Kerodon

Example 6.3.1.18. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, which we identify with edges of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with respect to $W$. Then the induced functor $\operatorname{\mathcal{C}}\simeq \mathrm{h} \mathit{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \xrightarrow { \mathrm{h} \mathit{F} } \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ exhibits the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ as a $1$-categorical localization of $\operatorname{\mathcal{C}}$ with respect to $W$, in the sense of in the sense of Definition 6.3.0.6.