Kerodon

Proof. Fix a small regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits (this condition is satisfied, for example, if $\operatorname{\mathcal{C}}$ is accessible). Using Proposition 9.4.6.16, we can choose a small regular cardinal $\lambda > \kappa $ such that $K$ is $\lambda $-small and $\operatorname{\mathcal{C}}$ is $\lambda $-accessible. We will complete the proof by showing that $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is $\lambda $-accessible. Let $\operatorname{\mathcal{C}}_{< \lambda }$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\lambda $-compact objects. Since $\operatorname{\mathcal{C}}_{< \lambda }$ is $\kappa $-sequentially cocomplete, Corollary 9.4.5.14 implies that the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is $\lambda $-compactly generated, and that the full subcategory of $\lambda $-compact objects coincides with $\operatorname{Fun}(K, \operatorname{\mathcal{C}}_{< \lambda } )$. To complete the proof, it will suffice to show that the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}}_{< \lambda } )$ is essentially small. This follows from Remark 4.7.5.11, since $\operatorname{\mathcal{C}}_{< \lambda }$ and $K$ are essentially small. $\square$

Proof. Using Variant 9.4.7.12, we can choose a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are $\kappa $-accessible and the functors $F_{-}$ and $F_{+}$ are $\kappa $-compact. Applying Corollary 9.4.3.6, we deduce that $\operatorname{\mathcal{C}}_{\pm }$ is $\kappa $-compactly generated. Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $\kappa $-compact objects, and define $\operatorname{\mathcal{C}}^{0}_{-}$, $\operatorname{\mathcal{C}}_{+}^{0}$, and $\operatorname{\mathcal{C}}^{0}_{\pm }$ similarly. We will complete the proof by showing that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }^{0}$ is essentially small. By assumption, the $\infty $-categories $\operatorname{\mathcal{C}}_{-}^{0}$, $\operatorname{\mathcal{C}}_{+}^{0}$, and $\operatorname{\mathcal{C}}^{0}$ are essentially small. Using Corollary 9.4.3.6, we can identify $\operatorname{\mathcal{C}}_{\pm }^{0}$ with the oriented fiber product $\operatorname{\mathcal{C}}_{-}^{0} \vec{\times }_{ \operatorname{\mathcal{C}}^{0} } \operatorname{\mathcal{C}}_{+}^{0}$, so the desired result is a special case of Proposition 4.7.5.15. $\square$

Proof. Combine Propositions 9.4.8.1 and 9.4.8.3 with the equivalences

supplied by Theorem 4.6.4.19. $\square$

Proof. Fix a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ admit small $\kappa $-filtered colimits which are preserved by the functors $F_{-}$ and $F_{+}$. Using Variant 9.4.7.12, we can choose a small regular cardinal $\lambda > \kappa $ such that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are $\lambda $-accessible and the functors $F_{-}$ and $F_{+}$ are $\lambda $-compact. Apply Corollary 9.4.4.10, we deduce that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $\lambda $-compactly generated. Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $\lambda $-compact objects, and define $\operatorname{\mathcal{C}}^{0}_{-}$, $\operatorname{\mathcal{C}}_{+}^{0}$, and $\operatorname{\mathcal{C}}^{0}_{\pm }$ similarly. We will complete the proof by showing that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }^{0}$ is essentially small. By assumption, the $\infty $-categories $\operatorname{\mathcal{C}}_{-}^{0}$, $\operatorname{\mathcal{C}}_{+}^{0}$, and $\operatorname{\mathcal{C}}^{0}$ are essentially small. Using Corollary 9.4.4.10, we can identify $\operatorname{\mathcal{C}}_{\pm }^{0}$ with the homotopy fiber product $\operatorname{\mathcal{C}}_{-}^{0} \times ^{\mathrm{h}}_{ \operatorname{\mathcal{C}}^{0} } \operatorname{\mathcal{C}}_{+}^{0}$, so the desired result is a special case of Corollary 4.7.5.16. $\square$

Proof. This is a reformulation of Proposition 9.4.8.8 (together with Remark 9.4.8.9). $\square$

Proof. By virtue of Proposition 7.6.6.15 (together with Remarks 7.6.6.16 and Exercise 7.6.6.17), it will suffice to show that the $\infty $-category $\operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ admits pullbacks and small products which are preserved by the inclusion functor $\iota $. For pullbacks this is a reformulation of Corollary 9.4.8.10 (see Example 7.6.3.4), and for small products it follows from Corollary 9.4.6.19 and Example 9.4.7.4. $\square$

Proof. The implication $(1) \Rightarrow (2)$ is immediate, the implication $(2) \Rightarrow (3)$ follows from Remark 9.4.6.14 and Warning 9.4.8.13, and the implication $(3) \Rightarrow (4)$ follows from Proposition 9.4.6.2. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied and choose an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\kappa $-accessible. Our assumption that $U$ is $\kappa $-compactly generated guarantees that $\operatorname{\mathcal{E}}_{\sigma }$ is $\kappa $-compactly generated. Let $\operatorname{\mathcal{E}}'_{\sigma } \subseteq \operatorname{\mathcal{E}}_{\sigma }$ be the full subcategory spanned by the $\kappa $-compact objects. We will complete the proof by showing that $\operatorname{\mathcal{E}}'_{\sigma }$ is essentially small (see Proposition 9.4.6.2). This follows from assumption $(4)$, since $\operatorname{\mathcal{E}}'_{\sigma }$ can be identified with the fiber product $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ (see the proof of Proposition 9.4.5.11). $\square$

Proof. The implication $(1) \Rightarrow (2)$ is immediate from the definition. The converse follows by combining Propositions 9.4.8.19, 9.4.5.11, and 4.7.9.7. $\square$

Proof. Combine Proposition 9.4.8.19 with Corollary 9.4.5.8 and Proposition 5.1.5.14. $\square$

Proof. Combine Proposition 9.4.8.19 with Corollary 9.4.5.9 and Proposition 5.1.5.14. $\square$

Proof. Combine Corollary 9.4.8.24 with Variant 9.4.7.12. $\square$

Proof. Let $\operatorname{\mathcal{E}}' \subseteq \operatorname{\mathcal{E}}$ be the full simplicial subset spanned by those vertices $X \in \operatorname{\mathcal{E}}$ which are $\lambda $-compact when viewed as objects of the fiber $\{ U(X) \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Then $U$ restricts to an essentially small inner fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ (Proposition 9.4.6.2), so the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}' )$ is essentially small (Variant 4.7.9.11). Applying Theorem 9.4.5.13, we see that $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is an $\operatorname{Ind}_{\lambda }$-completion of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}' )$, and is therefore $\lambda $-accessible. $\square$

Proof. Using Remark 9.4.8.16, we can choose a small regular cardinal $\kappa $ such that $U$ is edgewise $\kappa $-accessible. Let $\lambda $ be another small regular cardinal satisfying $\kappa \triangleleft \lambda $. Then the inner fibration $U$ is also edgewise $\lambda $-accessible (Proposition 9.4.6.5). Applying Proposition 9.4.8.28, we conclude that $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is $\lambda $-accessible. $\square$