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


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 $\operatorname{\mathcal{C}}$.


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 $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits (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 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, so that $S = \bigcup _{n} \operatorname{sk}_ n(S)$. It follows from Proposition 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 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 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 By virtue of Proposition, 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 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 $\square$