$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.6.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda $ be an infinite cardinal which is not regular. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-complete.
- $(2)$
For every infinite cardinal $\kappa < \lambda $, the $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-complete.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda ^{+}$-complete, where $\lambda ^{+}$ denotes the successor cardinal of $\lambda $.
Proof of Corollary 7.6.6.12.
The equivalence $(1) \Leftrightarrow (2)$ and the implication $(3) \Rightarrow (1)$ follow from Remark 7.6.6.2. We will complete the proof by showing that $(1)$ implies $(3)$. Assume that $\operatorname{\mathcal{C}}$ is $\lambda $-complete; we wish to show that it is $\lambda ^{+}$-complete. By virtue of Proposition 7.6.6.9, it will suffice to show that every collection of objects $\{ X_ i \} _{i \in I}$ admits a product in $\operatorname{\mathcal{C}}$, provided that the index set $I$ has cardinality $\leq \lambda $. Our assumption that $\lambda $ is not regular guarantees that we can decompose $I$ as a disjoint union of $\lambda $-small subsets $\{ I_ j \subseteq I \} _{j \in J}$, where the index set $J$ is $\lambda $-small. It follows from $(1)$ that $\operatorname{\mathcal{C}}$ admits $J$-indexed products and also that it admits $I_{j}$-indexed products for each $j \in J$, and therefore admits $I$-indexed products by virtue of Corollary 7.6.1.22.
$\square$