Corollary 9.3.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. Then $f$ is $n$-truncated if and only if it satisfies the following condition for every positive integer $m \geq n+4$:
- $(\ast _ m)$
If $\sigma : \Lambda ^{m}_{m} \rightarrow \operatorname{\mathcal{C}}$ is a diagram having the property that the composite map
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \hookrightarrow \Lambda ^{m}_{m} \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]is equal to $f$, then $\sigma $ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.