locally compact groupoid

Definition 0.1   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}$.

Remarks: 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.

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

Groupoids and Topological Groupoids

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

$\displaystyle r,s~:~ \xymatrix{ {\mathsf{G}}\ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)} }$ (0.1)

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

$\displaystyle \circ~:~ {\mathsf{G}}^{(2)}: = {\mathsf{G}}\times_{{\mathsf{G}}^{...
...{\mathsf{G}}~:~ s(\gamma_1) = r(\gamma_2)~ \}~ {\longrightarrow}~{\mathsf{G}}~,$ (0.2)

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].

Bibliography

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



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