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In [[mathematics]], a '''Lie groupoid''' is a [[groupoid]] where the set <math>Ob</math> of [[object (category theory)|object]]s and the set <math>Mor</math> of [[morphism]]s are both [[manifold]]s, the source and target operations


:<math>s,t : Mor \to Ob </math>


are  [[submersion (mathematics)|submersion]]s, and all the [[category (mathematics)|category]] operations (source and target, composition, and identity-assigning map) are smooth.
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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 (mathematics)|group]]. Just as every Lie group has a [[Lie algebra]], every Lie groupoid has a [[Lie algebroid]].
 
==Examples==
 
*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 <math>M</math>, there is a Lie groupoid called the pair groupoid, with <math>M</math> 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 <math>M \times M</math>.
 
*Given a Lie group <math>G</math> acting on a manifold <math>M</math>, there is a Lie groupoid called the [[translation groupoid]] with one morphism for each triple <math>g \in G, x,y \in M</math> with <math>gx = y</math>.
 
*Any [[foliation]] gives a Lie groupoid.
 
*Any [[principal bundle]] <math>P\to M</math> with structure group ''G'' gives a groupoid, namely <math>P\times P/G</math> over ''M'', where ''G'' acts on the pairs componentwise. Composition is defined via compatible representatives as in the pair groupoid.
 
==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 <math>\{U_\alpha\}</math> an open cover of ''M''.  Define <math>G_0:=\bigsqcup_\alpha U_\alpha</math> the disjoint union with the obvious submersion <math>p:G_0\to M</math>. In order to encode the structure of the manifold ''M'' define the set of morphisms <math>G_1:=\bigsqcup_{\alpha,\beta}U_{\alpha\beta}</math> where <math>U_{\alpha\beta}=U_\alpha \cap U_\beta\subset M</math>. The source and target map are defined as the embeddings <math>s:U_{\alpha\beta}\to U_\alpha</math> and <math>t:U_{\alpha\beta}\to U_\beta</math>.  And multiplication is the obvious one if we read the <math>U_{\alpha\beta}</math> as subsets of ''M'' (compatible points in <math>U_{\alpha\beta}</math> and <math>U_{\beta\gamma}</math> actually are the same in ''M'' and also lie in <math>U_{\alpha\gamma}</math>).
 
This Čech groupoid is in fact the [[pullback groupoid]] of  <math>M\Rightarrow M</math>, 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 <math>G_1\Rightarrow G_0</math> and <math>H_1\Rightarrow H_0</math> are Morita equivalent iff there exists a third groupoid <math>K_1\Rightarrow K_0</math> 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 <math>G_0/G_1 = H_0/H_1</math> and secondly the stabilizer groups <math>G_p\cong H_q</math> for corresponding points <math>p\in G_0</math> and <math>q\in H_0</math>.
 
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]].
 
===Examples===
*The notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks.
*But also [[orbifold]]s are smooth stacks, namely (equivalence classes of) [[Etale#.C3.89tale_morphisms_and_the_inverse_function_theorem|étale]] groupoids.
*Orbit spaces of foliations are another class of examples
 
==External links==
 
Alan Weinstein, Groupoids: unifying internal and external
symmetry, ''AMS Notices'', '''43''' (1996), 744-752. Also available as [http://arxiv.org/abs/math/9602220 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
 
[[Category:Differential geometry]]
[[Category:Lie groupoids]]
[[Category:Manifolds]]
[[Category:Symmetry]]

Latest revision as of 06:15, 5 August 2014


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