Kerodon

Example 4.8.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category (Definition 1.4.5.3). Then the comparison map $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ of Construction 1.4.5.6 exhibits $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ as a homotopy $1$-category of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.4.1. This is a reformulation of Proposition 1.4.5.7 (see Example 4.8.1.3). Stated more informally, there is a canonical isomorphism of simplicial sets $\mathrm{h}_{\mathit{\leq 1}}\mathit{(\operatorname{\mathcal{C}})} \simeq \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$. We will sometimes abuse notation by identifying the homotopy $1$-category $\mathrm{h}_{\mathit{\leq 1}}\mathit{(\operatorname{\mathcal{C}})}$ with the ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.