Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 2.3.6.7. For every strict $2$-category $\operatorname{\mathcal{C}}$, there is a canonical isomorphism of simplicial sets

\[ \operatorname{Sing}_{\bullet }^{\operatorname{Path}_{(2)}[\bullet ] }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}), \]

given on $n$-simplices by composition with the lax functor $T_{[n]}: [n] \rightarrow \operatorname{Path}[n]$ of Construction 2.3.6.4. In other words, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is given by

\[ \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}}) \simeq \{ \textnormal{Strict functors $\operatorname{Path}_{(2)}[n] \rightarrow \operatorname{\mathcal{C}}$} \} . \]