Corollary 4.8.2.15 (058J)

Corollary 4.8.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category. The following conditions are equivalent:

$(1)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $0$-truncated.
$(2)$: The comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}})$ is an equivalence of $\infty $-categories.
$(3)$: The comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is a trivial Kan fibration.
$(4)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to (the nerve of) an ordinary category.

Proof. The implication $(1) \Rightarrow (2)$ follows from Example 4.8.2.11 and Proposition 4.8.2.14. Since the comparison map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an isofibration (Corollary 4.4.1.9), the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20. The implication $(2) \Rightarrow (4)$ is clear, and the implication $(4) \Rightarrow (1)$ follows from Example 4.8.2.2. $\square$