Corollary 5.4.8.8. Let $\operatorname{\mathcal{C}}$ be a simplicial category having 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. Let $\operatorname{\mathcal{C}}'$ denote the simplicial subcategory of $\operatorname{\mathcal{C}}$ having the same objects, with morphism simplicial sets given by $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y)_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }^{\simeq }$. Then the inclusion of simplicial categories $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}') \simeq \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) )$.
Proof. Let $\sigma $ be an $n$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which we identify with a simplicial functor $F: \operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ carrying each $i \in [n]$ to an object $C_{i} \in \operatorname{\mathcal{C}}$. If $T \subseteq [n]$ is a nonempty subset having smallest element $i$ and largest element $k$, let us write $F(T)$ for the corresponding vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet }$. If $S \subseteq T$ is a subset containing $i$ and $k$, let us write $F(S \subseteq T): F(T) \rightarrow F(S)$ for the corresponding edge of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet }$. Let us abuse notation by identifying $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$ with a simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that $\sigma $ is contained in $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$ if and only if the following condition is satisfied:
- $(1)$
For every inclusion $S \subseteq T$ of nonempty subsets of $[n]$ having the same smallest element $i$ and largest element $k$, the edge $F(S \subseteq T): F(T) \rightarrow F(S)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet }$.
Using the thinness criterion of Proposition 5.4.8.7, we see that $\sigma $ belongs to the pith $\operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}))$ if and only if the following a priori weaker condition is satisfied:
- $(2)$
For every triple of elements $0 \leq i \leq j \leq k \leq n$, the edge
\[ F( \{ i,k \} \subseteq \{ i,j,k\} ): F( \{ i, j, k\} ) \rightarrow F( \{ i, k \} ) \]is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet })$.
To complete the proof, it will suffice to show that $(2) \Rightarrow (1)$. Assume that $(2)$ is satisfied, and suppose that we are given nonempty subsets $S \subseteq T$ of $[n]$ having the same smallest element $i$ and largest element $k$. We wish to show that $F(S \subseteq T)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet }$. Since the collection of isomorphisms contains all identity morphisms and is closed under composition (Remark 1.4.6.3), we may assume without loss of generality that the difference $T \setminus S$ contains exactly one element $j$. Set $S_{-} = \{ s \in S: s < j \} $ and $S_{+} = \{ s \in S: s > j \} $. Let $i'$ be the largest element of $S_{-}$, and let $k'$ denote the smallest element of $S_{+}$. Unwinding the definitions, we see that the edge $F(S \subseteq T)$ is the image of $F( \{ i',k' \} \subseteq \{ i',j,k'\} )$ under the functor
and is therefore an isomorphism by virtue of assumption $(2)$. $\square$