Example 8.5.9.8 (0431)

Example 8.5.9.8. Let $\mathrm{Dy}_{> 0}$ denote the full subcategory of the Thompson groupoid $\mathrm{Dy}$ spanned by the intervals $[0,s]$ where $s > 0$. Note that the action of $\mathrm{Dy}$ on itself (via concatenation) restricts to an action of $\mathrm{Dy}$ on the groupoid $\mathrm{Dy}_{> 0}$, and therefore determines a (strict) functor of $2$-categories

\[ B\mathrm{Dy} \rightarrow \{ \textnormal{Groupoids} \} \quad \quad X \mapsto \mathrm{Dy}_{> 0}. \]

Passing to nerves, we obtain a functor of $\infty $-categories

\[ \operatorname{N}_{\bullet }^{\operatorname{D}}( B \mathrm{Dy} ) \rightarrow \operatorname{\mathcal{S}}\quad \quad X \mapsto \operatorname{N}_{\bullet }( \mathrm{Dy}_{> 0} ), \]

which carries the $1$-morphism $\overline{e}$ of $B \mathrm{Dy}$ to the homotopy idempotent endomorphism

\[ \operatorname{N}_{\bullet }( \mathrm{Dy}_{> 0} ) \rightarrow \operatorname{N}_{\bullet }( \mathrm{Dy}_{> 0} ) \quad \quad [0,s] \mapsto [0,1] \circledast [0,s] = [0, 1+s]. \]

Note that, up to isomorphism, this coincides with the homotopy idempotent endomorphism constructed in Proposition 8.5.7.14; this follows from the observation that the diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ B \operatorname{Aut}_{ \mathrm{Dy} }( [0,1] ) \ar [r]^-{\alpha } \ar [d]^{\sim } & B \operatorname{Aut}_{ \mathrm{Dy} }( [0,1] ) \ar [d]^{\sim } \\ \mathrm{Dy}_{> 0} \ar [r]^-{ [0,1] \circledast } & \mathrm{Dy}_{ > 0} } \]

commutes up to isomorphism (where the vertical maps are the inclusion functors and $\alpha $ is the homomorphism of Construction 8.5.7.13).