nLab
Lie groupoid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A Lie groupoid is a groupoid internal to smooth manifolds. This is a joint generalization of smooth manifolds and Lie groups to higher differential geometry.

Regarded in the more general context of smooth groupoids/smooth stacks, Lie groupoids present certain well-behaved such objects, often called differentiable stacks.

Definition

A Lie groupoid XX is an internal groupoid in the category Diff of smooth manifolds.

Since Diff does not have all pullbacks, to ensure that this definition makes sense, one needs to ensure that the space Mor(X)× s,tMor(X)Mor(X) \times_{s,t} Mor(X) 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 s,t:Mor(X)Obj(X)s,t : Mor(X) \to Obj(X) are submersions. This ensures the pullback exists to define said manifold or composable pairs.

But for most practical purposes, the apparently evident 2-category Grpd(Diff)Grpd(Diff) 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 XYX \to Y 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 Grpd(Diff)Grpd(Diff).

See the section 2-Category of Lie groupoids below.

A bit more general than a Lie groupoid is a diffeological groupoid.

Terminology

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?)

Specialisations

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 XX, let X 1 isoX_1^{iso} 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)

Definition

A topological groupoid X 1X 0X_1 \rightrightarrows X_0 is locally trivial if for every point pX 0p\in X_0 there is a neighbourhood UU of pp and a lift of the inclusion {p}×UX 0×X 0\{p\} \times U \hookrightarrow X_0 \times X_0 through (s,t):X 1 isoX 0×X 0(s,t):X_1^{iso}\to X_0 \times X_0.

Clearly for a Lie groupoid X 1 iso=X 1X_1^{iso} = X_1. It is simple to show from the definition that for a transitive Lie groupoid, (s,t)(s,t) 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.

The (2,1)-category of Lie groupoids

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.

One way to deal with this is to equip the 2-category with some structure of a homotopical category and allow morphisms of Lie groupoids to be anafunctors, i.e. spans of internal functors XX^YX \stackrel{\simeq}{\leftarrow} \hat X \to Y.

Such generalized morphisms – called Morita morphisms or generalized morphisms in the literature – are sometimes modeled as bibundles and then called Hilsum-Skandalis morphisms.

Another equivalent approach is to embed Lie groupoids into the context of 2-topos theory:

The (2,1)-topos Sh (2,1)(Diff)Sh_{(2,1)}(Diff) 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 DiffDiff into stacks on DiffDiff. this wider context contains for instance also diffeological groupoids.

Regarded inside this wider context, Lie groupoids are identified with differentiable stacks. The (2,1)-category of Lie groupoids is then the full sub-(2,1)(2,1)-category of Sh (2,1)(Diff)Sh_{(2,1)}(Diff) on differentiable stacks.

For more comments on this, see also the beginning of ∞-Lie groupoid.

Lie algebroids

As the infinitesimally approximation to a Lie group is a Lie algebra, so the infinitesimal approximation to a Lie groupoid is a Lie algebroid.

Higher Lie groupoids

See

Examples

References

Topological and differentiable (or smooth, “Lie”) groupoids (and more generally categories) were introduced in

  • Charles Ehresmann, Catégories topologiques et catégories différentiables Colloque de Géometrie Differentielle Globale (Bruxelles, 1958), 137–150, Centre Belge Rech. Math., Louvain, 1959;

Reviews and developments of the theory of Lie groupoids include

  • Pradines, ….

  • Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005, xxxviii + 501 pages (website)

  • Kirill Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (MathSciNet)

Discussion in the context of foliation theory (foliation groupoids) is in

  • Ieke Moerdijk, Janez Mrčun Introduction to Foliations and Lie Groupoids , Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)

The relation to differentiable stacks is discussed/reviewed in section 2 of

Lie groupoids as a source for groupoid convolution C*-algebras are discussed in

Expository discussion of various kinds of groupoids is also in

Revised on April 8, 2013 16:08:05 by Urs Schreiber (82.113.106.75)