Kerodon

Definition 9.3.4.26. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. A subobject of $X$ is a subterminal object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$: that is, an object which is given by a monomorphism $f: X_0 \hookrightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Remark 9.3.4.16). In this situation, we will sometimes abuse terminology by referring to $X_0$ as a subobject of $X$ and writing $X_0 \subseteq X$; in this case, we implicitly assume that a monomorphism $X_0 \hookrightarrow X$ has been specified.