A Lie groupoid is a groupoid with smooth structure The notion of Lie groupoid is the groupoid analog of Lie group.
A Lie groupoid is an internal groupoid in Diff. One can define a Lie groupoid to be an internal groupoid in the sense of Ehresmann?, which includes as data the manifold of composable pairs, or take the conventional route and specify that the source and target maps are submersions. This ensures the pullback exists to define said manifold or composable pairs.
Note that originally Lie groupoids were called differentiable groupoids (and also one considered differentiable categories). Sometime in the 1980s there was a change of terminology. (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 the infinitesimally approximation to a Lie group is a Lie algebra, so the infinitesimal approximation to a Lie groupoid is a Lie algebroid.
See
Topological and differentiable (or smooth, “Lie”) groupoids (and more generally categories) were introduced in
Reviews and developments of the theory of Lie groupoids include
Pradines, ….
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005, xxxviii + 501 pages (website)
K. C. H. 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)
John Baez talks about various kinds of Lie groupoids in TWF 256.
Revision on July 20, 2010 at 07:04:49 by Urs Schreiber See the history of this page for a list of all contributions to it.