Notation 8.5.9.12. Let $m$ and $n$ be nonnegative integers. We will say that a homeomorphism $f: [0,m] \xrightarrow {\sim } [0,n]$ is a dyadic contraction if, for every integer $0 \leq k < m$, the restriction of $f$ to the closed interval $[k, k+1]$ is given by the formula $f(x) = (x+a) / 2^ b$ for some integers $a$ and $b$ with $b \geq 0$.
We let $\mathrm{Dy}_{+}$ denote the subcategory of the Thompson groupoid $\mathrm{Dy}$ whose objects are intervals of the form $[0,m]$, where $m$ is a nonnegative integer, and whose morphisms are dyadic contractions.