Theorem 5.4.8.1. Let $\operatorname{\mathcal{C}}$ be a simplicial category. Suppose that, for every pair of objects $X$ and $Y$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Then the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category.
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Proof of Theorem 5.4.8.1. Let $\operatorname{\mathcal{C}}$ be a simplicial category with the property that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Using Example 5.4.8.3, we immediately deduce that every degenerate $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is thin, and that every morphism $\Lambda ^{2}_{1} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be extended to a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. We will complete the proof that $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category by showing that, if $n \geq 3$ and $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is a morphism of simplicial sets for which the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n \} )}$ is right-degenerate, then $\sigma _0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ (the dual assertion regarding extension of maps $\Lambda ^{n}_{0} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ follows by the same argument, applied to the opposite simplicial category $\operatorname{\mathcal{C}}^{\operatorname{op}}$). Let us identify $\sigma _0$ with a simplicial functor $F: \operatorname{Path}[ \Lambda ^{n}_{n} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$, carrying each element $i \in [n]$ to an object $C_ i \in \operatorname{\mathcal{C}}$.
Let $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the simplicial cube of dimension $(n-1)$ and let ${\boldsymbol {\sqcup }}^{n-1}_{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the hollow cube of Notation 2.4.5.5, so that Remark 2.4.5.4 and Proposition 5.4.8.4 supply isomorphisms
\[ \operatorname{Hom}_{\operatorname{Path}[n]}(0,n)_{\bullet } \simeq \operatorname{\raise {0.1ex}{\square }}^{n-1} \quad \quad \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{n} ] }( 0, n) \simeq {\boldsymbol {\sqcap }}^{n-1}_{n-1}. \]
Let $\lambda _0$ denote the composite map
\[ {\boldsymbol {\sqcup }}^{n-1}_{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{n} ] }(0, n)_{\bullet } \xrightarrow {F} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_ n)_{\bullet }. \]
Note that our degeneracy assumption on $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n \} )}$ guarantees that the functor $F$ induces an isomorphism $C_{n-1} \simeq C_{n}$ in the category $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.4.8.5, it will suffice to show that $\lambda _0$ can be extended to a morphism of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}C_0, C_ n)_{\bullet }$.
Let us identify ${\boldsymbol {\sqcup }}^{n-1}_{n-1}$ with the pushout
\[ (\operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \times \Delta ^1) {\coprod }_{ (\operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \times \{ 1\} ) } ( \operatorname{\raise {0.1ex}{\square }}^{n-2} \times \{ 1\} ). \]
Let $v$ be the final vertex of the cube $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2}$ (corresponding to the set $\{ 1, 2, \ldots , n-2 \} $, regarded as a subset of itself). Our assumption that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n \} ) }$ is right-degenerate guarantees that the composite map
\[ \{ v\} \times \Delta ^1 \hookrightarrow {\boldsymbol {\sqcup }}^{n-1}_{n-1} \xrightarrow {\lambda _0} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }. \]
is a degenerate edge of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_ n)_{\bullet }$; in particular, it is an isomorphism of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_ n)_{\bullet }$. Note that every simplex of $\operatorname{\raise {0.1ex}{\square }}^{n-2}$ which is not contained in the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2}$ has final vertex $v$. The existence of the desired extension $\lambda $ now follows by applying Proposition 4.4.5.8. $\square$