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