Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.3.6.7. Let $X$ be a simplicial set. Then the comparison map $\rho _{X}^{\infty }: X \rightarrow \operatorname{Ex}^{\infty }(X)$ is a weak homotopy equivalence.

Proof. By virtue of Proposition 3.2.7.3, it will suffice to show that for each $n \geq 0$, the composite map

\[ X \xrightarrow { \rho _{X} } \operatorname{Ex}(X) \xrightarrow { \rho _{ \operatorname{Ex}(X) } } \cdots \xrightarrow { \rho _{ \operatorname{Ex}^{n-1}(X)} } \operatorname{Ex}^ n(X) \]

is a weak homotopy equivalence. Proceeding by induction on $n$, we are reduced to showing that each of the comparison maps $\rho _{ \operatorname{Ex}^{n-1}(X) }: \operatorname{Ex}^{n-1}(X) \rightarrow \operatorname{Ex}^{n}(X)$ is a weak homotopy equivalence, which is a special case of Theorem 3.3.5.1. $\square$