A Lie groupoid is a groupoid such that both the space of arrows and the space of objects are smooth manifolds, all structure maps are smooth, and source and target maps are surjective submersions.
Since Diff does not have all pullbacks, to ensure that this definition makes sense, one needs to ensure that the space of composable morphisms is an object of Diff. This is achieved either by adopting the definition of internal groupoid in the sense of Ehresmann, which includes as data the smooth manifold of composable pairs, or by taking the conventional route and demanding that the source and target maps are submersions. This ensures the pullback exists to define said manifold or composable pairs. Therefore a definition used in most modern differential geometry literature is as we see above.
But for most practical purposes, the apparently evident 2-category of such internal groupoids, internal functors and internal natural transformations is not the correct 2-category to consider. One way to see this is that the axiom of choice fails in Diff, which means that an internal functor of internal groupoids which is essentially surjective and full and faithful may nevertheless not be an equivalence, in that it may not have a weak inverse in .
See the section 2-Category of Lie groupoids below.
A bit more general than a Lie groupoid is a diffeological groupoid.
A Lie group is a Lie groupoid with and a point. Composition of is provided by the multiplication of .
A manifold is a Lie groupoid with and . Source and target maps are identities and we only have identity arrows in this example.
Given a manifold and an open cover , we can form a Lie groupoid with and . Then for an element , , , and . This is sometimes called the Čech groupoid or covering groupoid.
Given a Lie group (right) action on a manifold , then we may form an associated action groupoid (or sometimes called transformation groupoid) as follows: and . For an element , we have , , and (we must have for the multiplication to happen). Action groupoid presents the quotient stack . Roughly speaking, it is a good replacement for quotient space even if the action is not as nice as you want.
Given a manifold , we may also form so-called pair groupoid: and . Source and target are projections, and multiplication is given by . Pair groupoid may be interpreted as the global object of tangent bundle (think why? see the section below on Lie algebroid).
Given a manifold , we have also an associated fundamental groupoid or homotopy groupoid : paths in homotopies, . Source and target are end points of a path. Multiplication is concatenation of paths (think why associative?).
Given a manifold with a foliation , we may form various groupoids associated with .
-pair groupoid: , . Source and target are obvious projections and multiplication is like in the case of pair groupoid. The problem for this groupoid is that it might not be a Lie groupoid. (why not? for counter example, we refer to Section 13.5 of Geometric Models for Noncommutative Algebras ).
monodromy groupoid (it is a foliation version of fundamental groupoid, thus it is also sometimes called -fundamental groupoid): leaf-wise paths leaf-wise homotopy, and the rest is like in the case of fundamental groupoid.
holonomy groupoid : leaf-wise paths holonomy, and the rest is like in the case of fundamental groupoid. Here, the holonomy of a path is defined to the germ of diffeomorphisms induced by between the transversals at the end points.
Among all possible Lie groupoids associated to a foliation, monodromy groupoid is the biggest and holonomy groupoid is the smallest.
Originally Lie groupoids were called (by Ehresmann) differentiable groupoids (and also one considered differentiable categories). Sometime in the 1980s there was a change of terminology to Lie groupoid and differentiable stacks. (reference?)
One definition which Ehresmann introduced in his paper Catégories topologiques et catégories différentiables (see below) is that of locally trivial groupoid. It is defined more generally for topological categories, and extends in an obvious way to topological groupoids, and Lie categories and groupoids. For a topological (resp. Lie) category , let denote the subspace (resp. submanifold) of invertible arrows . (This always exists, by general abstract nonsense - I should look up the reference, it’s in Bunge-Pare I think - DR)
A topological groupoid is locally trivial if for every point there is a neighbourhood of and a lift of the inclusion through .
Clearly for a Lie groupoid . It is simple to show from the definition that for a transitive Lie groupoid, has local sections. Ehresmann goes on to show a link between smooth principal bundles and transitive, locally trivial Lie groupoids. See locally trivial category for details.
As usual for internal categories, the naive 2-category of internal groupoids, internal functors and internal natural transformations is not quite “correct”. One sign of this is that the axiom of choice fails in Diff so that an internal functor which is an essentially surjective functor and a full and faithful functor may still not have an internal weak inverse.
Another equivalent approach is to embed Lie groupoids into the context of 2-topos theory:
The (2,1)-topos of stacks/2-sheaves on Diff may be understood as a nice 2-category of general groupoids modeled on smooth manifolds. The degreewise Yoneda embedding allows to emebed groupoids internal to into stacks on . this wider context contains for instance also diffeological groupoids.
For more comments on this, see also the beginning of ∞-Lie groupoid.
A Lie algebroid is a vector bundle together with a vector bundle morphism (called anchor map), and a Lie bracket on the space of sections of , satisfying the Leibniz rule
You would expect to preserve , wouldn’t you? It is actually automatic! (see Y. Kosmann-Shwarzbah and F. Magri. Poisson-Nijenhuis strutures. Ann. Inst. H. Poinaré Phys. Théor., 53(1):3581, 1990.)
Recent progress: it turns out that one may link Lie algebroid with -spaces (ask Owen Gwilliam for it)
A Lie algebra is a Lie algebroid with base space being a point.
-bundle over a manifold is certainly a Lie algebroid in a trivial way.
