Proposition 10.2.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The simplicial object $X_{\bullet }$ is essentially constant: that is, there exists an isomorphism of simplicial objects $\alpha : \underline{C} \rightarrow X_{\bullet }$ for some object $C \in \operatorname{\mathcal{C}}$.
- $(2)$
The functor $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carries each morphism in the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
- $(3)$
The functor $X_{\bullet }$ is left Kan extended from the full subcategory $\{ [0] \} \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$.
- $(4)$
For every simplicial object $Y_{\bullet }$ of $\operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}^{\operatorname{op}}), \operatorname{\mathcal{C}})}( X_{\bullet }, Y_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{0}, Y_0 ) \]is a homotopy equivalence.