Theorem 7.4.4.6 (Diffraction Criterion). Let $\kappa $ be an uncountable cardinal and 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 essentially $\kappa $-small cocartesian fibrations. The following conditions are equivalent:
- $(1)$
The restriction map
\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]
is an equivalence of $\infty $-categories.
- $(2)$
The covariant transport representation
\[ \operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}^{< \kappa } \]
of Notation 5.6.5.16 is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$.
Proof of Theorem 7.4.4.6.
Choose an uncountable cardinal $\lambda $ of exponential cofinality $\geq \kappa $, so that the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$ is locally $\lambda $-small (Remark 5.5.4.14). For every object $\operatorname{\mathcal{K}}\in \operatorname{\mathcal{QC}}^{< \kappa }$, let $h^{\operatorname{\mathcal{K}}}: \operatorname{\mathcal{QC}}^{< \kappa } \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ be the functor corepresented by $\operatorname{\mathcal{K}}$. By virtue of Proposition 5.6.6.17, we may assume that $h^{\operatorname{\mathcal{K}}}$ is obtained from the homotopy coherent nerve of the simplicial functor
\[ \operatorname{QCat}\rightarrow \operatorname{Kan}\quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}})^{\simeq }. \]
By virtue of Proposition 7.4.1.18, it will suffice to show that the following conditions are equivalent:
- $(1_{\operatorname{\mathcal{K}}} )$
The restriction map
\[ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) )^{\simeq } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}))^{\simeq } \]
is a homotopy equivalence of Kan complexes.
- $(2_{\operatorname{\mathcal{K}}} )$
The composition $h^{\operatorname{\mathcal{K}}} \circ \operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$.
Using Example 5.6.5.9, we can replace $U$ by the projection map
\[ \operatorname{\mathcal{C}}^{\triangleleft } \times _{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}^{\triangleleft } ) } \operatorname{Fun}( \operatorname{\mathcal{K}}, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{\mathcal{C}}^{\triangleleft } \]
and thereby reduce to proving the equivalence $(1_{\operatorname{\mathcal{K}}} ) \Leftrightarrow ( 2_{\operatorname{\mathcal{K}}} )$ in the special case where $\operatorname{\mathcal{K}}= \Delta ^{0}$. In this case, the desired result follows by applying Proposition 7.4.1.16 to the underlying left fibration of $U$ (see Example 5.6.5.8).
$\square$