# 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 $\U0001d5a6$ is a small category with inverses^{}
over its set of objects $X=Ob(\U0001d5a6)$ . One writes ${\U0001d5a6}_{x}^{y}$ for
the set of morphisms^{} in $\U0001d5a6$ from $x$ to $y$ .

*A topological groupoid* consists of a space $\U0001d5a6$, a distinguished subspace
${\U0001d5a6}^{(0)}=\mathrm{Ob}(\U0001d5a6)\subset \U0001d5a6$, called the space of objects of $\U0001d5a6$,
together with maps

$$r,s:$$ | (1.1) |