Kerodon

Proof. The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definitions and the equivalence $(2) \Leftrightarrow (3)$ follows from the description of $\operatorname{Ind}_{\kappa }^{\lambda }$-completions supplied by Theorem 9.3.4.14. The implication $(3) \Rightarrow (2)$ follows Proposition 9.4.1.11. Conversely, if $\operatorname{\mathcal{C}}$ is the $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$, then $\operatorname{\mathcal{C}}_{< \kappa }$ is an idempotent-completion of $\operatorname{\mathcal{C}}_0$ (Proposition 9.4.1.19) and is therefore essentially small (Proposition 8.5.5.7). $\square$

Proof. By assumption, $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$. Applying Theorem 9.3.6.4, we conclude that $\operatorname{\mathcal{C}}$ is also an $\operatorname{Ind}_{\lambda }$-completion of the $\infty $-category $\operatorname{\mathcal{C}}_1 = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}_0)$, which is essentially small by virtue of Variant 9.3.3.3. $\square$

Proof. Combine Proposition 9.4.6.5 with Example 9.1.7.10. $\square$

Proof. Enlarging $\lambda $ if necessary, we may assume that $\kappa \trianglelefteq \lambda $ (see Proposition 9.1.7.8). In this case, $\operatorname{\mathcal{C}}$ is $\lambda $-accessible (Proposition 9.4.6.5), so the desired result follows from Proposition 9.4.6.2. $\square$

Proof. Since $\operatorname{\mathcal{C}}$ is accessible, it is $\kappa $-accessible for some small regular cardinal $\kappa $. Combining Propositions 9.4.6.5 and 9.1.7.8, we conclude that $\operatorname{\mathcal{C}}$ is $\lambda $-accessible for every small regular cardinal $\lambda $ of exponential cofinality $\geq \kappa $ (for example, every cardinal of the form $(\mu ^{\kappa })^{+}$). $\square$

Proof. Using Variant 9.4.6.18, we can choose a small regular cardinal $\lambda $ such $I$ is $\lambda $-small and each $\operatorname{\mathcal{C}}_ i$ is $\lambda $-accessible. In this case, Remark 9.4.6.4 implies that $\prod _{i \in I} \operatorname{\mathcal{C}}_ i$ is $\lambda $-accessible. $\square$

Proof. Assume that $\operatorname{\mathcal{C}}$ is both accessible and $\kappa $-compactly generated; we wish to show that it is $\kappa $-accessible (the reverse implication is immediate from the definitions). By virtue of Proposition 9.4.6.2, it will suffice to show that the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects is essentially small. Using Proposition 9.4.6.16, we can choose a regular cardinal $\lambda \geq \kappa $ such that $\operatorname{\mathcal{C}}$ is $\lambda $-accessible. Then the $\infty $-category $\operatorname{\mathcal{C}}_{< \lambda }$ is essentially small, so the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}_{< \lambda }$ has the same property. $\square$

Proof. If $Q$ is small, then the $\infty $-category $\operatorname{\mathcal{C}}$ is essentially small and idempotent-complete (Example 8.5.4.6), hence accessible by virtue of Proposition 9.4.6.21. For the converse, suppose that $\operatorname{\mathcal{C}}$ is accessible. Then $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }$-completion of a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$, for some small regular cardinal $\kappa $. Let us identify $\operatorname{\mathcal{C}}_0$ with the nerve of a partially ordered set $Q_0 \subseteq Q$. It follows from Exercise 9.3.2.12 that $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ can be identified with the nerve of the partially ordered set of ideals in $Q_0$. In particular, $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is essentially small, so that $\operatorname{\mathcal{C}}\simeq \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}}_0)$ is also essentially small. $\square$

Proof. Assume otherwise. Then there exists a subset $Y \subseteq X$ having cardinality $\kappa $. For each $y \in Y$, we can choose an index $\alpha _ y \in A$ such that $y$ is the image of some element $\widetilde{y} \in X_{ \alpha _{y} }$. Since $Y$ is $\lambda $-small and $(A, \leq )$ is $\lambda $-directed, the set $\{ \alpha _{y}: y \in A \} $ has an upper bound $\alpha $ in $A$. It follows that $Y$ is contained in the image of the map $X_{\alpha } \rightarrow X$, contradicting our assumption that $X_{\alpha }$ is $\kappa $-small. $\square$

