Kerodon

Example 2.4.6.5. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category (Definition 2.2.0.1) and let $\operatorname{\mathcal{C}}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.8). Then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}_{\bullet }}$ of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Construction 2.4.6.1) can be identified with the coarse homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ of $\operatorname{\mathcal{C}}$ (in the sense of Construction 2.2.8.2).