Kerodon

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

Corollary 3.3.6.8. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Then $f$ is a weak homotopy equivalence if and only $\operatorname{Ex}^{\infty }(f)$ is a weak homotopy equivalence.

Proof. We have a commutative diagram

\[ \xymatrix { X \ar [d]^{ \rho ^{\infty }_ X} \ar [r]^-{f} & Y \ar [d]^{ \rho ^{\infty }_{Y} } \\ \operatorname{Ex}^{\infty }(X) \ar [r]^-{ \operatorname{Ex}^{\infty }(f) } & \operatorname{Ex}^{\infty }(Y), } \]

where the vertical maps are weak homotopy equivalences (Proposition 3.3.6.7). The desired result now follows from the two-out-of-three property (Remark 3.1.5.16). $\square$