Definition 9.1.9.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, where $\operatorname{\mathcal{C}}$ admits small filtered colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is finitary if $F$ preserves $\operatorname{\mathcal{K}}$-indexed colimits, for every small filtered $\infty $-category $\operatorname{\mathcal{K}}$. We let $\operatorname{Fun}^{\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the finitary functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
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9.1.9 Finitary Functors
We now study colimits indexed by filtered $\infty $-categories.
Remark 9.1.9.2. We will primarily make use of Definition 9.1.9.1 in situations where the $\infty $-category $\operatorname{\mathcal{D}}$ also admits small filtered colimits.
We will often use an infinitary version of Definition 9.1.9.1.
Definition 9.1.9.3. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits, and let $\operatorname{\mathcal{D}}$ be another $\infty $-category. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-finitary if it preserves $\operatorname{\mathcal{K}}$-indexed colimits, for every small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$. We let $\operatorname{Fun}^{\kappa -\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $\kappa $-finitary functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
Remark 9.1.9.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ admits small filtered colimits. Then $F$ is finitary (in the sense of Definition 9.1.9.3) if and only if it is $\aleph _0$-finitary (in the sense of Definition 9.1.9.3).
Example 9.1.9.5. If $K$ is a $\kappa $-small simplicial set, then the limit functor $\varprojlim : \operatorname{Fun}(K, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ is $\kappa $-finitary: this is a refomulation of Theorem 9.1.5.7 (see Proposition 9.1.4.1). In particular, if $K$ is a finite simplicial set, then the functor $\varprojlim : \operatorname{Fun}(K, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ is finitary.
It will sometimes be useful to consider a further generalization of Definition 9.1.9.1.
Definition 9.1.9.6. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-finitary if it preserves $\operatorname{\mathcal{K}}$-indexed colimits, for every $\infty $-category $\operatorname{\mathcal{K}}$ which is $\lambda $-small and $\kappa $-filtered. We let $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the $(\kappa ,\lambda )$-finitary functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
Warning 9.1.9.7. Definition 9.1.9.6 makes sense for every pair of regular cardinals $\kappa \leq \lambda $. However, it is well-behaved only in the case where $\kappa \trianglelefteq \lambda $, in the sense of Definition 9.1.7.5. Roughly speaking, this condition guarantees that there are “enough” examples of $\lambda $-small $\kappa $-filtered $\infty $-categories (see Corollary 9.1.7.16).
Remark 9.1.9.8. Following the convention of Remark 4.7.0.5, a cardinal $\kappa $ is small if it satisfies $\kappa < \operatorname{\textnormal{\cjRL {t}}}$, where $\operatorname{\textnormal{\cjRL {t}}}$ is some fixed strongly inaccessible cardinal. In this case, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-finitary (in the sense of Definition 9.1.9.3) if and only if it is $(\kappa ,\operatorname{\textnormal{\cjRL {t}}})$-finitary (in the sense of Definition 9.1.9.6). Note that in this case we automatically have $\kappa \triangleleft \operatorname{\textnormal{\cjRL {t}}}$ (Example 9.1.7.11). In particular, $F$ is finitary if and only if it is $(\aleph _0, \operatorname{\textnormal{\cjRL {t}}})$-finitary.
Remark 9.1.9.9. Let $\kappa \leq \lambda $ be regular cardinals, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. Let $\mu $ be an uncountable cardinal having cofinality $\geq \lambda $ and exponential cofinality $\geq \kappa $, so that the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$ admits $\lambda $-small colimits and $\kappa $-small limits (see Corollary 7.4.3.8 and Variant 7.4.1.15). Then the $\infty $-category $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ is closed under the formation of $\kappa $-small limits in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. See Proposition 9.1.5.8 (and Proposition 9.1.4.1). In particular:
If $\kappa $ is small and $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits, then the collection of $\kappa $-finitary functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is closed under $\kappa $-small limits.
If $\operatorname{\mathcal{C}}$ admits small filtered colimits, then the collection of finitary functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is closed under finite limits.
