Remark 9.3.2.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an object $X \in \operatorname{\mathcal{C}}$ is discrete if and only if it satisfies the following condition for every integer $m \geq 3$:
- $(\ast _ m)$
Every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.
In this case, $X$ is subterminal if and only if it also satisfies condition $(\ast _2)$. See Proposition 9.3.1.18.