Kerodon

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Example 2.4.3.9 (Vertices and Edges of the Homotopy Coherent Nerve). In the cases $Q = [0]$ and $Q = [1]$, the map $\pi : \operatorname{Path}[Q]_{\bullet } \rightarrow Q$ is an equivalence of simplicial categories (since a path in $Q$ is uniquely determined by its endpoints). It follows that for every simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, the comparison map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ of Remark 2.4.3.8 is bijective on vertices and edges. In particular:

• Vertices of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with objects $X$ of the underlying category $\operatorname{\mathcal{C}}$.

• Edges of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with morphisms $f: X \rightarrow Y$ of the underlying category $\operatorname{\mathcal{C}}$.

• The face maps $d_{0}, d_{1}: \operatorname{N}_{1}^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{0}^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carry a morphism $f: X \rightarrow Y$ to its target $Y = d_0(f)$ and source $f = d_1(f)$, respectively.

• The degeneracy map $s_0: \operatorname{N}_{0}^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{1}^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carries an object $X \in \operatorname{\mathcal{C}}$ to the identity morphism $\operatorname{id}_{X}: X \rightarrow X$.