Example 2.4.6.15 ($3$-Simplices of the Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. Using Proposition 2.4.6.10, we see that a map of simplicial sets $\sigma _0: \operatorname{\partial }\Delta ^3 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with the following data:
A collection of four objects $\{ X_ i \in \operatorname{\mathcal{C}}\} _{0 \leq i \leq 3}$.
A collection of six morphisms $\{ f_{ji} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_ i, X_ j) \} _{0 \leq i < j \leq 3}$.
A collection of four $1$-simplices $\{ h_{kji} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_ i, X_ k)_{1} \} _{ 0 \leq i < j < k \leq 3}$, where each $h_{kji}$ is a homotopy from $f_{kj} \circ f_{ji}$ to $f_{ki}$.
From this data, we can assemble a map of simplicial sets $\lambda _0: \operatorname{\partial \raise {0.1ex}{\square }}^{2} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_3)_{\bullet }$, which is represented by the diagram
Corollary 2.4.6.13 then asserts that extending $\sigma _0$ to a $3$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is equivalent to extending $\lambda _0$ to a map of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^2 \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_3)_{\bullet }$. Stated more informally, the map $\sigma _0$ supplies two potentially different paths from the composition $f_{32} \circ f_{21} \circ f_{10}$ to $f_{30}$ in the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_3)_{\bullet }$. To extend $\sigma _0$ to a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, one must supply additional data which “witnesses” that these paths are homotopic.