Proposition 9.4.2.18 (Transitivity). Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\mu )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\mu )$-cocomplete. Then every $(\kappa ,\mu )$-compact functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfies the following pair of conditions:
- $(1)$
The functor $F$ is $(\lambda ,\mu )$-compact. In particular, it restricts to a functor $F_{< \lambda }: \operatorname{\mathcal{C}}_{< \lambda } \rightarrow \operatorname{\mathcal{D}}_{< \lambda }$, where $\operatorname{\mathcal{C}}_{< \lambda }$ and $\operatorname{\mathcal{D}}_{< \lambda }$ denote the full subcategories of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ spanned by the $(\lambda ,\mu )$-compact objects.
- $(2)$
The functor $F_{< \lambda }$ is $(\kappa ,\lambda )$-compact.
The converse holds if the $\infty $-category $\operatorname{\mathcal{D}}$ is $(\lambda ,\mu )$-compactly generated.
Proof.
Assume first that $F$ is $(\kappa ,\mu )$-compact. To prove $(1)$, we must show that for every object $C \in \operatorname{\mathcal{C}}_{< \lambda }$, the image $F(C)$ belongs to $\operatorname{\mathcal{D}}_{< \lambda }$. Using Corollary 9.3.6.7, we can realize $C$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram $G: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, carrying each object of $\operatorname{\mathcal{K}}$ to a $(\kappa ,\mu )$-compact object of $\operatorname{\mathcal{D}}$. Since the functor $F$ is $(\kappa ,\mu )$-finitary, it follows that $F(C)$ is a colimit of the diagram $(F \circ G): \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$. By assumption, this diagram carries each object of $\operatorname{\mathcal{K}}$ to an object of $\operatorname{\mathcal{D}}$ which is $(\kappa ,\mu )$-compact, and therefore also $(\lambda ,\mu )$-compact. Since $\operatorname{\mathcal{K}}$ is $\lambda $-small, it follows that $F(C)$ is also $(\lambda ,\mu )$-compact (Proposition 9.2.5.24). This completes the proof of $(1)$. To prove $(2)$, we observe that if $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-compact, then $F(C)$ is $(\kappa ,\mu )$-compact as an object of $\operatorname{\mathcal{D}}$, and therefore also $(\kappa ,\lambda )$-compact as an object of the subcategory $\operatorname{\mathcal{D}}_{< \lambda }$.
We now prove the converse. Assume that $\operatorname{\mathcal{D}}$ is $(\lambda ,\mu )$-compactly generated and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor satisfying conditions $(1)$ and $(2)$; we wish to show that $F$ is $(\kappa ,\mu )$-compact. Since $\operatorname{\mathcal{C}}$ is $(\lambda ,\mu )$-compactly generated (Corollary 9.4.1.26), assumption $(1)$ guarantees that we can identify $F$ with the $\operatorname{Ind}_{\lambda }^{\mu }$-extension of the functor $F_{< \lambda }: \operatorname{\mathcal{C}}_{< \lambda } \rightarrow \operatorname{\mathcal{D}}$). Assumption $(2)$ guarantees that $F_{< \lambda }$ is $(\kappa ,\lambda )$-finitary when viewed as a functor from $\operatorname{\mathcal{C}}_{< \lambda }$ to $\operatorname{\mathcal{D}}_{< \lambda }$, and therefore also when viewed as a functor from $\operatorname{\mathcal{C}}_{< \lambda }$ to $\operatorname{\mathcal{D}}$ (Corollary 9.3.5.27). Applying Corollary 9.3.6.11, we deduce that $F$ is $(\kappa ,\mu )$-finitary. To complete the proof, we must show that for every $(\kappa ,\mu )$-compact object $C \in \operatorname{\mathcal{C}}$, the image $F(C) \in \operatorname{\mathcal{D}}$ is also $(\kappa ,\mu )$-compact. Assumption $(1)$ guarantees that $F(C)$ belongs to $\operatorname{\mathcal{D}}_{< \lambda }$, and assumption $(2)$ guarantees that $F(C)$ is $(\kappa ,\lambda )$-compact as an object of $\operatorname{\mathcal{D}}_{< \lambda }$. Applying Example 9.3.6.13, we conclude that $F(C)$ is $(\kappa ,\mu )$-compact when viewed as an object of $\operatorname{\mathcal{D}}\simeq \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{D}}_{< \lambda } )$.
$\square$