Corollary 7.2.4.12. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then $\operatorname{\mathcal{C}}$ is filtered if and only if it is contractible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. If $\operatorname{\mathcal{C}}$ is a contractible Kan complex, then there exists a categorical equivalence $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$, so that $\operatorname{\mathcal{C}}$ is filtered by virtue of Corollary 7.2.4.11. The converse is a special case of Proposition 7.2.4.9. $\square$