Definition 9.4.2.1. Let $\operatorname{\mathcal{C}}$ be a compactly generated $\infty $-category and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small filtered colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is compact if it is finitary (Definition 9.2.2.1) and carries compact objects of $\operatorname{\mathcal{C}}$ to compact objects of $\operatorname{\mathcal{D}}$. We let $\operatorname{Fun}^{\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the compact functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
9.4.2 Compact Functors
We now study functors between compactly generated $\infty $-categories.
Variant 9.4.2.2. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}$ be a $\kappa $-compactly generated $\infty $-category and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-compact if it is $\kappa $-finitary (Definition 9.2.2.3) and carries $\kappa $-compact objects of $\operatorname{\mathcal{C}}$ to $\kappa $-compact objects of $\operatorname{\mathcal{D}}$. We let $\operatorname{Fun}^{\kappa -\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $\kappa $-compact functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
Remark 9.4.2.3. Let $\operatorname{\mathcal{C}}$ be a compactly generated $\infty $-category and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small filtered colimits. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is compact (in the sense of Definition 9.4.2.1) if and only if it is $\aleph _0$-compact (in the sense of Variant 9.4.2.2).
Variant 9.4.2.4. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-compact if it is $(\kappa ,\lambda )$-finitary (Definition 9.2.2.6) and carries $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$ to $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{D}}$. We let $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{c}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $(\kappa ,\lambda )$-compact functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
Remark 9.4.2.5. Following the convention of Remark 4.7.0.5, a regular cardinal $\kappa $ is small if it satisfies $\kappa < \operatorname{\Omega }$, where $\operatorname{\Omega }$ is some fixed strongly inaccessible cardinal. In this case, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-compact (in the sense of Variant 9.4.2.2) if and only if it is $(\kappa ,\operatorname{\Omega })$-compact (in the sense of Variant 9.4.2.4).
Example 9.4.2.6. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated if and only if $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-compactly generated. If these conditions are satisfied, then the functor $F$ is automatically $(\kappa ,\lambda )$-compact. In particular, if $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, then the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ is $(\kappa ,\lambda )$-compact.
Example 9.4.2.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between idempotent-complete $\infty $-categories. Then $F$ is $(\kappa ,\kappa )$-compact for every regular cardinal $\kappa $. See Examples 9.2.2.16 and 9.2.5.15.
Example 9.4.2.8. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact (in the sense of Definition 9.2.5.12) if and only if the inclusion map $\{ C \} \hookrightarrow \operatorname{\mathcal{C}}$ is a $(\kappa ,\lambda )$-compact functor (in the sense of Variant 9.4.2.4).
Example 9.4.2.9. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated. For every simplicial set $K$, the diagram $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{C}})$ is $(\kappa ,\lambda )$-cocomplete and the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is $(\kappa ,\lambda )$-finitary (see Proposition 7.1.8.2). If $K$ is $\kappa $-small, then the functor $\delta $ is $(\kappa ,\lambda )$-compact (see Proposition 9.2.8.9).
Remark 9.4.2.10. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then the restriction functor is an equivalence of $\infty $-categories. Here $\operatorname{\mathcal{C}}_{< \kappa }$ and $\operatorname{\mathcal{D}}_{< \kappa }$ denote the full subcategories spanned by the $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. See Proposition 9.4.1.11.
Example 9.4.2.11. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and set $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\widehat{\operatorname{\mathcal{D}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$. If $\operatorname{\mathcal{D}}$ is idempotent-complete, then the construction $F \mapsto \operatorname{Ind}_{\kappa }^{\lambda }(F)$ induces an equivalence of $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{c}}( \widehat{\operatorname{\mathcal{C}}}, \widehat{\operatorname{\mathcal{D}}} )$. More precisely, if $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ and $j: \operatorname{\mathcal{D}}\rightarrow \widehat{\operatorname{\mathcal{D}}}$ are functors which exhibit $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ as $\operatorname{Ind}_{\kappa }^{\lambda }$-completions of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ (respectively), then the functors are fully faithful and have the same essential image.
