Definition 8.5.9.1 (The Thompson Groupoid). We define a category $\mathrm{Dy}$ as follows:
The objects of $\mathrm{Dy}$ are closed intervals of the form $[0,s]$, where $s \geq 0$ is a dyadic rational number.
If $s,t \geq 0$ are dyadic rational numbers, then a morphism from $[0,s]$ to $[0,t]$ in the category $\mathrm{Dy}$ is a dyadic homeomorphism $[0,s] \xrightarrow {\sim } [0,t]$ (see Definition 8.5.7.9).
The composition law on $\mathrm{Dy}$ is given by composition of dyadic homeomorphisms (which is well-defined by virtue of Exercise 8.5.7.11).
It follows from Exercise 8.5.7.10 that the category $\mathrm{Dy}$ is a groupoid. We will refer to $\mathrm{Dy}$ as the Thompson groupoid.