Definition 9.2.4.25. 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 9.2.4.16). 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.

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