Theorem 7.4.5.11 (Refraction Criterion). Let $\lambda $ be an uncountable cardinal and suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix { \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } } \]
where $U$ and $\overline{U}$ are essentially $\lambda $-small cocartesian fibrations. Let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$, and assume that the following condition is satisfied. Then the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright }}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$ is a colimit diagram if the following condition is satisfied:
- $(\ast )$
The covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect $W$.
The converse holds if $\operatorname{\mathcal{C}}$ is $\kappa $-small, where $\kappa = \mathrm{cf}(\lambda )$ is the cofinality of $\lambda $.
Proof of Theorem 7.4.5.11.
Let $\lambda $ be an uncountable cardinal and suppose we are given a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]
where $U$ and $\overline{U}$ are essentially $\lambda $-small cocartesian fibrations. Let $W$ denote the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$, let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$, let $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ be a covariant refraction diagram (Definition 7.4.5.6). Assume first that $\mathrm{Rf}$ exhibits the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. We wish to show that the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \lambda }$.
Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is an $\infty $-category. Note that the induced map $\operatorname{\mathcal{C}}^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}'^{\triangleright }$ is also inner anodyne (Proposition 4.3.6.4). Applying Corollary 5.6.7.3, we can realize $\overline{U}$ as the pullback of a cocartesian fibration of $\infty $-categories $\overline{U}': \overline{\operatorname{\mathcal{E}}}' \rightarrow \operatorname{\mathcal{C}}'^{\triangleright }$. Form a pullback diagram
7.55
\begin{equation} \begin{gathered}\label{equation:hard-half-of-detection} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] \ar [d] & \overline{\operatorname{\mathcal{E}}}' \ar [d]^{ \overline{U}' } \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}'^{\triangleright }, } \end{gathered} \end{equation}
and let $W'$ denote the collection of all $U'$-cocartesian morphisms of $\operatorname{\mathcal{E}}'$. Using Proposition 7.4.5.8, we can choose a covariant refraction diagram $\mathrm{Rf}': \operatorname{\mathcal{E}}' \rightarrow \overline{\operatorname{\mathcal{E}}}'_{{\bf 1}} = \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ for the cocartesian fibration $\overline{U}'$. Note that the restriction $\mathrm{Rf}|_{\operatorname{\mathcal{E}}}$ is a covariant refraction collapse diagram for the cocartesian fibration $\overline{U}$, and is therefore isomorphic to $\mathrm{Rf}$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$. It follows that $\mathrm{Rf}'|_{\operatorname{\mathcal{E}}}$ also exhibits the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$ (Exercise 6.3.1.11). Applying Lemma 7.4.6.3, we see that $\mathrm{Rf}$ exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}'$ with respect to $W$.
Enlarging $\lambda $ if necessary, we may assume that $\lambda $ is regular and that the $\infty $-categories $\operatorname{\mathcal{C}}'$ and $\overline{\operatorname{\mathcal{E}}}'$ are essentially $\lambda $-small. Using Corollary 5.6.5.13, we can extend $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }$ to a functor $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}}' / \operatorname{\mathcal{C}}'^{\triangleright } }: \operatorname{\mathcal{C}}'^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}^{< \lambda }$ which is a covariant transport representation for $\overline{U}'$. Applying Proposition 7.4.6.2 to the diagram of $\infty $-categories (7.55), we deduce that $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}}' / \operatorname{\mathcal{C}}'^{\triangleright } }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \lambda }$. Since the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}'$ is right cofinal (Proposition 7.2.1.3), it follows that $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }$ is also a colimit diagram in $\operatorname{\mathcal{QC}}^{< \lambda }$, as desired.
We now prove the converse. Let $\kappa = \mathrm{cf}(\lambda )$ be the cofinality of $\lambda $. Assume that the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small and that the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}^{< \lambda }$; we wish to show that the covariant refraction diagram $\mathrm{Rf}$ exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. By virtue of Proposition 7.4.5.16 (and Remark 7.4.5.17), we can choose another pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \operatorname{\mathcal{E}}^{+} \ar [d]^{ U^{+} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]
where $U^{+}$ is an essentially $\lambda $-small cocartesian fibration for which the covariant refraction diagram $\mathrm{Rf}^{+}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}^{+}_{ {\bf 1} }$ exhibits $\operatorname{\mathcal{E}}^{+}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. Applying Corollary 5.6.5.13, we see that $U^{+}$ admits a covariant transport representation $\operatorname{Tr}_{ \operatorname{\mathcal{E}}^{+} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}^{<\lambda }$ satisfying $(\operatorname{Tr}_{ \operatorname{\mathcal{E}}^{+} / \operatorname{\mathcal{C}}^{\triangleright } })|_{\operatorname{\mathcal{C}}} = (\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } })|_{\operatorname{\mathcal{C}}}$. The first part of the proof shows that $\operatorname{Tr}_{ \operatorname{\mathcal{E}}^{+} / \operatorname{\mathcal{C}}^{\triangleright } }$ is also a colimit diagram, and is therefore isomorphic to $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{QC}})$. Using Theorem 5.6.0.2, we see $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ and $U^{+}: \operatorname{\mathcal{E}}^{+} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ are equivalent as cocartesian fibrations over $\operatorname{\mathcal{C}}^{\triangleright }$. Applying Exercise 7.4.5.14, we conclude that $\mathrm{Rf}$ also exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$, as desired.
$\square$