Warning 7.2.5.7. The converse of Corollary 7.2.5.6 is false. For example, if $\operatorname{\mathcal{C}}$ is a simply connected Kan complex, then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is automatically filtered. However, $\operatorname{\mathcal{C}}$ is filtered if and only if it is contractible (Corollary 7.2.4.12).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$