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~{}:~{}\hbox{}

(1.1)