Construction 5.3.2.20 (036D)

Construction 5.3.2.20. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$. For each object $C \in \operatorname{\mathcal{C}}$, we let

\[ f_{C}: \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ) \times \mathscr {F}(C) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \]

denote the morphism of simplicial sets given on $n$-simplices by the formula $f_{C}( \sigma , \tau ) = ( \overline{\sigma }, \overline{\tau } )$, where $\overline{\sigma }$ denotes the image of $\sigma $ in $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\overline{\tau }$ denote the image of $\tau $ under the map $\mathscr {F}(C) \rightarrow \mathscr {F}( \overline{\sigma }(0) )$. Note that we can identify $f_{C}$ with a morphism of simplicial sets

\[ u_{\mathscr {F},C}: \mathscr {F}(C) \rightarrow \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/}), \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) ) = \operatorname{wTr}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) / \operatorname{\mathcal{C}}}(C). \]

This morphism depends functorially on $C$: that is, the collection $u_{\mathscr {F}} = \{ u_{\mathscr {F},C} \} _{C \in \operatorname{\mathcal{C}}}$ is a natural transformation from $\mathscr {F}$ to the weak transport representation $\operatorname{wTr}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) / \operatorname{\mathcal{C}}}$.