Lie groupoid

Definition 0.1.

A Lie groupoid is is a categoryMathworldPlanetmath 𝒢L=(G0,G1) in which every arrow or morphism is invertible, and also such that the following conditions are satisfied:

  1. 1.

    The space of objects G0 and the space of arrows G1 are both smooth manifoldsMathworldPlanetmath

  2. 2.

    Both structure mapsPlanetmathPlanetmathPlanetmath s,t:G1G0 are smooth

  3. 3.

    All structure maps are submersionsMathworldPlanetmath:





Notes: A Lie groupoid can be considered as a generalization of a Lie groupMathworldPlanetmath, but it does have the additional requirements for the groupoidPlanetmathPlanetmathPlanetmath’s structure maps that do not have corresponding conditions in the simpler case of the Lie group structure. Because the object space G0 of a Lie groupoid 𝒢L is a smooth manifold, G0 is denoted in this case as M.

Title Lie groupoid
Canonical name LieGroupoid
Date of creation 2013-03-22 19:19:21
Last modified on 2013-03-22 19:19:21
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 15
Author bci1 (20947)
Entry type Definition
Classification msc 22E70
Classification msc 22E60
Classification msc 20F40
Classification msc 22A22
Classification msc 20L05
Related topic Groupoid
Related topic GroupoidRepresentation4
Related topic RepresentationsOfLocallyCompactGroupoids
Related topic FunctorMathworldPlanetmath