Corollary 2.3.4.6 (02BG)

Corollary 2.3.4.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its coarse homotopy category, and let $F: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be the functor of Proposition 2.2.8.3. Then the induced map of simplicial sets

\[ \operatorname{N}_{\bullet }^{\operatorname{D}}(F): \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \]

exhibits $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy category of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, in the sense of Definition 1.3.6.1.

Proof. Let $\operatorname{\mathcal{D}}$ be a category, which we regard as a $2$-category having only identity morphisms. We wish to show that every morphism of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ factors uniquely through the morphism $\operatorname{N}_{\bullet }^{\operatorname{D}}(F)$. By virtue of Theorem 2.3.4.1, this is equivalent to the assertion that every strictly unitary lax functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ factors uniquely through $F$, which follows from Proposition 2.2.8.3. $\square$