Recall that an $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if every small diagram in $\operatorname{\mathcal{C}}$ admits a colimit (Definition 7.6.6.1). It will sometimes be helpful to break this down into two separate conditions:
Proposition 9.2.1.1. An $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if and only if it satisfies both of the following conditions:
The $\infty $-category $\operatorname{\mathcal{C}}$ is finitely cocomplete: that is, it admits finite colimits.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits small filtered colimits.
Proof. Assume that conditions $(a)$ and $(b)$ are satisfied; we will show that $\operatorname{\mathcal{C}}$ is cocomplete (the converse is immediate from the definitions). Let $K$ be a small simplicial set; we wish to show that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits. Let $\{ K_{\alpha } \} _{\alpha \in A}$ be the collection of all finite simplicial subsets of $K$. For $\alpha ,\beta \in A$, let us write $\alpha \leq \beta $ if $K_{\alpha }$ is contained in $K_{\beta }$. Then $(A, \leq )$ is a small directed partially ordered set, so $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(A)$-indexed colimits by virtue of condition $(b)$. It follows from condition $(a)$ that $\operatorname{\mathcal{C}}$ also admits $K_{\alpha }$-indexed colimits, for each $\alpha \in A$. Applying Corollary 9.1.6.6, we conclude that $\operatorname{\mathcal{C}}$ admits colimits indexed by $K = \varinjlim _{\alpha \in A} K_{\alpha }$, as desired. $\square$
Remark 9.2.1.2. The criterion of Proposition 9.2.1.1 is quite useful in practice, because the kinds of arguments used to verify conditions $(a)$ and $(b)$ often have a different flavor from one another:
To show that an $\infty $-category $\operatorname{\mathcal{C}}$ admits colimits of finite diagrams, we can reduce to the case of diagrams of very specific type: for example, $\operatorname{\mathcal{C}}$ is finitely cocomplete if and only if it admits pushouts and has an initial object (Corollary 7.6.2.30).
Many mathematical constructions preserve the formation of filtered colimits, but not general colimits. For this reason, it is often easy to check that some “forgetful” functor $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ creates filtered colimits (in the sense of Definition 7.1.4.18), so that the existence of filtered colimits in $\operatorname{\mathcal{C}}$ can be reduced to existence of filtered colimits in some (potentially simpler) $\infty $-category $\operatorname{\mathcal{D}}$.
Proposition 9.2.1.1 has a counterpart for $\kappa $-cocomplete $\infty $-categories, where $\kappa $ is any regular cardinal. Before formulating it, it will be convenient to introduce some terminology.
Definition 9.2.1.3. Let $\kappa \leq \lambda $ be regular cardinals. We will say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete if it admits $\operatorname{\mathcal{K}}$-indexed colimits for every $\infty $-category $\operatorname{\mathcal{K}}$ which is $\lambda $-small and $\kappa $-filtered.
Warning 9.2.1.4. Definition 9.2.1.3 makes sense for every pair of regular cardinals $\kappa \leq \lambda $. However, it is well behaved under the additional assumption $\kappa \trianglelefteq \lambda $ (see Definition 9.1.7.5). Roughly speaking, this condition guarantees that there are “enough” examples of $\lambda $-small $\kappa $-filtered $\infty $-categories.
Remark 9.2.1.5 (Monotonicity). Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then, for any regular cardinals $\kappa \leq \kappa ' \leq \lambda ' \leq \lambda $, the $\infty $-category $\operatorname{\mathcal{C}}$ is also $(\kappa ', \lambda ')$-cocomplete.
Example 9.2.1.6. Let $X$ be a Kan complex. Then $X$ is $(\kappa ,\lambda )$-cocomplete when viewed as an $\infty $-category, for every pair of regular cardinals $\kappa \leq \lambda $. This follows from Corollary 7.2.3.5, since every filtered $\infty $-category is weakly contractible (Proposition 9.1.1.18).
Proposition 9.2.1.7. 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 both $\kappa $-cocomplete and $(\kappa ,\lambda )$-cocomplete.
Remark 9.2.1.8. Following the convention of Remark 4.7.0.5, Proposition 9.2.1.1 can be viewed as a special case where of Proposition 9.2.1.7, where we take $\kappa = \aleph _0$ and $\lambda = \operatorname{\Omega }$ to be a strongly inaccessible cardinal.
