Kerodon

Example 2.4.6.14 ($1$-Simplices of the Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. By definition, giving a map of simplicial sets $\sigma _0: \operatorname{\partial }\Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is equivalent to giving a pair of objects $X_0 = \sigma _0(0)$ and $X_1=\sigma _0(1)$ of the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$. Applying Corollary 2.4.6.13, we deduce that extending $\sigma _0$ to a $1$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is equivalent to supplying a morphism $f: X_0 \rightarrow X_1$ in the category $\operatorname{\mathcal{C}}$ (see Example 2.4.3.9).