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# 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 $M$.

Keywords:

groupoid, Lie structures, smooth manifold

Related:

Groupoid,GroupoidRepresentation4,RepresentationsOfLocallyCompactGroupoids,Functor

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

22E70*no label found*22E60

*no label found*20F40

*no label found*22A22

*no label found*20L05

*no label found*

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