Definition 9.2.3.1. Let $\kappa $ be a regular cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete if it admits $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$-indexed colimits. Here $\mathrm{Ord}_{< \kappa }$ denotes the collection of ordinals which are strictly smaller than $\kappa $ (so that $\mathrm{Ord}_{< \kappa }$ is a linearly ordered set of order type $\kappa $: see Proposition 4.7.1.22). If this condition is satisfied, then we say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-sequential colimits if it preserves $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$-indexed colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.2.3 Example: Sequential Cocompleteness
Recall that an $\infty $-category $\operatorname{\mathcal{C}}$ is sequentially cocomplete if every diagram
\[ X_0 \rightarrow X_1 \rightarrow X_2 \rightarrow \cdots \]
admits a colimit in $\operatorname{\mathcal{C}}$ (Definition 7.6.5.1). In this section, we study a transfinite variant of this condition.
Example 9.2.3.2. An $\infty $-category $\operatorname{\mathcal{C}}$ is sequentially cocomplete (in the sense of Definition 7.6.5.1) if and only if it is $\aleph _0$-sequentially cocomplete (in the sense of Definition 9.2.3.1). If this condition is satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves sequential colimits if and only if it preserves $\aleph _0$-sequential colimits.
Warning 9.2.3.3. Let $\kappa $ be a regular cardinal. There is a slight tension between the terminological conventions of Definitions 7.6.6.6 and Definition 9.2.3.1:
According to Definition 7.6.6.6, an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it admits $K$-indexed colimits for every $\kappa $-small simplicial set $K$.
According to Definition 9.2.3.1, an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete if it admits colimits indexed by the $\infty $-category $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$.
Neither of these conditions implies the other (the linearly ordered set $\mathrm{Ord}_{< \kappa }$ has cardinality $\kappa $, and is therefore not $\kappa $-small). Instead, they should be viewed as complementary to one another: see Corollary 9.2.3.9.
Notation 9.2.3.4. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.5.4.1). For every regular cardinal $\kappa $, we define a subcategory $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{seq}} \subseteq \operatorname{\mathcal{QC}}$ as follows:
An object $\operatorname{\mathcal{C}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{ \kappa -\mathrm{seq} }$ if and only if it is $\kappa $-sequentially cocomplete.
A morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ of $\operatorname{\mathcal{QC}}$ belongs to $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{seq}}$ if and only if it preserves $\kappa $-sequential colimits, in the sense of Definition 9.2.2.6.
More generally, if $\lambda $ is any uncountable cardinal, we let $\operatorname{\mathcal{QC}}^{ \kappa -\mathrm{seq} }_{< \lambda }$ denote the subcategory of $\operatorname{\mathcal{QC}}_{< \lambda }$ whose objects are $\lambda $-small $\infty $-categories which are $\kappa $-sequentially cocomplete, and whose morphisms are functors which preserve $\kappa $-seuqential colimits.
To formulate Definition 9.2.3.1 it is not necessary to assume that $\kappa $ is regular, or even that it is a cardinal. However, this does not really lead to any additional generality:
Proposition 9.2.3.5. Let $(Q, \leq )$ be a nonempty linearly ordered set with no largest element, so that the cofinality $\kappa = \mathrm{cf}(Q)$ is a regular cardinal (Example 4.7.3.15). Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete (in the sense of Definition 9.2.3.1) if and only if it admits $\operatorname{N}_{\bullet }(Q)$-indexed colimits. If this condition is satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-sequentially cocontinuous if and only if it preserves $\operatorname{N}_{\bullet }(Q)$-indexed colimits.
