9.5.2 Cocontinuous Functors
Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small colimits. Recall that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is cocontinuous if it preserves small colimits (Definition 7.6.6.4). By virtue of Corollary 9.2.2.22, this is equivalent to the following pair of a priori weaker conditions:
- $(a)$
The functor $F$ is $\kappa $-finitary: that is, it preserves small $\kappa $-filtered colimits.
- $(b)$
The functor $F$ is $\kappa $-cocontinuous: that is, it preserves $\kappa $-small colimits.
Our goal in this section is to show that, if the $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-presentable, then condition $(b_{\kappa })$ can be weakened further: it is enough to assume that $F$ preserves $\kappa $-small colimits when restricted to the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects (Corollary 9.5.2.6). We will deduce this from the following more general result:
Proposition 9.5.2.1. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-cocomplete, so that $f$ admits an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ (Definition 9.3.1.12). The following conditions are equivalent:
- $(1)$
The functor $f$ is $\kappa $-right exact.
- $(2)$
The functor $F$ is $\kappa $-right exact.
- $(3)$
The functor $F$ is $\lambda $-right exact.
Proof.
Fix an object $D \in \operatorname{\mathcal{D}}$. Using the criterion of Theorem 9.3.5.15, it will suffice to prove the equivalence of the following conditions:
- $(1_ D)$
The $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is $\kappa $-filtered.
- $(2_ D)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is $\kappa $-filtered.
- $(3_ D)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is $\lambda $-filtered.
Note that the $\infty $-category $\operatorname{\mathcal{E}}= \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ admits $\lambda $-small $\kappa $-filtered colimits which are preserved by the right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ (Proposition 7.1.9.8). Applying Proposition 9.3.1.16, we can identify $\operatorname{\mathcal{E}}$ with the $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$. The equivalence of $(1_ D)$, $(2_ D)$, and $(3_ D)$ now follows from Corollary 9.3.7.6.
$\square$
Corollary 9.5.2.2. Let $\kappa \trianglelefteq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, so that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\lambda $-cocomplete (Corollary 9.5.1.10). Suppose we are given a functor $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-cocomplete, and let $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ be the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$. The following conditions are equivalent:
- $(1)$
The functor $f$ is $\kappa $-cocontinuous.
- $(2)$
The functor $F$ is $\kappa $-cocontinuous.
- $(3)$
The functor $F$ is $\lambda $-cocontinuous.
Proof.
Combine Proposition 9.5.2.1 with Corollary 9.3.5.24.
$\square$
Corollary 9.5.2.3. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then the restriction functor
\[ \operatorname{Fun}^{\lambda -\mathrm{cocont}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
is an equivalence of $\infty $-categories.
Proof.
By virtue of the universal property of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, this is a reformulation of Corollary 9.5.2.2.
$\square$
Corollary 9.5.2.4. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $\kappa $-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\lambda $-cocontinuous if and only if it satisfies the following pair of conditions:
- $(a)$
The functor $F$ is $(\kappa ,\lambda )$-finitary.
- $(b^{-})$
The restriction $F|_{ \operatorname{\mathcal{C}}_{< \kappa } }$ is $\kappa $-cocontinuous, where $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denotes the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$.
Proof.
The necessity of conditions $(a)$ and $(b^{-})$ is clear. The sufficiency follows from Corollary 9.5.2.2, since $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_{< \kappa }$ (Proposition 9.4.1.11).
$\square$
Corollary 9.5.2.5. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $\kappa $-compactly generated, and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then the restriction functor
\[ \operatorname{Fun}^{\lambda -\mathrm{cocont}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{ \kappa -\mathrm{cocont} }( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}}) \]
is an equivalence of $\infty $-categories.
Corollary 9.5.2.6. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is cocomplete and $\kappa $-compactly generated (for example, an $\infty $-category which is $\kappa $-presentable), and let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. Then a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is cocontinuous if and only if it is $\kappa $-finitary and the restriction $F|_{ \operatorname{\mathcal{C}}_{< \kappa } }: \operatorname{\mathcal{C}}_{<\kappa } \rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-cocontinuous.
Proof.
Apply Corollary 9.5.2.4 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal.
$\square$
We conclude with an application of the preceding results.
Proposition 9.5.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is presentable if and only if it satisfies the following conditions:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small.
- $(b)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete.
- $(c)$
There exists a small collection of objects $\{ C_ i \} _{i \in I}$ which generates $\operatorname{\mathcal{C}}$ under small colimits in the following sense: if $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a full subcategory which contains each $C_ i$ and is closed under small colimits, then $\operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}$.
- $(d)$
For every object $C \in \operatorname{\mathcal{C}}$, there exists a small regular cardinal $\kappa $ such that $C$ is $\kappa $-compact.
The proof of Proposition 9.5.2.7 will require some preliminaries. We begin by establishing “cocontinuous” variants of Propositions 9.3.2.1 and 9.3.2.3.
