Construction 8.3.6.5. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote its homotopy coherent nerve. Let $J$ be a linearly ordered set and let $\overline{J}$ denote its opposite; for each element $j \in J$, we write $\overline{j}$ for the corresponding element of $\overline{J}$. Suppose we are given a morphism of simplicial sets $\sigma : \operatorname{N}_{\bullet }(J) \rightarrow \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) )$, which we identify with a simplicial functor $f: \operatorname{Path}[ \overline{J} \star J ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ (see Warning 8.1.1.9 and Proposition 2.4.4.15). Note that the composition
can be identified with a simplicial functor $F_{\sigma }: \operatorname{Path}[J]_{\bullet } \rightarrow \operatorname{Kan}$, given on objects by the formula $F_{\sigma }(j) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( f( \overline{j} ), f(j) )_{\bullet }$ (see Proposition 2.4.4.15). Let $J^{\triangleleft } = \{ x\} \star J$ denote the linearly ordered set obtained from $J$ by adding a new smallest element $x$. We extend $F_{\sigma }$ to a simplicial functor $\widetilde{F}_{\sigma }: \operatorname{Path}[ J^{\triangleleft } ]_{\bullet } \rightarrow \operatorname{Kan}$ as follows:
- $(a)$
The functor $\widetilde{F}_{\sigma }$ carries the element $x \in J^{\triangleleft }$ to the Kan complex $\Delta ^0$.
- $(b)$
Let $j$ be an element of $J$. Let us identify $\operatorname{Hom}_{ \operatorname{Path}[ J^{\triangleleft } ] }( x, j)_{\bullet }$ with the nerve $\operatorname{N}_{\bullet }( Q)$, where $Q$ is the collection of finite subsets $I \subseteq J$ satisfying $\mathrm{max}(I) = j$ (partially ordered by reverse inclusion). Similarly, we identify $\operatorname{Hom}_{ \operatorname{Path}[ \overline{J} \star J] }( \overline{j}, j )_{\bullet }$ with the nerve $\operatorname{N}_{\bullet }( Q' )$, where $Q'$ is the collection of finite subsets $I' \subseteq \overline{J} \star J$ satisfying $\mathrm{max}(I') = j$ and $\mathrm{min}(I') = \overline{j}$ (partially ordered by reverse inclusion). Then $\widetilde{F}_{\sigma }$ is defined on the morphism space $\operatorname{Hom}_{ \operatorname{Path}[ J^{\triangleleft } ] }(x,j)_{\bullet }$ by the composition
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Path}[ J^{\triangleleft } ] }(x,j)_{\bullet } & \simeq & \operatorname{N}_{\bullet }(Q) \\ & \xrightarrow { I \mapsto \overline{I} \cup I } & \operatorname{N}_{\bullet }(Q') \\ & \simeq & \operatorname{Hom}_{ \operatorname{Path}[ \overline{J} \star J] }( \overline{j}, j )_{\bullet } \\ & \xrightarrow {f} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( f(\overline{j}), f(j) )_{\bullet } \\ & \simeq & \operatorname{Fun}( \widetilde{F}_{\sigma }(x), \widetilde{F}_{\sigma }(j) ). \end{eqnarray*}
In the special case where $J$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $, we can identify $\widetilde{F}_{\sigma }$ with an $n$-simplex of the $\infty $-category of pointed spaces $\operatorname{\mathcal{S}}_{\ast } = \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})_{ \Delta ^0 / }$. The assignment $\sigma \mapsto \widetilde{F}_{\sigma }$ depends functorially on $[n]$, and therefore determines a functor $\widetilde{\mathscr {H}}_{\operatorname{\mathcal{C}}}: \operatorname{Tw}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$. By construction, this functor fits into a commutative diagram
where the left vertical map is the twisted arrow fibration of Proposition 8.1.1.11 and the right vertical map is the forgetful functor.