Kerodon

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Notation 9.2.4.26. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We let $\operatorname{Sub}(X)$ denote the set $\operatorname{Sub}( \operatorname{\mathcal{C}}_{/X} )$ of isomorphism classes of subterminal objects of $\operatorname{\mathcal{C}}_{/X}$ (see Notation 9.2.2.18). If $f: X_0 \hookrightarrow X$ is a monomorphism, we write $[X_0] \in \operatorname{Sub}(X)$ for the isomorphism class of $f$. We will sometimes abuse notation by identifying the isomorphism class $[X_0]$ with the object $X_0$ itself: by virtue of Remark 9.2.2.19, this identification is essentially harmless provided that $X_0$ is understood as an object of the slice $\infty$-category $\operatorname{\mathcal{C}}_{/X}$ (that is, provided that we remember the data of the monomorphism $f$). We will refer to $\operatorname{Sub}(X)$ as the set of subobjects of $X$, and we endow it with the partial ordering described in Notation 9.2.2.18: that is, if $f_0: X_0 \hookrightarrow X$ and $f_1: X_1 \hookrightarrow X$ are monomorphisms, we write $[X_0] \subseteq [X_1]$ if there exists a $2$-simplex

$\xymatrix@R =50pt@C=50pt{ X_0 \ar [dr]_{f_0} \ar [rr]^{g} & & X_1 \ar [dl]^{f_1} \\ & Y & }$

in the $\infty$-category $\operatorname{\mathcal{C}}$. In this case, $g$ is automatically a monomorphism (see Remark 9.2.4.15).