Proof. Recall that the cofinality $\kappa = \mathrm{cf}(Q)$ is characterized by the requirement that there exists a cofinal map of linearly ordered sets $f: \mathrm{Ord}_{< \kappa } \rightarrow Q$. Then, for each $q \in Q$, there exists an ordinal $\alpha < \kappa $ satisfying $q \leq f( \alpha )$. Let $g(q)$ be the smallest such ordinal, so that the construction $q \mapsto g(q)$ determines a cofinal map of linearly ordered sets $g: Q \rightarrow \mathrm{Ord}_{< \kappa }$. Invoking Corollary 7.2.3.4, we deduce that the functors
\[ \operatorname{N}_{\bullet }(f): \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } ) \rightarrow \operatorname{N}_{\bullet }(Q) \quad \quad \operatorname{N}_{\bullet }(g): \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } ) \]
are right cofinal. The desired results now follow from Corollaries 7.2.2.12 and 7.2.2.8. $\square$
Definition 9.2.3.1 can be viewed as a special case of Definition 9.2.1.3:
Proposition 9.2.3.6. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete (in the sense of Definition 9.2.3.1).
The $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa , \kappa ^{+} )$-cocomplete (in the sense of Definition 9.2.1.3), where $\kappa ^{+}$ is the successor of $\kappa $.
Proof. The implication $(2) \Rightarrow (1)$ is immediate, since the $\infty $-category $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$ is $\kappa ^{+}$-small and $\kappa $-filtered (Example 9.1.1.9). The converse follows from Corollary 7.2.2.12, since every $\kappa ^{+}$-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ admits a right cofinal functor $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } ) \rightarrow \operatorname{\mathcal{K}}$ (Corollary 9.1.8.9). $\square$
Remark 9.2.3.7. In the situation of Proposition 9.2.3.6, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-sequential colimits (in the sense of Definition 9.2.3.1) if and only if it is $(\kappa , \kappa ^{+})$-finitary (in the sense of Definition 9.2.2.6). In other words, we have an equality $\operatorname{\mathcal{QC}}^{\kappa -\mathrm{seq}} = \operatorname{\mathcal{QC}}^{ (\kappa ,\kappa ^{+} )-\mathrm{ccomp} }$ of subcategories of $\operatorname{\mathcal{QC}}$ (see Notations 9.2.3.4 and 9.2.2.14).
Corollary 9.2.3.8. An $\infty $-category $\operatorname{\mathcal{C}}$ is $(\aleph _0, \aleph _1)$-cocomplete (in the sense of Definition 9.2.1.3) if and only if it is sequentially cocomplete (in the sense of Definition 7.6.5.1). If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\aleph _0, \aleph _1)$-finitary if and only if it preserves sequential colimits.
Proof. Apply Proposition 9.2.3.6 (and Remark 9.2.3.7) in the special case $\kappa = \aleph _0$ (see Example 9.2.3.2). $\square$
Corollary 9.2.3.9. Let $\kappa $ be a regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa ^{+}$-cocomplete if and only if it is both $\kappa $-cocomplete and $\kappa $-sequentially cocomplete. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa ^{+}$-cocontinuous if and only if it is $\kappa $-cocontinuous and preserves $\kappa $-sequential colimits.
Proof. Combine Proposition 9.2.3.6 with Proposition 9.2.1.7 (for the first assertion) and Proposition 9.2.2.21 (for the second). $\square$
Corollary 9.2.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is $\aleph _1$-cocomplete (that is, it admits countable colimits) if and only if it is finitely cocomplete and sequentially cocomplete. If these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\aleph _1$-cocontinuous (that is, it preserves countable colimits) if and only if it preserves finite colimits and sequential colimits.
Proof. Apply Corollary 9.2.3.9 in the special case $\kappa = \aleph _0$ (see Example 9.2.3.2). $\square$
Corollary 9.2.3.11. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits if and only if it admits small $\kappa ^{+}$-filtered colimits and is $\kappa $-sequentially cocomplete. 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 $\kappa ^{+}$-finitary and preserves $\kappa $-sequential colimits.
Proof. Combine Proposition 9.2.3.6 with Corollary 9.2.1.16 (for the first assertion) and Corollary 9.2.2.30 (for the second). $\square$
Corollary 9.2.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits small filtered colimits if and only if it is sequentially cocomplete and admits small $\aleph _1$-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 $\aleph _1$-finitary and preserves sequential colimits.
Proof. Apply Corollary 9.2.3.11 in the special case $\kappa = \aleph _0$. $\square$