Remark 9.3.1.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. The following conditions are equivalent:
- $(1)$
The object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated, in the sense of Definition 9.3.1.1.
- $(2)$
The constant map $\operatorname{\partial \Delta }^{n+2} \rightarrow \{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ exhibits $X$ as a power of itself by $\operatorname{\partial \Delta }^{n+2}$, in the sense of Definition 7.1.2.1.
- $(3)$
The constant map
\[ ( \operatorname{\partial \Delta }^{n+2} )^{\triangleright } \simeq \Lambda ^{n+3}_{n+3} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}} \]is a limit diagram in $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.3.4.
The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 3.5.9.22, and the equivalence $(2) \Leftrightarrow (3)$ from Remark 7.1.2.6.