Corollary 3.5.1.33. A simplicial set $X$ is weakly contractible if and only if it is nonempty and the diagonal map $\delta _{X}: X \hookrightarrow X \times X$ is a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.5.1.33. A simplicial set $X$ is weakly contractible if and only if it is nonempty and the diagonal map $\delta _{X}: X \hookrightarrow X \times X$ is a weak homotopy equivalence.
Proof. Combine Remark 3.5.1.19 with Corollary 3.5.1.32. $\square$