Example 4.8.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is essentially $n$-categorical.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. Moreover, if $n \leq -2$, then $\operatorname{\mathcal{C}}$ is nonempty.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to an $(n,1)$-category.
- $(4)$
For $m \geq n+2$, every morphism $\operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $m$-simplex of $\operatorname{\mathcal{E}}$.
The equivalence $(1) \Leftrightarrow (2)$ follows from Remark 4.8.5.13, the equivalence $(2) \Leftrightarrow (3)$ from Corollary 4.8.3.3, and the equivalence $(2) \Leftrightarrow (4)$ from Corollary 4.8.3.11.