Proposition 9.3.1.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. Then $X$ is $n$-truncated if and only if it satisfies the following condition for each $m \geq n+3$:
- $(\ast _ m)$
Every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.