Kerodon

Remark 9.5.4.5. In the situation of Proposition 9.5.4.4, the homotopy fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is accessible (Proposition 9.4.8.9). If the functors $F_{-}$ and $F_{+}$ are both cocontinuous, then $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is closed under small colimits in the presentable $\infty $-category $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ (Remark 7.1.9.6), and is therefore also presentable. If $F_{-}$ and $F_{+}$ are both continuous, then $\operatorname{\mathcal{C}}_{\pm }$ is closed under small limits in $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$, and therefore also presentable.

Remark 9.5.4.9. In the situation of Proposition 9.5.4.7, suppose that $\operatorname{\mathcal{D}}$ is a cocomplete $\infty $-category. Then a functor $T: \operatorname{\mathcal{D}}\rightarrow \varprojlim (\mathscr {F})$ is cocontinuous if and only if, for each $C \in \operatorname{\mathcal{C}}$, the composite functor $\operatorname{\mathcal{D}}\xrightarrow {T} \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$ is cocontinuous. See Proposition 7.6.6.24.

Remark 9.5.4.11. In the situation of Variant 9.5.4.10, suppose that $\operatorname{\mathcal{D}}$ is a complete $\infty $-category. Then a functor $T: \operatorname{\mathcal{D}}\rightarrow \varprojlim (\mathscr {F})$ is continuous if and only if, for each $C \in \operatorname{\mathcal{C}}$, the composite functor

is continuous. Similarly, if $\operatorname{\mathcal{D}}$ is accessible, then $T$ is accessible if and only if each of the functors $T_{C}$ is accessible. See Proposition 7.6.6.24 and Remark 9.4.8.10.

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 $U$.

Remark 9.5.4.16 (Ambidexterity). Suppose we are given a small diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\operatorname{LPr}}$, which we identify with the covariant transport representation of a presentable fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Using Corollary 7.4.4.2 and Remark 9.5.4.15, we see that the limit and colimit of $\mathscr {F}$ are given by the $\infty $-categories $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of cocartesian and cartesian sections of $U$, respectively. If $\operatorname{\mathcal{C}}$ is a Kan complex, every section of $U$ is both cartesian and cocartesian. In this case, we obtain a canonical equivalence

Remark 9.5.4.20. In the situation of Lemma 9.5.4.19, if $U$ is a locally cartesian fibration then small colimits in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ are computed levelwise: that is, they are preserved by the evaluation functor

for each vertex $C \in \operatorname{\mathcal{C}}$. See Corollary 7.1.10.5. Similarly, if $U$ is a locally cocartesian fibration, then small limits in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ are computed levelwise.