Lemma 9.5.2.8. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which satisfies the following conditions:
- $(0)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits $\lambda $-small colimits.
- $(1)$
The functor $h$ is fully faithful and preserves $\kappa $-small colimits.
- $(2)$
For each object $C \in \operatorname{\mathcal{C}}$, the image $h(C)$ is a $(\kappa ,\lambda )$-compact object of $\widehat{\operatorname{\mathcal{C}}}$.
Then the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $h$ is a fully faithful functor $H: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \widehat{\operatorname{\mathcal{C}}}$, whose essential image is the smallest full subcategory which contains the essential image of $h$ and is closed under $\lambda $-small colimits.
Proof.
Proposition 9.3.2.1 guarantees that $H$ is fully faithful, and therefore restricts to an equivalence of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ with a replete full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$. Since $h$ is $\kappa $-cocontinuous, Corollary 9.5.2.2 guarantees that $H$ is $\lambda $-cocontinuous: that is, the full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under $\lambda $-small colimits. We conclude by observing that every object of $\widehat{\operatorname{\mathcal{C}}}'$ can be realized as the colimit of a $\lambda $-small diagram in $\operatorname{\mathcal{C}}$ (which we can even take to be $\kappa $-filtered; see Corollary 9.3.4.17).
$\square$
Lemma 9.5.2.9. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. Then $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ if and only if it satisfies conditions $(0)$, $(1)$, and $(2)$ of Lemma 9.5.2.8, together with the following additional condition:
- $(3)$
The objects $\{ h(C) \} _{C \in \operatorname{\mathcal{C}}}$ generate $\widehat{\operatorname{\mathcal{C}}}$ under $\lambda $-small colimits. That is, if a full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ contains the essential image of $h$ and is closed under $\lambda $-small colimits, then $\widehat{\operatorname{\mathcal{C}}}' = \widehat{\operatorname{\mathcal{C}}}$.
Proof.
The sufficiency of conditions $(0)$, $(1)$, $(2)$, and $(3)$ follows from Lemma 9.5.2.8. Necessity follows by combining Proposition 9.3.2.3 with Corollaries 9.5.1.10 and 9.3.5.26.
$\square$
Example 9.5.2.10. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete, and let $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denote the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated: that is, every object of $\operatorname{\mathcal{C}}$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}_{< \kappa }$ (Variant 9.4.1.7).
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is generated by $\operatorname{\mathcal{C}}_{< \kappa }$ under $\lambda $-small colimits.
The implication $(1) \Rightarrow (2)$ is trivial, and the reverse implication follows by applying Lemma 9.5.2.9 to the inclusion functor $\operatorname{\mathcal{C}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{C}}$.
We now formulate a more precise version of Proposition 9.5.2.7.
Proposition 9.5.2.11. Let $\kappa $ be a small regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-presentable if and only if it satisfies the following conditions:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small.
- $(b)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete.
- $(c_{\kappa })$
There exists a small collection of $\kappa $-compact objects $\{ C_{i} \} _{i \in I}$ which generates $\operatorname{\mathcal{C}}$ under small colimits.
Proof.
If $\operatorname{\mathcal{C}}$ is $\kappa $-presentable, then assertions $(b)$ and $(c_{\kappa })$ are immediate and $(a)$ follows from Remark 9.4.6.14. Conversely, suppose that conditions $(a)$, $(b)$, and $(c_{\kappa })$ are satisfied. Let $\{ C_ i \} _{i \in I}$ be a small collection of $\kappa $-compact objects which generates $\operatorname{\mathcal{C}}$ under small colimits, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the objects $C_ i$. It follows from assumption $(a)$ that $\operatorname{\mathcal{C}}_0$ is essentially small. Enlarging $\operatorname{\mathcal{C}}_0$ if necessary, we can assume that it is closed under $\kappa $-small colimits (Proposition 9.2.5.24), so that $\operatorname{\mathcal{C}}_0$ is $\kappa $-cocomplete and the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is $\kappa $-cocontinuous. Applying Lemma 9.5.2.9, we conclude that $\iota $ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}_0$, and is therefore $\kappa $-presentable.
$\square$
Proof of Proposition 9.5.2.7.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which satisfies the hypotheses of Proposition 9.5.2.7; we wish to show that $\operatorname{\mathcal{C}}$ is presentable (the converse follows from Remark 9.4.6.14 and 9.4.6.8). By assumption, $\operatorname{\mathcal{C}}$ is generated under small colimits by a small collection of objects $\{ C_ i \} _{i \in I}$. Moreover, each $C_ i$ is $\kappa _ i$-compact for some small regular cardinal $\kappa _ i$. Choose a small regular cardinal $\kappa $ which is an upper bound for the collection $\{ \kappa _ i \} _{i \in I}$. Then each of the objects $C_ i$ is $\kappa $-compact. Applying Proposition 9.5.2.11, we conclude that $\operatorname{\mathcal{C}}$ is $\kappa $-presentable. In particular, it is presentable.
$\square$