Proof of Proposition 9.2.1.7. The implication $(1) \Rightarrow (2)$ is trivial (and does not require the assumption $\kappa \trianglelefteq \lambda $). To prove the converse, we proceed as in the proof of Proposition 9.2.1.1. Assume that $\operatorname{\mathcal{C}}$ is both $\kappa $-cocomplete and $(\kappa ,\lambda )$-cocomplete; we wish to show that every $\lambda $-small diagram $K \rightarrow \operatorname{\mathcal{C}}$ admits a colimit. Using Lemma 9.1.7.18, we can realize $K$ as the colimit of a diagram of $\kappa $-small simplicial subsets $\{ K_{\alpha } \} _{\alpha \in A}$ indexed by a $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$. Since $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete, it admits $K_{\alpha }$-indexed colimits for each $\alpha \in A$. Our assumption that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete guarantees that it also admits $\operatorname{N}_{\bullet }(A)$-indexed colimits. Applying Corollary 9.1.6.6, we conclude that $\operatorname{\mathcal{C}}$ admits colimits indexed by $K = \varinjlim _{\alpha \in A} K_{\alpha }$, as desired. $\square$
Corollary 9.2.1.9. 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.
Proof. Apply Proposition 9.2.1.7 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal (see Example 9.1.7.11). $\square$
Corollary 9.2.1.10. Let $\lambda $ be a regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-cocomplete if and only if it is finitely cocomplete and admits $\lambda $-small filtered colimits.
Proof. Apply Proposition 9.2.1.7 in the special case where $\kappa = \aleph _0$ (see Example 9.1.7.10). $\square$
Definition 9.2.1.3 can often be simplified.
Proposition 9.2.1.11. 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 $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$.
Proof. The implication $(1) \Rightarrow (2)$ follows from Example 9.1.1.8, and the reverse implication follows by combining Theorem 9.1.8.7 with Corollary 7.2.2.12. $\square$
Corollary 9.2.1.12. Let $\kappa $ be a small regular 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 $(A, \leq )$.
Proof. Apply Proposition 9.2.1.11 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal (see Example 9.1.7.11). $\square$
Corollary 9.2.1.13. Let $\lambda $ be an uncountable regular cardinal. Then $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small filtered colimits if and only if it admits $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every $\lambda $-small directed partially ordered set $(A, \leq )$.
Proof. Apply Proposition 9.2.1.11 in the special case $\kappa = \aleph _0$ (see Example 9.1.7.10). $\square$
Beware that Proposition 9.2.1.11 is not quite correct in the degenerate case $\kappa = \lambda $. Instead we have the following:
Proposition 9.2.1.14. Let $\kappa $ be a regular cardinal. Then:
If $\kappa $ is uncountable, then an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\kappa )$-cocomplete if and only if it is idempotent-complete.
If $\kappa = \aleph _0$, then every $\infty $-category is $(\kappa ,\kappa )$-cocomplete.
Proof. We will assume that $\kappa $ is uncountable (the case $\kappa = \aleph _0$ follows from Corollary 9.1.8.12). In this case, the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{Idem})$ of Construction 8.5.2.7 is both $\kappa $-small and $\kappa $-filtered (Example 9.1.1.10), so every $(\kappa ,\kappa )$-cocomplete $\infty $-category is idempotent-complete (Proposition 8.5.4.10). Conversely, suppose that $\operatorname{\mathcal{C}}$ is idempotent-complete; we wish to show that $\operatorname{\mathcal{C}}$ admits $\operatorname{\mathcal{K}}$-indexed colimits for every $\infty $-category $\operatorname{\mathcal{K}}$ which is both $\kappa $-small and $\kappa $-filtered. Proposition 9.1.8.11 guarantees the existence of a right cofinal functor $\operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{K}}$, so the desired result is follows from Corollary 7.2.2.12. $\square$
Proposition 9.2.1.15 (Transitivity). Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda \trianglelefteq \mu $. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-cocomplete if and only if it is both $(\kappa ,\lambda )$-cocomplete and $(\lambda ,\mu )$-cocomplete.
Proof. We will assume $\kappa < \lambda $ (otherwise, there is nothing to prove). Assume that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete and $(\lambda ,\mu )$-cocomplete; we wish to show that it is $(\kappa ,\mu )$-cocomplete (the converse follows immediately from Remark 9.2.1.5, and requires only the assumption $\kappa \leq \lambda \leq \mu $). This 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 9.2.1.16. 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 is $(\kappa ,\lambda )$-cocomplete.
Proof. Apply Proposition 9.2.2.29 in the special case where $\lambda = \operatorname{\Omega }$ is a fixed strongly inaccessible cardinal (see Example 9.1.7.11). $\square$
Corollary 9.2.1.17. 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.
Proof. Apply Corollary 9.2.1.16 in the special case $\kappa = \aleph _0$ (see Example 9.1.7.10). $\square$