Proposition 8.5.7.14. Let $\operatorname{Aut}_{ \mathrm{Dy} }([0,1])$ be the Thompson group of Definition 8.5.7.12 and let $X = B_{\bullet } \operatorname{Aut}_{ \mathrm{Dy} }([0,1])$ denote its classifying simplicial set (Construction 1.3.2.5). Then the homomorphism $\alpha $ of Construction 8.5.7.13 induces a homotopy idempotent endomorphism $e: X \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{S}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We wish to show that the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{e} & \\ X \ar [ur]^{e} \ar [rr]^{e } & & X } \]
commutes up to homotopy. By virtue of Proposition 1.5.3.3, this is equivalent to the assertion that the homomorphisms $\alpha , \alpha ^2: \operatorname{Aut}_{ \mathrm{Dy} }([0,1]) \rightarrow \operatorname{Aut}_{ \mathrm{Dy} }([0,1])$ are conjugate: that is, there exists an element $g \in \operatorname{Aut}_{ \mathrm{Dy} }([0,1])$ satisfying the identity $\alpha (f) \circ g = g \circ \alpha ^2(f)$ for every element $f \in \operatorname{Aut}_{ \mathrm{Dy}}( [0,1] )$. Concretely, we can take $g$ to be any dyadic homeomorphism satisfying the identity $g(x) = 2x$ for $0 \leq x \leq 1/4$. $\square$