Proof. Enlarging $\mu $ if necessary, we may assume that $\lambda \trianglelefteq \mu $. In this case, Theorem 9.1.8.7 guarantees the existence of a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{K}}$, where $(A, \leq )$ is a $\mu $-small $\lambda $-directed partially ordered set. By virtue of Corollary 5.6.5.18, we may assume that $\mathscr {F}$ is obtained from (strictly) commutative diagram of Kan complexes

In this case, we can use Variant 9.1.6.4 to identify the colimit $\varinjlim (\mathscr {F})$ (formed in the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$) with the colimit $X = \varinjlim _{\alpha } X_{\alpha }$ (formed in the ordinary category of Kan complexes). We will complete the proof by showing that $X$ is essentially $\kappa $-small.

It follows from Lemma 9.4.6.23 that the set of connected components $\pi _0(X)$ is $\kappa $-small. By virtue of Proposition 4.7.7.1, it will suffice to show that for every vertex $x \in X$ and every integer $n > 0$, the homotopy group $G = \pi _{n}(X,x)$ is $\kappa $-small. Choose an index $\alpha \in A$ such that $x$ lifts to a vertex $x_{\alpha } \in X_{\alpha }$. For each $\beta \geq \alpha $, let $x_{\beta }$ denote the image of $x_{\alpha }$ in the Kan complex $X_{\beta }$. Then $G$ can be identified with the $\lambda $-directed colimit $\varinjlim _{\beta \geq \alpha } \pi _{n}( X_{\beta }, x_{\beta } )$, which is $\kappa $-small by virtue of Lemma 9.4.6.23. $\square$

Proof. By virtue of Theorem 9.3.4.14, we can identify $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ with the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ spanned by those functors which are $\lambda $-flat and $h_{\bullet }$ with the covariant Yoneda embedding $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C)$. To complete the proof, we must show that every $\lambda $-flat functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is retract of a representable functor. Equivalently, we must show that for each object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is essentially $\lambda $-small (Remark 9.3.4.12). In fact, we claim that $\mathscr {F}(C)$ is essentially $\kappa $-small. To prove this, we can use Lemma 9.3.4.21 to realize $\mathscr {F}(C)$ as the colimit of a $\mu $-small $\lambda $-filtered diagram

which is $\kappa $-small by virtue of Lemma 9.4.6.24 (since $\operatorname{\mathcal{C}}$ is locally $\kappa $-small). $\square$

Proof. Let $\operatorname{\mathcal{K}}$ be a $\lambda $-filtered $\infty $-category, and fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{K}}$ is $\mu $-small. Applying Proposition 9.4.6.25, we see that the tautological map $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories. Since $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ admits $\operatorname{\mathcal{K}}$-indexed colimits, it follows that $\operatorname{\mathcal{C}}$ admits $\operatorname{\mathcal{K}}$-indexed colimits. We will complete the proof by showing that every functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\operatorname{\mathcal{K}}$-indexed colimits. Enlarging $\operatorname{\mathcal{D}}$ if necessary, we may assume that it admits $\mu $-small $\lambda $-filtered colimits (for example, we can replace $\operatorname{\mathcal{D}}$ by the $\infty $-category $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{D}})$). In this case, $F$ admits an $\operatorname{Ind}_{\lambda }^{\mu }$-extension $\widehat{F}: \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ (Definition 9.3.1.12). Then $\widehat{F}$ preserves $\operatorname{\mathcal{K}}$-indexed colimits and $F$ is isomorphic to the composite functor $\widehat{F} \circ h$. Since $h$ is an equivalence of $\infty $-categories, it follows that $F$ also preserves $\operatorname{\mathcal{K}}$-indexed colimits. $\square$

Proof of Proposition 9.4.6.21. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category which is idempotent-complete; we wish to show that $\operatorname{\mathcal{C}}$ is accessible (the reverse implication follows from Remark 9.4.6.14). Let $\kappa $ be a small uncountable regular cardinal for which $\operatorname{\mathcal{C}}$ is locally $\kappa $-small, and let $\lambda > \kappa $ be a small regular cardinal for which $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Applying Proposition 9.4.6.25, we deduce that $\operatorname{\mathcal{C}}$ can be identified with an $\operatorname{Ind}_{\lambda }$-completion of itself. In particular, $\operatorname{\mathcal{C}}$ is $\lambda $-accessible. $\square$