Remark 9.3.4.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is a monomorphism if and only if it satisfies the following condition for each $m \geq 3$:
- $(\ast _ m)$
Let $\sigma : \Lambda ^{m}_{m} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the composition
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \subset \Lambda ^{m}_{m} \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]coincides with $f$. Then $\sigma $ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.