Example 2.4.3.12 (00KZ): The Case of an Ordinary Category

Example 2.4.3.12 (The Case of an Ordinary Category). Let $\operatorname{\mathcal{C}}$ be an ordinary category, regarded as a constant simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ via the construction of Example 2.4.2.4. Combining Examples 2.3.1.3 and Examples 2.4.3.11, we obtain isomorphisms

\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \underline{\operatorname{\mathcal{C}}} ). \]

Unwinding the definitions, we see that the composite isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{hc}}_{\bullet }( \underline{\operatorname{\mathcal{C}}})$ is the comparison map of Remark 2.4.3.8. In other words, when restricted to constant simplicial categories, the homotopy coherent nerve of Definition 2.4.3.5 reduces to the classical nerve of Construction 1.3.1.1.