Example 2.4.3.11 (00KY): Comparison with the Duskin Nerve

Example 2.4.3.11 (Comparison with the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a strict $2$-category and let $\operatorname{\mathcal{C}}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.8). For any partially ordered set $Q$, Remark 2.4.2.9 and Theorem 2.3.5.6 supply bijections

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[Q]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) & \simeq & \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{Str}}}( \operatorname{Path}_{(2)}[Q], \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{ULax}}}( Q, \operatorname{\mathcal{C}}). \end{eqnarray*}

Restricting to partially ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain an isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is the homotopy coherent nerve of Definition 2.4.3.5 and $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is the Duskin nerve of Construction 2.3.1.1.