Kerodon

Remark 2.3.2.10. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{Pith}(\operatorname{\mathcal{C}}) )$ is an $\infty $-category (Theorem 2.3.2.1). Unwinding the definitions, we see that $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{Pith}(\operatorname{\mathcal{C}}) )$ can be identified with the largest simplicial subset $X_{\bullet }$ of $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{\mathcal{C}})$ having the property that each $2$-simplex of $X_{\bullet }$ is thin when regarded as a $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ (so that an $n$-simplex $\sigma \in \operatorname{N}_{n}^{\operatorname{D}}(\operatorname{\mathcal{C}})$ belongs to $\operatorname{N}^{\operatorname{D}}_{\bullet }( \operatorname{Pith}(\operatorname{\mathcal{C}}) )$ if and only if, for every map $\Delta ^2 \rightarrow \Delta ^ n$, the composition $\Delta ^2 \rightarrow \Delta ^ n \xrightarrow {\sigma } \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is thin).