Remark 3.2.4.18. Let $f: X \rightarrow Y$ be a weak homotopy equivalence of simplicial sets. Then $X$ is weakly contractible if and only if $Y$ is weakly contractible (see Remark 3.1.6.16). If $f$ is a homotopy equivalence, then $X$ is contractible if and only if $Y$ is contractible (see Remark 3.1.6.7).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$