Definition 9.4.7.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be accessible $\infty $-categories. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is accessible if there exists a small regular cardinal $\kappa $ such that $F$ preserves small $\kappa $-filtered colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.4.7 Accessible Functors
We now study functors between accessible $\infty $-categories.
Remark 9.4.7.2. In the situation of Definition 9.4.7.1, if the functor $F$ is preserves small $\kappa $-filtered collimates, then it also preserves small $\lambda $-filtered colimits for every small regular cardinal $\lambda \geq \kappa $. In particular, we can arrange that the $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-accessible (Proposition 9.4.6.16), so that $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\lambda }$-completion of the full subcategory $\operatorname{\mathcal{C}}_{< \lambda }$ spanned by the $\lambda $-compact objects, and $F$ is an $\operatorname{Ind}_{\lambda }$-extension of the functor $F_{< \lambda } = F|_{ \operatorname{\mathcal{C}}_{< \lambda } }$ (Definition 9.3.1.12).
Example 9.4.7.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be accessible $\infty $-categories. If $\operatorname{\mathcal{C}}$ is essentially small, then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is accessible. See Corollary 9.4.6.26.
Example 9.4.7.4. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category and let $\{ \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a collection of accessible $\infty $-categories indexed by a small set $I$, so that the product $\operatorname{\mathcal{D}}= \prod _{i \in I} \operatorname{\mathcal{D}}_ i$ is also an accessible $\infty $-category (Corollary 9.4.6.19). Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is accessible if and only if, for each $i \in I$, the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_ i$ is accessible.
Example 9.4.7.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{C}}$ is accessible if and only if $\operatorname{\mathcal{D}}$ is accessible. If these conditions are satisfied, then $F$ is an accessible functor. In particular, for every accessible $\infty $-category $\operatorname{\mathcal{C}}$, the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is accessible.
Remark 9.4.7.6 (Retracts). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be accessible $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an accessible functor. If another functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a retract of $F$ (in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), then $G$ is also accessible. In particular, if two functors $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ are isomorphic, then $F$ is accessible if and only if $G$ is accessible.
Remark 9.4.7.7 (Composition). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be accessible $\infty $-categories. If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ are accessible functors, then the composite functor $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is also accessible.
Notation 9.4.7.8. Following the convention of Remark 4.7.0.5, an $\infty $-category is essentially small if it is essentially $\operatorname{\Omega }$-small, where $\operatorname{\Omega }$ denotes some fixed strongly inaccessible cardinal. In this case, every accessible $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\operatorname{\Omega }^{+}$-small (Remark 9.4.6.14). We let $\operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ denote the subcategory of $\operatorname{\mathcal{QC}}_{\leq \operatorname{\Omega }}$ whose objects are accessible $\infty $-categories and whose morphisms are accessible functors. We will refer to $\operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ as the $\infty $-category of accessible $\infty $-categories.
Proposition 9.4.7.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be accessible $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
The functor $F$ is accessible: that is, there exists a small regular cardinal $\kappa $ such that $F$ preserves small $\kappa $-filtered colimits.
There exists a small regular cardinal $\lambda $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\lambda $-accessible and the functor $F$ is $\lambda $-compact (see Variant 9.4.2.2).
Proof. Assume that $F$ is accessible; we will show that condition $(2)$ is satisfied (the converse is immediate from the definitions). Using Remark 9.4.7.2, we can choose a small regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\kappa $-accessible and the functor $F$ is $\kappa $-finitary. Let $\operatorname{\mathcal{C}}_{< \kappa }$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects. It follows from Proposition 9.4.6.2 that $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially small. Using Remark 9.4.6.8, we can choose a small regular cardinal $\lambda \geq \kappa $ such that $F( \operatorname{\mathcal{C}}_{< \kappa } ) \subseteq \operatorname{\mathcal{D}}_{< \lambda }$: that is, $F$ carries $\kappa $-compact objects of $\operatorname{\mathcal{C}}$ to $\lambda $-compact objects of $\operatorname{\mathcal{D}}$. Enlarging $\lambda $ if necessary, we may assume that $\kappa \trianglelefteq \lambda $ (see Proposition 9.1.7.8). We will complete the proof by showing that the functor $F$ is $\lambda $-compact. Since the functor $F$ is $\kappa $-finitary, it is also $\lambda $-finitary. It will therefore suffice to show that for every $\lambda $-compact object $C \in \operatorname{\mathcal{C}}$, the image $F(C) \in \operatorname{\mathcal{D}}$ is also $\lambda $-compact. Using Corollary 9.3.6.7, we can realized $C$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram $G: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$. Since the functor $F$ is $\kappa $-finitary, the object $F(C)$ is a colimit of the composite functor
\[ \operatorname{\mathcal{K}}\xrightarrow {G} \operatorname{\mathcal{C}}_{< \kappa } \xrightarrow {F} \operatorname{\mathcal{D}}_{< \lambda } \subseteq \operatorname{\mathcal{D}}, \]
and is therefore $\lambda $-compact by virtue of Corollary 9.2.5.25. $\square$
Remark 9.4.7.10. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ be a functor between essentially small $\infty $-categories. For any small regular cardinal $\lambda $, the induced functor is an accessible functor between accessible $\infty $-categories. Proposition 9.4.7.9 asserts the converse: up to equivalence, every accessible functor can be obtained in this way.
