Kerodon

Remark 9.5.4.15 (Computing Colimits of Presentable $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a small simplicial set, and suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$. Our proof of Corollary 9.5.4.13 supplies a concrete description of the colimit $\varinjlim (\mathscr {F} )$. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration with covariant transport representation $\mathscr {F}$ (for example, we can take $\operatorname{\mathcal{E}}$ to be the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ introduced in Definition 5.6.2.1). Then $U$ is a presentable cocartesian fibration (Definition 9.5.3.5). It is therefore a cartesian fibration which admits a contravariant transport representation $\mathscr {F}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{\operatorname{RPr}}$ (Corollary 9.5.3.8). In this case, we can identify the colimit $\varinjlim ( \mathscr {F} )$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{LPr}}$) with the limit $\varprojlim ( \mathscr {F}' )$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}_{\leq \Omega }$). Applying Proposition 8.6.8.13, this limit can be identified with the $\infty $-category $\operatorname{Fun}^{\operatorname{Cart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ spanned by the cartesian sections of $U$.