Example 4.8.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then:
For every integer $n \leq -2$, the unique functor $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ exhibits $\Delta ^0$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$.
If $\operatorname{\mathcal{C}}$ is nonempty, then $F$ also exhibits $\Delta ^0$ as a homotopy $(-1)$-category of $\operatorname{\mathcal{C}}$.
If $\operatorname{\mathcal{C}}$ is empty, then the identity map $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \emptyset $ exhibits the empty simplicial set as a homotopy $(-1)$-category of $\operatorname{\mathcal{C}}$.