Kerodon

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

Proposition 3.1.5.12. Let $f: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets. Then $f$ is a weak homotopy equivalence.

Proof of Proposition 3.1.5.12. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets; we wish to show that $i$ is a weak homotopy equivalence. Let $X_{}$ be any Kan complex. It follows from Corollary 3.1.3.6 that the restriction map $\theta : \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}(A_{}, X_{} )$ is a trivial Kan fibration. In particular, $\theta $ is a homotopy equivalence (Proposition 3.1.5.9), and therefore induced a bijection on connected components $\pi _0( \operatorname{Fun}( B_{}, X_{} ) ) \rightarrow \pi _0( \operatorname{Fun}( A_{}, X_{} ) )$ (Remark 3.1.5.5). $\square$