Lie groupoid
Encyclopedia
In mathematics
, a Lie groupoid is a groupoid
where the set
of objects and the set 
of morphisms are both manifolds
, the source and target operations
are submersion
s, 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
.
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
.
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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Lie groupoid is a groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
where the set
of objects and the set of morphisms are both manifoldsManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, the source and target operations
are submersion
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
s, and all the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, 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
Lie algebroid
In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
.
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
, there is a Lie groupoid called the pair groupoid, with 
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 
.
- Given a Lie group
acting on a manifold 
, there is a Lie groupoid called the translation groupoid with one morphism for each triple 
with 
.
- Any foliationFoliationIn mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
gives a Lie groupoid.
- Any principal bundlePrincipal bundleIn mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
with structure group G gives a groupoid, namely 
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 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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
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
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
.
Examples
- The notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks.
- But also orbifoldOrbifoldIn the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
s are smooth stacks, namely (equivalence classes of) étale groupoids. - Orbit spaces of foliations are another class of examples
External links
Alan Weinstein, Groupoids: unifying internal and externalsymmetry, 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