\[ \operatorname{Ind}_{\lambda }( F_0 ): \operatorname{Ind}_{\lambda }( \operatorname{\mathcal{C}}_0 ) \rightarrow \operatorname{Ind}_{\lambda }( \operatorname{\mathcal{D}}_0 ) \]
Remark 9.4.7.11. In the situation of Proposition 9.4.7.9, let $\lambda $ be a small regular cardinal for which $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\lambda $-accessible and the functor $F$ is $\lambda $-compact. Then, for every small regular cardinal $\mu $ satisfying $\lambda \trianglelefteq \mu $, the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\mu $-accessible and the functor $F$ is $\mu $-compact. In particular, there are arbitrarily large (small) regular cardinals $\mu $ which satisfy this condition. See Propositions 9.4.6.5 and 9.4.2.18.
Variant 9.4.7.12. Let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a small collection of accessible functors between accessible $\infty $-categories. Then there exists a small regular cardinal $\lambda $ such that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ and $\operatorname{\mathcal{D}}_{i}$ is $\lambda $-accessible, and each of the functors $F_ i$ is $\lambda $-compact. Moreover, $\lambda $ can be chosen arbitrarily large.
Example 9.4.7.13. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category. Then, for every object $C \in \operatorname{\mathcal{C}}$, the corepresentable functor is accessible. This is a reformulation of Remark 9.4.6.8.
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}} \]
Proposition 9.4.7.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are accessible. The following conditions are equivalent:
The functor $F$ is accessible.
For every object $D \in \operatorname{\mathcal{D}}$, the composite functor
is accessible.
\[ \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{D}}}(D, \bullet ) } \operatorname{\mathcal{S}} \]
Proof. The implication $(1) \Rightarrow (2)$ follows from Example 9.4.7.13 and Remark 9.4.7.7. Conversely, suppose that condition $(2)$ is satisfied. Since $\operatorname{\mathcal{D}}$ is accessible, it admits a dense full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ which is essentially small (for example, if $\operatorname{\mathcal{D}}$ is $\kappa $-accessible, we can take $\operatorname{\mathcal{D}}_0$ to be the full subcategory spanned by the $\kappa $-compact objects). Applying Proposition 8.4.1.8, we see that the restricted Yoneda embedding
\[ h: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}_{0}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \bullet , Y ) \]
is fully faithful. It will therefore suffice to show that there exists a small regular cardinal $\lambda $ for which the composite functor $h \circ F$ preserves small $\lambda $-filtered colimits. By virtue of Proposition 7.1.8.2, this is equivalent to the requirement that the composite functor $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{D}}}(D, \bullet ) } \operatorname{\mathcal{S}}$ preserves small filtered colimits for each $D \in \operatorname{\mathcal{D}}_0$. The existence of $\lambda $ follows by combining $(2)$ with our assumption that $\operatorname{\mathcal{D}}_0$ is essentially small. $\square$
Corollary 9.4.7.15 (Adjoints are Accessible). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be accessible $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. Then $F$ and $G$ are accessible functors.
Proof. The functor $F$ preserves all colimits which exist in $\operatorname{\mathcal{C}}$ (Corollary 7.1.4.23) and is therefore accessible. To show that $G$ is accessible, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ) } \operatorname{\mathcal{S}} \]
is accessible (Proposition 9.4.7.14). Since $G$ is right adjoint to $F$, this composite functor is corepresented by the object $F(C) \in \operatorname{\mathcal{D}}$, so the desired result follows from Example 9.4.7.13. $\square$