Lie groupoid
Definition 0.1.
A Lie groupoid is is a category^{} ${\mathcal{G}}_{L}=({G}_{0},{G}_{1})$ in which every arrow or morphism is invertible, and also such that the following conditions are satisfied:

1.
The space of objects ${G}_{0}$ and the space of arrows ${G}_{1}$ are both smooth manifolds^{}

2.
Both structure maps^{} $s,t:{G}_{1}\u27f6{G}_{0}$ are smooth

3.
All structure maps are submersions^{}:
$$s,t:{G}_{1}\u27f6{G}_{0},$$ $$u:{G}_{0}\u27f6{G}_{1},$$ $$i:{G}_{1}\u27f6{G}_{1},$$ and
$$m:{G}_{1}{\times}_{s,t}{G}_{1}\u27f6{G}_{1}$$ .
Notes: A Lie groupoid can be considered as a generalization of a Lie group^{}, but it does have the additional requirements for the groupoid^{}’s structure maps that do not have corresponding conditions in the simpler case of the Lie group structure. Because the object space ${G}_{0}$ of a Lie groupoid ${\mathcal{G}}_{L}$ is a smooth manifold, ${G}_{0}$ is denoted in this case as $M$.
Title  Lie groupoid 
Canonical name  LieGroupoid 
Date of creation  20130322 19:19:21 
Last modified on  20130322 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  Functor^{} 