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Example ($3$-Simplices of the Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. Using Proposition, 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

\[ \xymatrix@C =50pt@R=50pt{ f_{32} \circ f_{21} \circ f_{10} \ar [r]^-{ h_{321} \circ \operatorname{id}_{f_{10}} } \ar [d]^-{\operatorname{id}_{f_{32}} \circ h_{210} } & f_{31} \circ f_{10} \ar [d]^-{h_{310}} \\ f_{32} \circ f_{20} \ar [r]^-{ h_{320} } & f_{30}. } \]

Corollary 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.