Kerodon

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Remark 2.4.6.20. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. The comparison map $V: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$ of Remark 2.4.6.19 is always bijective at the level of vertices (which can be identified with the objects of the category $\operatorname{\mathcal{C}}_0$ underlying $\operatorname{\mathcal{C}}_{\bullet }$) and edges (which can be identified with morphisms of $\operatorname{\mathcal{C}}_0$). Suppose that, for every pair of objects $C,D \in \operatorname{\mathcal{C}}_0$, the simplicial set $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$ is an $\infty $-category. In this case, the map $V$ is also surjective (but not necessarily injective) at the level of $2$-simplices. By virtue of Example 2.3.1.15, we can identify $2$-simplices $\overline{\sigma }$ of $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )$ with diagrams

\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{[\mu ]} & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z, } \]

where $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ are morphisms in $\operatorname{\mathcal{C}}_0$, and $[\mu ]: g \circ f \rightarrow h$ is a morphism in the homotopy category of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$. To lift $\overline{\sigma }$ to a $2$-simplex $\sigma $ of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, one must choose a morphism $\mu : g \circ f \rightarrow h$ in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$ which represents the homotopy class $[\mu ]$ (see Example 2.4.3.10). Such a representative always exists, but is not necessarily unique.