Lie Groupoid - Morita Morphisms and Smooth Stacks

Morita Morphisms and Smooth Stacks

Beside isomorphism of groupoids there is a more coarse notation of equivalence, the so called Morita equivalence. A quite general example is the Morita-morphism of the Čech groupoid which goes as follows. Let M be a smooth manifold and an open cover of M. Define the disjoint union with the obvious submersion . In order to encode the structure of the manifold M define the set of morphisms where . The source and target map are defined as the embeddings and . And multiplication is the obvious one if we read the as subsets of M (compatible points in and actually are the same in M and also lie in ).

This Čech groupoid is in fact the pullback groupoid of, i.e. the trivial groupoid over M, under p. That is what makes it Morita-morphism.

In order to get the notion of an equivalence relation we need to make the construction symmetric and show that it is also transitive. In this sense we say that 2 groupoids and are Morita equivalent iff there exists a third groupoid together with 2 Morita morphisms from G to K and H to K. Transitivity is an interesting construction in the category of groupoid principal bundles and left to the reader.

It arises the question of what is preserved under the Morita equivalence. There are 2 obvious things, one the coarse quotient/ orbit space of the groupoid and secondly the stabilizer groups for corresponding points and .

The further question of what is the structure of the coarse quotient space leads to the notion of a smooth stack. We can expect the coarse quotient to be a smooth manifold if for example the stabilizer groups are trivial (as in the example of the Čech groupoid). But if the stabilizer groups change we cannot expect a smooth manifold any longer. The solution is to revert the problem and to define:

A smooth stack is 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. As an example consider the Lie groupoid cohomology.