Corollary 9.3.3.10 (05FH)

Corollary 9.3.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq 0$ be an integer, let $\operatorname{\raise {0.1ex}{\square }}^{n+1}$ denote the simplicial cube of dimension $n+1$ (Notation 2.4.5.2), and let $y \in \operatorname{\raise {0.1ex}{\square }}^{n+1}$ be the final vertex. Let $Q: \operatorname{\raise {0.1ex}{\square }}^{n+1} \rightarrow \Delta ^1$ be the morphism given on vertices by

\[ Q(v) = \begin{cases} 1 & \text{ if $v = y$ } \\ 0 & \text{ otherwise. } \end{cases} \]

Then a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is $(n-2)$-truncated if and only if the composite map

\[ \operatorname{\raise {0.1ex}{\square }}^{n+1} \xrightarrow {Q} \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}} \]

is a limit diagram in $\operatorname{\mathcal{C}}$.

Proof. Let us identify $\operatorname{\raise {0.1ex}{\square }}^{n+1}$ with the iterated join $\{ x \} \star \operatorname{Sd}( \operatorname{\partial \Delta }^ n ) \star \{ y\} $, where $\operatorname{Sd}(\operatorname{\partial \Delta }^ n)$ denotes the subdivision of $\operatorname{\partial \Delta }^ n$ (see Proposition 3.3.3.17). Using Remark 7.1.3.11, we see that $f \circ Q$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if the constant map

\[ \{ x\} \star \operatorname{Sd}(\operatorname{\partial \Delta }^ n) \rightarrow \{ f \} \hookrightarrow \operatorname{\mathcal{C}}_{/Y} \]

is a limit diagram in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. The desired result now follows by combining Proposition 9.3.3.7 with Remark 9.3.1.11. $\square$