Notation 8.5.9.4. Let $B\mathrm{Dy}$ denote the (strict) $2$-category obtained by delooping $\mathrm{Dy}$ (see Example 2.2.0.8). Since $\mathrm{Dy}$ is a groupoid, $B\mathrm{Dy}$ is a $(2,1)$-category. It follows that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ is an $\infty $-category (Theorem 2.3.2.1). We can describe the low-dimensional simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ explicitly as follows:
The $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ has a unique object, which we will denote by $\overline{X}$.
Morphisms from $\overline{X}$ to itself in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ can be identified with nonnegative dyadic rational numbers $s$ (corresponding to the closed interval $[0,s]$, regarded as an object of the Thompson groupoid $\mathrm{Dy}$).
Suppose we are given dyadic rational numbers $s,t,u \geq 0$. Then $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ with boundary indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & \overline{X} \ar [dr]^{ s } & \\ \overline{X} \ar [ur]^{t} \ar [rr]^{u} & & \overline{X} } \]can be identified with dyadic homeomorphisms $[0, s+t ] \xrightarrow {\sim } [0,u]$.
Let $\overline{e}: \overline{X} \rightarrow \overline{X}$ denote the morphism in $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ corresponding to the object $[0,1] \in \mathrm{Dy}$, and let $\overline{\sigma }$ be the $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ corresponding to the dyadic homeomorphism
Then the triple $(\overline{X}, \overline{e}, \overline{\sigma })$ can then be viewed as a partial idempotent $\iota : \operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} )$ (see Example 8.5.8.3).