Construction 5.3.1.1 (035E)

Construction 5.3.1.1. Let $\operatorname{Cat}$ denote the ordinary category whose objects are (small) categories and whose morphisms are functors. If $\operatorname{\mathcal{C}}$ is a category, then the construction $C \mapsto \operatorname{\mathcal{C}}_{C/}$ determines a functor $\operatorname{\mathcal{C}}\rightarrow (\operatorname{Cat}_{ /\operatorname{\mathcal{C}}})^{\operatorname{op}}$, carrying each morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$ to the functor

\[ \operatorname{\mathcal{C}}_{D/} \rightarrow \operatorname{\mathcal{C}}_{C/} \quad \quad ( g: D \rightarrow E ) \mapsto ( (g \circ f): C \rightarrow E). \]

For any morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, we let $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor given on objects by the formula

\[ \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}). \]

We will refer to $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the weak transport representation of $U$.