Definition 9.3.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_0 \rightarrow X$ be a morphism of $\operatorname{\mathcal{C}}$. We say that $f$ is a monomorphism if, for every object $C \in \operatorname{\mathcal{C}}$, the composition map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X_0) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \]
induces a homotopy equivalence of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X_0)$ with a summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$.