Example 9.4.2.12. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, suppose we are given a $\kappa $-small collection of functors $\{ F_ i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$, where each of the $\infty $-categories $\operatorname{\mathcal{D}}_{i}$ is $(\kappa ,\lambda )$-cocomplete. If the induced functor is $(\kappa ,\lambda )$-compact, then each of the functors $F_ i$ is $(\kappa ,\lambda )$-compact. The converse holds if the index set $I$ is $\kappa $-small. See Proposition 9.2.8.6.
Remark 9.4.2.13 (Retracts). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be $(\kappa ,\lambda )$-compact functor. If $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a retract of $F$ (in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), then $G$ is also $(\kappa ,\lambda 0$-compact (see Remark 9.2.5.18). In particular, if $F$ and $G$ are isomorphic, then $F$ is $(\kappa ,\lambda )$-compact if and only if $G$ is $(\kappa ,\lambda )$-compact.
Remark 9.4.2.14 (Composition). Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ are $(\kappa ,\lambda )$-compact functors, then the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-compact.
Notation 9.4.2.15. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). For every pair of regular cardinals $\kappa \leq \lambda $, we define a subcategory $\operatorname{\mathcal{QC}}^{ \mathrm{CG}(\kappa ,\lambda ) } \subseteq \operatorname{\mathcal{QC}}$ as follows:
An object $\operatorname{\mathcal{C}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{ \mathrm{CG}(\kappa ,\lambda ) }$ if and only if it is a $(\kappa ,\lambda )$-compactly generated.
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{ \mathrm{CG}(\kappa ,\lambda ) }$ if and only if it is a $(\kappa ,\lambda )$-compact functor.
More generally, if $\mu $ is any uncountable cardinal, we let $\operatorname{\mathcal{QC}}^{ \mathrm{CG}(\kappa ,\lambda ) }_{< \mu }$ denote the subcategory of $\operatorname{\mathcal{QC}}_{< \mu }$ whose objects are $\mu $-small $\infty $-categories which are $(\kappa ,\lambda )$-compactly generated, and whose morphisms are $(\kappa ,\lambda )$-compact functors. In practice, we will be primarily interested in the case where $\mu $ is much larger than $\lambda $ (so that there are plenty of examples of such $\infty $-categories).
Remark 9.4.2.16. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{QC}}^{ \mathrm{IC} }$ denote the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by the idempotent complete $\infty $-categories. If $\lambda $ is small and uncountable, then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ induces an equivalence from $\operatorname{\mathcal{QC}}^{\mathrm{IC} }$ to the subcategory $\operatorname{\mathcal{QC}}^{ \mathrm{CG}(\kappa ,\lambda ) }$ of Notation 9.4.2.15. More generally, assume that $\lambda $ is uncountable and let $\mu > \lambda $ be a cardinal of exponential cofinality $\geq \lambda $, so that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ carries essentially $\mu $-small $\infty $-categories to essentially $\mu $-small $\infty $-categories (see Proposition 9.3.3.7). Then the formation of $\operatorname{Ind}_{\kappa }^{\lambda }$-completions induces an equivalence from $\operatorname{\mathcal{QC}}_{< \mu }^{\mathrm{IC} }$ to the subcategory $\operatorname{\mathcal{QC}}^{ \mathrm{CG}(\kappa ,\lambda ) }_{< \mu } \subseteq \operatorname{\mathcal{QC}}_{< \mu }$. This follows from Proposition 9.4.1.22 and Remark 9.4.2.10.
Example 9.4.2.17. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated and $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-cocomplete. Suppose that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then:
For a partial converse, see Proposition 
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:
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.
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 9.3.5.26). 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 $\operatorname{\mathcal{D}}\simeq \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{D}}_{< \lambda } )$. $\square$
Corollary 9.4.2.19. Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. If $\operatorname{\mathcal{D}}$ is idempotent-complete, then the construction $F \mapsto \operatorname{Ind}_{\lambda }^{\mu }(F)$ induces an equivalence of $\infty $-categories
Proof. For $\kappa = \lambda $, this is a special case of Example 9.4.2.11 (which does not require the assumption $\lambda \trianglelefteq \mu $). We may therefore assume that $\lambda > \kappa $. In this case, $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are idempotent-complete, and can therefore be identified with the full subcategories of $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ and $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{D}})$ spanned by the $(\lambda ,\mu )$-compact objects (Corollary 9.4.1.21). In this case, the desired result is a reformulation of Proposition 9.4.2.18. $\square$