Topological groups, Lie groups Definition

Lie groupoid

Builds on Groupoid Smooth manifold Smooth function Lie group Lie group action

Definition

A Lie groupoid $G_1 \rightrightarrows G_0$ is a groupoid with set of objects $G_0$ and set of arrows $G_1$ , with source and target maps $s, t \colon G_1 \to G_0$ , such that:

$G_0$ and $G_1$ are smooth manifolds;
$s$ and $t$ are smooth submersions;
the composition $m \colon G_1 \times_{G_0} G_1 \to G_1$ , defined on pairs $(h, g)$ with $s(h) = t(g)$ , is smooth;
the unit map $u \colon G_0 \to G_1$ , sending $x$ to the identity arrow at $x$ , and the inversion $i \colon G_1 \to G_1$ are smooth.

Since $s$ and $t$ are submersions, the set $G_1 \times_{G_0} G_1 = \set{(h, g) \in G_1 \times G_1 : s(h) = t(g)}$ is an embedded submanifold of $G_1 \times G_1$ , so clause 3 makes sense.

Remarks

Example: a Lie group $G$ is a Lie groupoid $G \rightrightarrows \set{*}$ with one object; the maps $s$ and $t$ to a point are submersions, and composition is the group product.

Example: for a smooth action of a Lie group $G$ on a manifold $M$ , the action groupoid has $G_0 = M$ , $G_1 = G \times M$ , $s(g, x) = x$ , $t(g, x) = g \cdot x$ , and $(h, g \cdot x)(g, x) = (hg, x)$ .

Non-example: the compact group $(\Z/2)^{\N}$ , viewed as a groupoid with one object, is not a Lie groupoid. Its space of arrows is not a manifold: it is totally disconnected but has no isolated points, so it is not locally Euclidean of any dimension.

Some authors allow $G_1$ to be a non-Hausdorff manifold, which is needed for groupoids arising from foliations.

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Groupoid concept
Lie group concept
Lie group action concept
Smooth function concept
Smooth manifold concept
Lie groupoid definition

Tag	`KCP52TPREW`
URI	`m2://KCP52TPREW`
Kind	definition
Writing	W0 stub
MSC	22A22, 58H05
Defines	Lie groupoid
Uses	Groupoid, Smooth manifold, Smooth function, Lie group, Lie group action

Lie groupoid

Definition

Remarks

Reading path

Further reading

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