Locally compact groupoids

Let us first 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: categorical definitions

Recall that a groupoid is a small category with inverses over its set of objects  . One writes for the set of morphisms in from $x$ to $y$ .

A topological groupoid consists of a space , a distinguished subspace , called the space of objects of , together with maps

(1.1)

(1.2)

(1)

$s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ \gamma_2) = r(\gamma_1)$ , for all  .

(2)

$s(x) = r(x) = x$ , for all  .

(3)

$\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma$ , for all  .

(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 the set of objects of  . For , the set of arrows $u \lra u$ forms a group , called the isotropy group of 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].

Locally compact and analytic groupoids

Definition 1.1   A locally compact groupoid 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 as well as the unit space is closed in .

Bibliography

1

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