Kerodon

Notation 4.8.4.2. Let $n$ be a nonnegative integer. We will see in a moment that for every $\infty $-category $\operatorname{\mathcal{C}}$, there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ which exhibits $\operatorname{\mathcal{C}}'$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Corollary 4.8.4.16). It follows immediately from the definition that the simplicial set $\operatorname{\mathcal{C}}'$ is unique up to (canonical) isomorphism and depends functorially on $\operatorname{\mathcal{C}}$. To emphasize this dependence, we will often denote $\operatorname{\mathcal{C}}'$ by $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ and refer to it as the homotopy $n$-category of $\operatorname{\mathcal{C}}$. For a more explicit description of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (at least for $n > 0$), see Construction 4.8.4.9 (and Proposition 4.8.4.15).