Remark 5.5.5.8 (025K): Passage to the Homotopy Category

Remark 5.5.5.8 (Passage to the Homotopy Category). Let $\operatorname{Cat}_{\bullet }$ denote the simplicial category associated to the strict $2$-category $\mathbf{Cat}$ (see Example 2.4.2.8). For every pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, Corollary 1.5.3.5 supplies a comparison map

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} ) ) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{\mathcal{D}}} ) ). \]

This construction is compatible with composition, and therefore determines a functor of simplicial categories

\[ \operatorname{\mathbf{QCat}}\rightarrow \operatorname{Cat}_{\bullet } \quad \quad \operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}. \]

Passing to homotopy coherent nerves (and invoking Example 2.4.3.11), we obtain a functor of $(\infty ,2)$-categories

\[ \operatorname{ \pmb {\mathcal{QC}} }= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathbf{QCat}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Cat}_{\bullet } ) \simeq \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathbf{Cat} ). \]

Stated more informally, the construction $\operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ determines a functor from the $(\infty ,2)$-category $\operatorname{ \pmb {\mathcal{QC}} }$ to the ordinary $2$-category $\mathbf{Cat}$.