Example 7.5.6.2 (03C5)

Example 7.5.6.2. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. Then the diagram

\[ \mathscr {F}_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {F}_{\operatorname{\mathcal{E}}}(C) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]

is projectively cofibrant, in the sense of Definition 7.5.6.1. To prove this, we must show that for every levelwise trivial Kan fibration $\mathscr {G}' \rightarrow \mathscr {G}$ between functors $\mathscr {G}', \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, the induced map

\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}_{\operatorname{\mathcal{E}}}, \mathscr {G}') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {G}_{\operatorname{\mathcal{E}}}, \mathscr {G} ) \]

is surjective. Using Proposition 5.3.3.24, we can identify $\theta $ with a pullback of the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G}'}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{E}}, \operatorname{N}_{\bullet }^{\mathscr {G} }(\operatorname{\mathcal{C}}) )$, which is surjective by virtue of Exercise 5.3.3.11.