Example 10.2.5.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $C_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The augmented simplicial object $C_{\bullet }$ is $(-1)$-coskeletal, in the sense of Definition 10.2.5.10.
- $(2)$
The augmented simplicial object $C_{\bullet }$ is essentially constant: that is, it is isomorphic to a constant functor from $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} )$ to $\operatorname{\mathcal{C}}$.
- $(3)$
The functor $C_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carries each morphism in the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}_{+}$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.
- $(4)$
For every augmented simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the restriction map
\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})}( X_{\bullet }, C_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{-1}, C_{-1} ) \]is a homotopy equivalence.
The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ are special cases of Corollary 7.3.3.14, since $[-1]$ is an initial object of the category $\operatorname{{\bf \Delta }}_{+}$. The equivalence $(1) \Leftrightarrow (4)$ follows from Corollary 7.3.6.13.