Corollary 4.8.2.17 (058L)

Corollary 4.8.2.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be an integer. The following conditions are equivalent:

$(1)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $(n-2)$-truncated.
$(2)$: The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories.
$(3)$: There exists an $n$-coskeletal $\infty $-category $\operatorname{\mathcal{D}}$ which is equivalent to $\operatorname{\mathcal{C}}$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Propositions 4.8.2.13 and 4.8.2.14. The implication $(2) \Rightarrow (3)$ is clear (since $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is an $\infty $-category), and the implication $(3) \Rightarrow (1)$ follows from Proposition 4.8.2.8 (together with Remark 4.8.2.3). $\square$