# Kerodon

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Proposition 7.6.7.9. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\kappa$ be an infinite cardinal. Then $\operatorname{\mathcal{C}}$ is $\kappa$-complete if and only if it satisfies the following conditions:

$(1)$

The $\infty$-category $\operatorname{\mathcal{C}}$ admits $\kappa$-small products. That is, every collection of objects $\{ X_ j \} _{j \in J}$ indexed by a $\kappa$-small set $J$ admits a product in $\operatorname{\mathcal{C}}$.

$(2)$

The $\infty$-category $\operatorname{\mathcal{C}}$ admits finite limits.

Proof. Assume that $\operatorname{\mathcal{C}}$ satisfies conditions $(1)$ and $(2)$; we wish to show that $\operatorname{\mathcal{C}}$ is $\kappa$-complete (the converse is immediate from the definitions). Let $S$ be a $\kappa$-small simplicial set; we wish to show that $\operatorname{\mathcal{C}}$ admits $S$-indexed limits. If $\kappa = \aleph _0$, this follows immediately from assumption $(2)$ (Example 7.6.7.3). We may therefore assume that $\kappa$ is uncountable, so that $\operatorname{\mathcal{C}}$ admits countable products.

For each $n \geq 0$, let $\operatorname{sk}_{n}(S)$ denote the $n$-skeleton of $S$ (Construction 1.1.4.1), so that $S = \bigcup _{n} \operatorname{sk}_ n(S)$. It follows from Proposition 7.6.6.16 that $\operatorname{\mathcal{C}}$ admits sequential limits. Consequently, to show that $\operatorname{\mathcal{C}}$ admits $S$-indexed limits, it will suffice to show that it admits $\operatorname{sk}_ n(S)$-indexed limits, for each $n \geq 0$ (Corollary 7.6.6.14). We may therefore assume without loss of generality that the simplicial set $S$ has finite dimension. We proceed by induction on the dimension $n$ of $S$. If $n=-1$, then $S$ is empty and the desired result is immediate. Assume that $n \geq 0$ and let $\{ \sigma _ j \} _{j \in J}$ denote the collection of nondegenerate $n$-simplices of $S$, so that Proposition 1.1.4.12 supplies a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \underset {j \in J}{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset {j \in J}{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}(S) \ar [r] & \operatorname{sk}_{n}(S). }$

Since the horizontal maps in this diagram are monomorphisms, it is also a categorical pushout square (Example 4.5.4.12). By virtue of Proposition 7.6.3.26, it will suffice to show that $\operatorname{\mathcal{C}}$ admits limits indexed by the simplicial sets $\operatorname{sk}_{n-1}(S)$, $J \times \operatorname{\partial \Delta }^{n}$, and $J \times \Delta ^ n$. In the first two cases, this follows from our inductive hypothesis. To handle the third case, we can use assumption $(1)$ and Corollary 7.6.1.20 to reduce to showing that the $\infty$-category $\operatorname{\mathcal{C}}$ admits $\Delta ^{n}$-indexed limits. This is clear, since the simplicial set $\Delta ^ n$ is an $\infty$-category containing an initial object (see Corollary 7.2.2.12). $\square$