Construction 4.6.6.4 (04J1)

Construction 4.6.6.4. Let $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and set $f_{-} = f_{\pm }|_{K_{-}}$ and $f_{+} = f_{\pm }|_{ K_{+} }$. Let $K$ be another simplicial set, set $M = \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} } )$. Let $\operatorname{ev}: M \times K \rightarrow \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} }$ be the evaluation map, and let $\pi _{-}: M \times K_{-} \rightarrow K_{-}$ and $\pi _{+}: M \times K_{+} \rightarrow K_{+}$ be given by projection onto the second factor. Then the composition

\begin{eqnarray*} M \times (K_{-} \star K \star K_{+} ) & \rightarrow & (M \times K_{-} ) \star (M \times K) \star ( M \times K_{+} ) \\ & \xrightarrow {\pi _{-} \star \operatorname{ev}\star \pi _{+}} & K_{-} \star \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} } \star K_{+} \\ & \rightarrow & \operatorname{\mathcal{C}}\end{eqnarray*}

classifies a morphism of simplicial sets $M \rightarrow \operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}})$, whose composition with the restriction map $\operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}})$ is the constant map taking the value $f_{\pm }$. We therefore obtain a comparison map

\[ \theta : \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} } ) \rightarrow \operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}}) } \{ f_{\pm } \} . \]