# 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. 1.

The space of objects $G_{0}$ and the space of arrows $G_{1}$ are both smooth manifolds

2. 2.

Both structure maps $s,t:G_{1}\longrightarrow G_{0}$ are smooth

3. 3.

All structure maps are submersions:

 $s,t:G_{1}\longrightarrow G_{0},$
 $u:G_{0}\longrightarrow G_{1},$
 $i:G_{1}\longrightarrow G_{1},$

and

 $m:G_{1}\times_{s,t}G_{1}\longrightarrow G_{1}$

.

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 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 Functor