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Proposition 5.3.8.7. 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. Let $\sigma $ be a $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which we identify with a (not necessarily commutative) diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]

in $\operatorname{\mathcal{C}}$ together with an edge $\mu : g \circ f \rightarrow h$ in the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$. Then $\sigma $ is thin if and only if $\mu $ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$.

Proof. It follows from Proposition 5.3.8.2 that if $\mu $ is an isomorphism, then $\sigma $ is thin. Conversely, assume that $\sigma $ is thin; we wish to show that $\mu $ is an isomorphism. Define a strict $2$-category $\operatorname{\mathcal{E}}$ as follows:

  • The objects of $\operatorname{\mathcal{E}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $A,B \in \operatorname{\mathcal{C}}$, we define $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(A,B)$ to be the homotopy category of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)_{\bullet }$.

  • For every triple of objects $A,B,C \in \operatorname{\mathcal{C}}$, we define the composition law

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(B,C) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(A,B) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(A,C) \]

    to be the functor of homotopy categories induced by the composition law

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,C)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,C) \]

    of the simplicial category $\operatorname{\mathcal{C}}$.

Let $\operatorname{\mathcal{D}}$ denote the simplicial category obtained by applying the construction of Example 2.4.2.7 to the strict $2$-category $\operatorname{\mathcal{E}}$: the simplicial category $\operatorname{\mathcal{D}}$ has the same objects as $\operatorname{\mathcal{C}}$, with simplicial morphism spaces given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(A,B)_{\bullet } = \operatorname{N}_{\bullet }( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(A,B) ) = \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)}_{\bullet } ). \]

There is an evident functor of simplicial categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, which is the identity on objects and which induces the unit map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)}_{\bullet } )$ on simplicial morphism spaces. Invoking Proposition 5.3.8.6, we see that the induced map $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ carries $\sigma $ to a thin $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{D}})$, which we can identify with the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{E}})$ of the $2$-category $\operatorname{\mathcal{E}}$ (Example 2.4.3.11). Using the description of the thin simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{E}})$ supplied by Theorem 2.3.2.5, we conclude that the homotopy class $[\mu ]$ is an isomorphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}( X,Z) = \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)}_{\bullet }$, so that $\mu $ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$. $\square$