Proposition 9.1.9.10. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \triangleleft \lambda $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$.
Moreover, if these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-finitary if and only if it preserves $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$.
Proof. The implication $(1) \Rightarrow (2)$ follows from Example 9.1.1.7, and the reverse implication follows by combining Theorem 9.1.8.7 with Corollary 7.2.2.12. The final assertion follows from Corollary 7.2.2.3. $\square$
Variant 9.1.9.11. An $\infty $-category $\operatorname{\mathcal{C}}$ admits $\aleph _1$-small, $\aleph _0$-filtered colimits if and only if it admits sequential colimits, in the sense of Definition 7.6.5.1. In this case, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $( \aleph _0, \aleph _1)$-finitary if and only if it preserves sequential colimits. See Corollary 9.1.8.9.
Corollary 9.1.9.12. Let $\kappa $ be a small cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits if and only if it admits $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every small $\kappa $-directed partially ordered set $(A, \leq )$. In this case, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-finitary if and only if preserves $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every small $\kappa $-directed partially ordered set $(A, \leq )$.
Proof. Apply Proposition 9.1.9.10 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a fixed strongly inaccessible cardinal (see Example 9.1.7.11). $\square$
Corollary 9.1.9.13. An $\infty $-category $\operatorname{\mathcal{C}}$ admits small filtered colimits if and only if it admits $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every small directed partially ordered set $(A, \leq )$. In this case, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is finitary if and only if it preserves $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every small directed partially ordered set $(A, \leq )$.
Proof. Apply Corollary 9.1.9.12 in the special case $\kappa = \aleph _0$ (see Example 9.1.7.10). $\square$
Example 9.1.9.14. Let $\operatorname{Kan}$ denote the ordinary category of Kan complexes and let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces. Then the inclusion functor is finitary: this follows by combining Corollary 9.1.9.13 with Variant 9.1.6.4. More generally, if $\lambda $ is an uncountable regular cardinal and $\operatorname{Kan}_{< \lambda }$ denote the category of $\lambda $-small Kan complexes, then the inclusion functor $\operatorname{N}_{\bullet }( \operatorname{Kan}_{< \lambda } ) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{< \lambda } ) = \operatorname{\mathcal{S}}_{< \lambda } )$ is $(\aleph _0, \lambda )$-finitary: that is, it preserves $\lambda $-small filtered colimits.
\[ \operatorname{N}_{\bullet }(\operatorname{Kan}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}} \]
Example 9.1.9.15. The inclusion fucntor $\operatorname{N}_{\bullet }(\operatorname{Set}) \hookrightarrow \operatorname{\mathcal{S}}$ is finitary. For a more general statement, see Variant 9.1.10.3.
Example 9.1.9.16. Let $\operatorname{QCat}$ denote the full subcategory of $\operatorname{Set_{\Delta }}$ spanned by the $\infty $-categories, and let $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ be its homotopy coherent nerve (see Construction 5.5.4.1). Then the inclusion functor is finitary: this follows by combining Corollary 9.1.9.13 with Corollary 9.1.6.3. More generally, if $\lambda $ is an uncountable regular cardinal and $\operatorname{QCat}_{< \lambda }$ denote the category of $\lambda $-small $\infty $-categories, then the inclusion functor $\operatorname{N}_{\bullet }( \operatorname{QCat}_{< \lambda } ) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{< \lambda } ) = \operatorname{\mathcal{QC}}_{< \lambda } )$ is $(\aleph _0, \lambda )$-finitary.
\[ \operatorname{N}_{\bullet }(\operatorname{QCat}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]
Proposition 9.1.9.17. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then:
If $\kappa $ is uncountable, then $\operatorname{\mathcal{C}}$ admits $\kappa $-small $\kappa $-filtered colimits if and only if it is idempotent complete.
If $\kappa = \aleph _0$, then $\operatorname{\mathcal{C}}$ automatically admits $\kappa $-small $\kappa $-filtered colimits.
In either case, any functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa , \kappa )$-finitary.
