Remark 4.7.6.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq -2$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}}) \]
is surjective for every integer $m \geq n+3$. This follows immediately from Proposition 4.7.1.16. For a stronger statement, see Corollary 4.7.6.25 below.