Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.4.3.14. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be a simplicial set which is essentially $\kappa $-small, and suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ with the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is essentially $\kappa $-small. Then the colimit $\varinjlim (\mathscr {F})$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) is essentially $\kappa $-small.

Proof. Without loss of generality, we may assume that $\mathscr {F}$ is a covariant transport representation for a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, so that the colimit $\varinjlim (\mathscr {F})$ can be identified with the localization $\operatorname{\mathcal{E}}[W^{-1}]$, where $W$ is the collection of $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ (Corollary 7.4.3.11). By virtue of Variant 6.3.2.6, it will suffice to show that the simplicial set $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small, which follows from Corollary 5.4.8.9. $\square$