.
 * Two unit groupoids are Morita equivalent if and only if the base manifolds are diffeomorphic.
 * Any transitive Lie groupoid is Morita equivalent to its isotropy groups.
 * Given a Lie groupoid $$G\rightrightarrows M$$ and a surjective submersion $$\mu: N\to M$$, the pullback groupoid $$\mu^*G \rightrightarrows N$$ is Morita equivalent to $$G\rightrightarrows M$$.
 * Given a free and proper Lie group action of $$G$$ on $$M$$ (therefore the quotient $$M/G$$ is a manifold), the action groupoid $$G \times M \rightrightarrows M$$ is Morita equivalent to the unit groupoid $$u(M/G) \rightrightarrows M/G$$.
 * A Lie groupoid $$G$$ is Morita equivalent to an étale groupoid if and only if all isotropy groups of $$G$$ are discrete.

A concrete instance of the last example goes as follows. Let M be a smooth manifold and $$\{U_\alpha\}$$ an open cover of $$M$$. Its Čech groupoid $$G_1\rightrightarrows G_0$$ is defined by the disjoint unions $$G_0:=\bigsqcup_\alpha U_\alpha$$ and $$G_1:=\bigsqcup_{\alpha,\beta}U_{\alpha\beta}$$, where $$U_{\alpha\beta}=U_\alpha \cap U_\beta\subset M$$. The source and target map are defined as the embeddings $$s:U_{\alpha\beta}\to U_\alpha$$ and $$t:U_{\alpha\beta}\to U_\beta$$, and the multiplication is the obvious one if we read the $$U_{\alpha\beta}$$ as subsets of M (compatible points in $$U_{\alpha\beta}$$ and $$U_{\beta\gamma}$$ actually are the same in $$M$$ and also lie in $$U_{\alpha\gamma}$$). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion $$p:G_0\to M$$, of the unit groupoid $$M\rightrightarrows M$$. As such, Čech groupoids associated to different open covers of $$M$$ are Morita equivalent.

Relations with differentiable stacks
Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack. For instance, the orbit space is a smooth manifold if the isotropy groups are trivial (as in the example of the Čech groupoid), but it is not smooth in general. The solution is to revert the problem and to define a smooth stack as a Morita-equivalence class of Lie groupoids. The natural geometric objects living on the stack are the geometric objects on Lie groupoids invariant under Morita-equivalence: an example is the Lie groupoid cohomology.

Since the notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks. Other classes of examples include orbifolds, which are (equivalence classes of) proper étale Lie groupoids, and orbit spaces of foliations.

Symplectic Morita equivalence and Poisson dual pairs
A symplectic Morita equivalence between symplectic groupoids is blabla

This is the same thing as a Poisson dual pair between Poisson manifolds, i.e. a couple of

Extra structure, e.g. diffeological Morita equivalence

Morita equivalence is an equivalence relation only between integrable Poisson manifolds