Corollary 11.5.0.13 (04VC)

Corollary 11.5.0.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$: There exists a partially ordered set $A$ and an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(A)$.
$(2)$: For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is either empty or contractible.
$(3)$: Every object of $\operatorname{\mathcal{C}}$ is subterminal.

Proof. The implications $(1) \Rightarrow (2)$ and $(2) \Leftrightarrow (3)$ follow immediately from the definitions. We conclude by observing that if condition $(3)$ is satisfied, then the construction $X \mapsto [X]$ induces a trivial Kan fibration $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Sub}(\operatorname{\mathcal{C}}) )$ (Proposition 11.5.0.12). $\square$