Construction 7.4.4.9 (02TD): Covariant Diffraction

Construction 7.4.4.9 (Covariant Diffraction). Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }, } \]

where $U$ and $\overline{U}$ are cocartesian fibrations. Let $\overline{\operatorname{\mathcal{E}}}_{ {\bf 0} }$ denote the fiber of $\overline{U}$ over the cone point ${\bf 0} \in \operatorname{\mathcal{C}}^{\triangleleft }$. We then have restriction maps

\[ \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \xleftarrow {\operatorname{ev}} \operatorname{Fun}_{/ \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \xrightarrow {\theta } \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}), \]

where $\operatorname{ev}$ is a trivial Kan fibration (Corollary 5.3.1.23). Composing $\theta $ with a section of $\operatorname{ev}$, we obtain a functor of $\infty $-categories $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \rightarrow \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ which is well-defined up to isomorphism. We will refer to $\mathrm{Df}$ as the covariant diffraction functor associated to the cocartesian fibration $\overline{U}$.