$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. The following conditions are equivalent:


The projection map $\operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is essentially $n$-categorical.


The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. Moreover, if $n \leq -2$, then $\operatorname{\mathcal{C}}$ is nonempty.


The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to an $(n,1)$-category.


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, the equivalence $(2) \Leftrightarrow (3)$ from Corollary, and the equivalence $(2) \Leftrightarrow (4)$ from Corollary