Lie groupoid

• Any Lie group gives a Lie groupoid with one object, and conversely. So, the theory of Lie groupoids includes the theory of Lie groups.
• Given any manifold ${\displaystyle M}$, there is a Lie groupoid called the pair groupoid, with ${\displaystyle M}$ as the manifold of objects, and precisely one morphism from any object to any other. In this Lie groupoid the manifold of morphisms is thus ${\displaystyle M\times M}$.
• Given a Lie group ${\displaystyle G}$ acting on a manifold ${\displaystyle M}$, there is a Lie groupoid called the translation groupoid with one morphism for each triple ${\displaystyle g\in G,x,y\in M}$ with ${\displaystyle gx=y}$.
• Any foliation gives a Lie groupoid.
• Any principal bundle ${\displaystyle P\to M}$ with structure group G gives a groupoid, namely ${\displaystyle P\times P/G}$ over M, where G acts on the pairs componentwise. Composition is defined via compatible representatives as in the pair groupoid.

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 ${\displaystyle \{U_{\alpha }\}}$ an open cover of M. Define ${\displaystyle G_{0}:=\bigsqcup _{\alpha }U_{\alpha }}$ the disjoint union with the obvious submersion ${\displaystyle p:G_{0}\to M}$. In order to encode the structure of the manifold M define the set of morphisms ${\displaystyle G_{1}:=\bigsqcup _{\alpha ,\beta }U_{\alpha \beta }}$ where ${\displaystyle U_{\alpha \beta }=U_{\alpha }\cap U_{\beta }\subset M}$. The source and target map are defined as the embeddings ${\displaystyle s:U_{\alpha \beta }\to U_{\alpha }}$ and ${\displaystyle t:U_{\alpha \beta }\to U_{\beta }}$. And multiplication is the obvious one if we read the ${\displaystyle U_{\alpha \beta }}$ as subsets of M (compatible points in ${\displaystyle U_{\alpha \beta }}$ and ${\displaystyle U_{\beta \gamma }}$ actually are the same in M and also lie in ${\displaystyle U_{\alpha \gamma }}$).

This Čech groupoid is in fact the pullback groupoid of ${\displaystyle M\Rightarrow M}$, 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 ${\displaystyle G_{1}\Rightarrow G_{0}}$ and ${\displaystyle H_{1}\Rightarrow H_{0}}$ are Morita equivalent iff there exists a third groupoid ${\displaystyle K_{1}\Rightarrow K_{0}}$ 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 ${\displaystyle G_{0}/G_{1}=H_{0}/H_{1}}$ and secondly the stabilizer groups ${\displaystyle G_{p}\cong H_{q}}$ for corresponding points ${\displaystyle p\in G_{0}}$ and ${\displaystyle q\in H_{0}}$.

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.

• Alan Weinstein, Groupoids: unifying internal and external symmetry, AMS Notices, 43 (1996), 744-752. Also available as arXiv:math/9602220

• Kirill Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge U. Press, 1987.

• Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge U. Press, 2005