Lie groupoid

In mathematics, a Lie groupoid is a groupoid where the set ${\displaystyle Ob}$ of objects and the set ${\displaystyle Mor}$ of morphisms are both manifolds, the source and target operations

are submersions, and all the category operations (source and target, composition, and identity-assigning map) are smooth.

A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Just as every Lie group has a Lie algebra, every Lie groupoid has a Lie algebroid.

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 ${\displaystyle U_{\alpha \gamma }}$).

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