Kerodon

Example 9.5.6.17 (Presentable Colimits as Bousfield Localizations). Let $\operatorname{\mathcal{C}}$ be a small simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a presentable cocartesian fibration, so that $U$ admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$ (Corollary 9.5.3.8). Recall that $U$ is a cartesian fibration, and that colimit $\varinjlim ( \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$) can be identified with the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{Cart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cartesian sections of $U$ (Remark 9.5.4.15). It follows from Proposition 9.5.6.15 that $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{Cart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a Bousfield localization of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of all sections of $U$ (which is also presentable; see Proposition 9.5.4.17).