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Homelocally compact groupoids

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# locally compact groupoids

# 1 Locally compact groupoids

This is a specific topic entry defining the basics of locally compact groupoids and related concepts.

Let us first recall the related concepts of groupoid and *topological groupoid*,
together with the appropriate notations needed to define a *locally compact groupoid*.

# 1.0.1 Groupoids and topological groupoids: categorical definitions

Recall that a groupoid ${\mathsf{G}}$ is a small category with inverses over its set of objects $X=Ob({\mathsf{G}})$ . One writes ${\mathsf{G}}^{y}_{x}$ for the set of morphisms in ${\mathsf{G}}$ from $x$ to $y$ .

*A topological groupoid* consists of a space ${\mathsf{G}}$, a distinguished subspace
${\mathsf{G}}^{{(0)}}={\rm Ob(\mathsf{G)}}\subset{\mathsf{G}}$, called the space of objects of ${\mathsf{G}}$,
together with maps

$r,s~{}:~{}\xymatrix{{\mathsf{G}}\ar@<1ex>[r]^{r}\ar[r]_{s}&{\mathsf{G}}^{{(0)}}}$ | (1.1) |

called the range and source maps respectively, together with a law of composition

$\circ~{}:~{}{\mathsf{G}}^{{(2)}}:={\mathsf{G}}\times_{{{\mathsf{G}}^{{(0)}}}}{% \mathsf{G}}=\{~{}(\gamma_{1},\gamma_{2})\in{\mathsf{G}}\times{\mathsf{G}}~{}:~% {}s(\gamma_{1})=r(\gamma_{2})~{}\}~{}{\longrightarrow}~{}{\mathsf{G}}~{},$ | (1.2) |

such that the following hold :

- (1)
$s(\gamma_{1}\circ\gamma_{2})=r(\gamma_{2})~{},~{}r(\gamma_{1}\circ\gamma_{2})=% r(\gamma_{1})$ , for all $(\gamma_{1},\gamma_{2})\in{\mathsf{G}}^{{(2)}}$ .

- (2)
$s(x)=r(x)=x$ , for all $x\in{\mathsf{G}}^{{(0)}}$ .

- (3)
$\gamma\circ s(\gamma)=\gamma~{},~{}r(\gamma)\circ\gamma=\gamma$ , for all $\gamma\in{\mathsf{G}}$ .

- (4)
$(\gamma_{1}\circ\gamma_{2})\circ\gamma_{3}=\gamma_{1}\circ(\gamma_{2}\circ% \gamma_{3})$ .

- (5)
Each $\gamma$ has a two–sided inverse $\gamma^{{-1}}$ with $\gamma\gamma^{{-1}}=r(\gamma)~{},~{}\gamma^{{-1}}\gamma=s(\gamma)$ .

Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call ${\mathsf{G}}^{{(0)}}=Ob({\mathsf{G}})$ the set of objects of ${\mathsf{G}}$ . For $u\in Ob({\mathsf{G}})$, the set of arrows $u{\longrightarrow}u$ forms a group ${\mathsf{G}}_{u}$, called the

*isotropy group of ${\mathsf{G}}$ at $u$*.

Thus, as is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalize bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to ref. [1].

# 1.1 Locally compact and analytic groupoids

###### Definition 1.1.

A *locally compact groupoid* ${\mathsf{G}}_{{lc}}$ is defined as a groupoid that has also the topological structure of a second countable, locally compact Hausdorff space, and if the product and also inversion maps are continuous. Moreover, each ${\mathsf{G}}_{{lc}}^{u}$ as well as the unit space ${\mathsf{G}}_{{lc}}^{0}$ is closed in ${\mathsf{G}}_{{lc}}$.

###### Remark 1.1.

The locally compact Hausdorff second countable spaces are analytic.

One can therefore say also that ${\mathsf{G}}_{{lc}}$ is analytic.

When the groupoid ${\mathsf{G}}_{{lc}}$ has only one object in its object space, that is, when it becomes a group, the above
definition is restricted to that of a *locally compact topological group*; it is then a special case of a one-object category with all of its morphisms being invertible, that is also endowed with a locally compact, topological structure.

# References

- 1
R. Brown. (2006).
*Topology and Groupoids*. BookSurgeLLC

## Mathematics Subject Classification

46M20*no label found*81R15

*no label found*46L05

*no label found*81R50

*no label found*18B40

*no label found*55U40

*no label found*22A22

*no label found*

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