Proof. Combine Proposition 9.1.8.10, Corollary 9.1.8.11, and Corollary 8.5.3.12. $\square$
Remark 9.1.9.18 (Monotonicity). Let $\kappa \leq \lambda $ be regular cardinals, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. Then, for any regular cardinals $\kappa \leq \kappa ' \leq \lambda ' \leq \lambda $, the $\infty $-category $\operatorname{\mathcal{C}}$ also admits $\lambda '$-small, $\kappa '$-filtered colimits. Moreover, every $(\kappa ,\lambda )$-finitary functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is also $(\kappa ', \lambda ')$-finitary.
Remark 9.1.9.18 admits the following partial converse:
Proposition 9.1.9.19 (Transitivity). Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $ and $\lambda \trianglelefteq \mu $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\mu $-small, $\kappa $-filtered colimits.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\mu $-small, $\lambda $-filtered colimits and $\lambda $-small, $\kappa $-filtered colimits.
Moreover, if these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa , \mu )$-finitary if and only if it is both $(\kappa , \lambda )$-finitary and $(\lambda ,\mu )$-finitary.
Proof. Without loss of generality we may assume $\kappa \neq \lambda $ (otherwise, there is nothing to prove). The implication $(1) \Rightarrow (2)$ follows from Remark 9.1.9.18 (and requires only the assumption $\kappa \leq \lambda \leq \mu $). The converse follows from Corollary 9.1.6.6, since every $\mu $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ can be realized as a $\mu $-small, $\lambda $-filtered colimit of $\lambda $-small, $\kappa $-filtered $\infty $-categories (Corollary 9.1.7.16). $\square$
Corollary 9.1.9.20. Let $\kappa $ and $\lambda $ be small regular cardinals satisfying $\kappa \trianglelefteq \lambda $. Then an $\infty $-category $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits if and only if it admits small $\lambda $-filtered colimits and $\lambda $-small $\kappa $-filtered colimits. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-finitary if and only if it is both $\lambda $-finitary and $(\kappa , \lambda )$-finitary.
Proof. Apply Proposition 9.1.9.19 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a fixed strongly inaccessible cardinal (see Example 9.1.7.11). $\square$
Corollary 9.1.9.21. Let $\lambda $ be a small regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ admits small filtered colimits if and only if it admits small $\lambda $-filtered colimits and $\lambda $-small filtered colimits. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is finitary if and only if it is both $\lambda $-finitary and $(\aleph _0, \lambda )$-finitary.
Proof. Apply Corollary 9.1.9.20 in the special case $\kappa = \aleph _0$ (see Example 9.1.7.10). $\square$
The preceding results have counterparts for non-filtered colimits.
Proposition 9.1.9.22. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete and admits $\lambda $-small $\kappa $-filtered colimits.
Moreover, if these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\lambda $-small colimits if and only if it is $(\kappa ,\lambda )$-finitary and preserves $\kappa $-small colimits.
Proof. The implication $(1) \Rightarrow (2)$ is trivial (and does not require the assumption $\kappa \trianglelefteq \lambda $). The reverse impllication follows from Corollary 9.1.6.6, since every $\lambda $-small $\infty $-category can be realized as a $\lambda $-small, $\kappa $-filtered colimit of $\kappa $-small $\infty $-categories (Variant 9.1.7.21). $\square$
Corollary 9.1.9.23. Let $\kappa $ be a small regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if and only if it is $\kappa $-cocomplete and admits small $\kappa $-filtered colimits. If these conditions are satisfied, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves small colimits if and only if it preserves both $\kappa $-small colimits and small $\kappa $-filtered colimits.
Proof. Apply Proposition 9.1.9.22 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a fixed strongly inaccessible cardinal (see Example 9.1.7.11). $\square$
Corollary 9.1.9.24. An $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if and only if it admits finite colimits and small filtered colimits. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves small colimits if and only if it preserves both finite colimits and small filtered colimits.
Proof. Apply Corollary 9.1.9.23 in the special case $\kappa = \aleph _0$. $\square$