Kerodon

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

Example 3.3.6.3. Let $X$ be a discrete simplicial set (Definition 1.1.4.9). Invoking Example 3.3.4.6 repeatedly, we deduce that the transition maps in the diagram

\[ X \xrightarrow { \rho _{X} } \operatorname{Ex}(X) \xrightarrow { \rho _{ \operatorname{Ex}(X) } } \operatorname{Ex}^2(X) \xrightarrow { \rho _{ \operatorname{Ex}^2(X)} } \operatorname{Ex}^3(X) \rightarrow \cdots , \]

are isomorphisms. It follows that the comparison map $\rho _{X}^{\infty }: X \rightarrow \operatorname{Ex}^{\infty }(X)$ is also an isomorphism.