# Kerodon

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Example 7.6.7.7. Let $\lambda$ be an uncountable regular cardinal and let $\kappa = \mathrm{cf}(\lambda )$ denote its cofinality. Then the $\infty$-categories $\operatorname{\mathcal{S}}^{< \lambda }$ and $\operatorname{\mathcal{QC}}^{< \lambda }$ admit $\kappa$-small colimits. Moreover, the inclusion maps

$\operatorname{\mathcal{S}}^{ < \lambda } \hookrightarrow \operatorname{\mathcal{S}}\quad \quad \operatorname{\mathcal{QC}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}$

preserve $\kappa$-small colimits. See Corollary 7.4.3.15 and Remark 7.4.5.7. In particular, if $\kappa = \lambda$ is an uncountable regular cardinal, then the $\infty$-categories $\operatorname{\mathcal{S}}^{< \kappa }$ and $\operatorname{\mathcal{QC}}^{< \kappa }$ admit $\kappa$-small colimits.