Example 7.6.6.4. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality (Definition 4.7.3.16). Let $\operatorname{\mathcal{S}}^{< \lambda }$ denote the $\infty $-category of $\lambda $-small spaces (Variant 5.5.4.13) and let $\operatorname{\mathcal{QC}}^{ < \lambda }$ denote the $\infty $-category of $\lambda $-small $\infty $-categories (Variant 5.5.4.11). Then the $\infty $-categories $\operatorname{\mathcal{S}}^{< \lambda }$ and $\operatorname{\mathcal{QC}}^{< \lambda }$ are $\kappa $-complete. Moreover, the inclusion maps
preserve $\kappa $-small limits. See Variants 7.4.1.15 and 7.4.4.14. In particular, if $\kappa = \lambda $ is a strongly inaccessible cardinal, then the $\infty $-categories $\operatorname{\mathcal{S}}^{< \kappa }$ and $\operatorname{\mathcal{QC}}^{< \kappa }$ are $\kappa $-complete.