Tangent bundle is a Lie algebroid with and the usual Lie bracket for vector fields. See tangent Lie algebroid.
Given a Poisson manifold with Poisson bivector field , the cotangent bundle is equipped with a Lie algebroid structure: and (or you may have if you prefer to think in Poisson bracket). See Poisson Lie algebroid.
There are several versions of Lie groupoid morphisms, some of them are equivalent in a correct sense, some of them are not.
strict morphism: a strict morphism from Lie groupoid to is a functor from to as categories and preserving the smooth structures.
A Lie groupoid functor is a weak equivalence if it is
essentially surjective; that is, is a surjective submersion;
fully faithful; that is, .
Composition of generalised morphism is given by weak pullback of Lie groupoids (see also weak limit). Given (strict) morphisms and , the weak pullback of along is a groupoid with space of objects and space of morphisms . When is a weak equivalence, the weak pullback is a Lie groupoid thank to the property of essentially surjective. (Is this composition associative?)
Composition of anafunctors is given through strong pullback of Lie groupoids, that is level-wise pullback.
The last three morphisms are more or less equivalent, that is they give arise to equivalent 2-categories (in fact (2,1)-categories) of Lie groupoids. To make it explicit, we need to talk about 2-morphisms between them.
A 2-morphism between bibundle functors is simply a bibundle isomorphism (of course preserving all the structures of bibundles).
A strict 2-morphism from generalised morphism to is given by a morphism such that the following diagram commutes
The idea is that Bundlisation may extend to an equivalence of -categories between , the -category made by generalised morphisms, and . The inverse is given by the following construction: given a bibundle functor , we pull back along the map and obtain a Lie groupoid . Then the natural projection is an acyclic fibration. Thus we obtain a generalised morphism which is also an anafunctor from .
Even though contains more morphisms than , a generalised morphism maybe equivalently replaced by an anafunctor. In fact a generalised morphism gives arise to an anafunctor .
As a consequence of the universal property of the calculus of fractions, and are equivalent.
Morphisms of Lie algebroids are counter-intuitive: they are not morphisms of vector bundles which preserve the algebroid structure. To define a Lie algebroid morphism, we first need to introduce the Chevalley-Eilenberg algebra associated to a Lie algebroid .
We consider to be a trivially graded vector bundle, i.e. concentrated in degree . Then is concentrated in degree . The functions on are given as
where is considered to be of degree , to be of degree , and so forth.
Now we can define a degree-one derivation on as follows: For and , let
The condition is not automatically fulfilled: since , we have . The condition is actually equivalent to being a Lie algebroid; that is, it is fulfilled if and only if
(Proof: calculation gives the restriction and and .)
Example: For the tangent Lie algebroid , .
Then a morphism from a Lie algebroid to is a morphism of the associated differential graded commutative algebras
Such a morphism of c.d.g.a.‘s is determined by maps on degree and a map on degree . Thus a morphism of vector bundles give rise to a morphism of c.g.a. For to be a Lie algebroid morphism, we further need to satisfy additional conditions so that preserves the differential.
This way to explain morphisms of Lie algebroids is described in Kirill Mackenzie, chapter 4.3. If the Lie algebroids are over the same manifold , then a morphism from to can be described as a morphism of vector bundles that respects the anchor maps and the Lie bracket. If, however, is over a different manifold , this direct approach does not work. In this situation we have to pull back the Lie algebroid to (Note that this is not simply the vector bundle pullback of along , but a more involved construction, see Kirill Mackenzie). Using the defintion of a morphism on a common base manifold one arrives at two conditions on the bundle morphism to be a morphism of Lie algebroids. For details see the linked book.
Let be an interval with the tangent bundle Lie algebroid and an arbitrary Lie algebroid on . Then a path defines a map from which respects the commutative graded algebra structure. A function gets mapped to , where is the projection of . A section gets mapped to under , where is the canonical section of . Since is concentrated in degree and , the other degrees get mapped to .
For this map to be a morphism of Lie algebroids, it has to respect the differentials. As explained above this only needs to be checked for a smooth function in :
A quick calculation shows that this is true if and only if
Let the unit square in . Then a Lie algebroid morphism from to , where is a Lie algebra, is given by a morphism of c.d.g.a.
Two smooth maps give a map between c.g.a. spaces above. Here a section , i.e. an element of gets mapped to the one form
where are the coordinates on . A quick calculation shows that this map respects the differential if and only if
This formula also shows that is a flat connection on the trivial principal -bundle on . Therefore Lie algebroid morphisms open a way to talk about higher connections and flat conditions.
The inner automorphism 2-group is a Lie groupoid. There is an obvious morphism .
The fundamental groupoid of a smooth manifold is naturally a Lie groupoid.
An orbifold is a Lie groupoid.
The (1-categorical) pullback
is a Lie groupoid equivalent to this principal bundle .
Similarly an anafunctor from to is a connection on a bundle (see there for details).
Topological and differentiable (or smooth, “Lie”) groupoids (and more generally categories) were introduced in
Reviews and developments of the theory of Lie groupoids include
The relation to differentiable stacks is discussed/reviewed in section 2 of
Expository discussion of various kinds of groupoids is also in
Groupoids and their various morphisms between them in different categories, including in Diff, is also in