$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.3.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Let $X$ be an object of $\operatorname{\mathcal{C}}$, and let us abuse notation by identifying $X$ with its image in the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (Notation 4.8.4.2). Then $X$ is $(n-1)$-truncated if and only if the diagram of $\infty $-categories
9.23
\begin{equation} \begin{gathered}\label{equation:discrete-via-slicing} \xymatrix { \operatorname{\mathcal{C}}_{/X} \ar [r] \ar [d] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}_{/X} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} } \end{gathered} \end{equation}
is a categorical pullback square.
Proof.
Since the vertical maps in the diagram (9.23) are right fibrations (Proposition 4.3.6.1), it is a categorical pullback square if and only if, for every object $Y \in \operatorname{\mathcal{C}}$, the induced map of right-pinched morphism spaces
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( Y,X) = \{ Y\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \{ Y\} \times _{ \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} } (\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})})_{/X} = \operatorname{Hom}_{\mathrm{h}_{\mathit{\leq n}}\mathit{(()}\operatorname{\mathcal{C}})}^{\mathrm{R}}( Y,X) \]
is a homotopy equivalence (Corollary 5.1.6.4). It follows from Corollary 4.8.4.8 that $\theta $ exhibits $\operatorname{Hom}_{\mathrm{h}_{\mathit{\leq n}}\mathit{(()}\operatorname{\mathcal{C}})}^{\mathrm{R}}( Y,X)$ as an $(n-1)$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( Y,X)$. In particular, it is a homotopy equivalence if and only if $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( Y,X)$ is $(n-1)$-truncated. This condition is satisfied for every object $Y \in \operatorname{\mathcal{C}}$ if and only if the object $X$ is $(n-1)$-truncated (Remark 9.3.1.2).
$\square$