# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 7.6.7.11. 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 $\operatorname{\mathcal{C}}$ admits $\lambda$-small limits.

$(2)$

For every infinite cardinal $\kappa < \lambda$, the $\infty$-category $\operatorname{\mathcal{C}}$ admits $\kappa$-small limits.

$(3)$

The $\infty$-category $\operatorname{\mathcal{C}}$ admits $\lambda ^{+}$-small limits, where $\lambda ^{+}$ denotes the successor of $\lambda$.

Proof of Corollary 7.6.7.11. The equivalence $(1) \Leftrightarrow (2)$ and the implication $(3) \Rightarrow (1)$ follow from Remark 7.6.7.2. We will complete the proof by showing that $(1)$ implies $(3)$. Assume that $\operatorname{\mathcal{C}}$ admits $\lambda$-small limits; we wish to show that it admits $\lambda ^{+}$-small limits. By virtue of Proposition 7.6.7.8, 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 7.6.1.19. $\square$