Proposition 9.3.7.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa \leq \lambda $ be regular cardinals. Each of the following conditions implies the previous:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-filtered.
- $(2)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\kappa $-filtered.
- $(3)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\lambda $-filtered.
- $(4)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ has a final object.
If $\lambda $ is uncountable and $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, then all four conditions are equivalent.
Proof.
The implication $(4) \Rightarrow (3)$ follows from Example 9.1.1.6 and the implication $(3) \Rightarrow (2)$ from Remark 9.1.1.11. If $\lambda $ is uncountable and $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, then the implication $(4) \Rightarrow (1)$ follows from Proposition 9.3.7.2. We will complete the proof by showing that $(2)$ implies $(1)$. In what follows, let us abuse notation by identifying $\operatorname{\mathcal{C}}$ with a full subcategory of $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Assume that $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa $-filtered; we will show that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered. Fix a $\kappa $-small diagram $F: K \rightarrow \operatorname{\mathcal{C}}$; we wish to show that there exists a natural transformation from $F$ to the constant diagram $\underline{X}: K \rightarrow \operatorname{\mathcal{C}}$, for some $X \in \operatorname{\mathcal{C}}$. Note that each object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 9.3.2.3). Applying Proposition 9.2.8.9, we conclude that $F$ is $(\kappa ,\lambda )$-compact when viewed as an object of the diagram $\infty $-category $\operatorname{Fun}(K, \widehat{\operatorname{\mathcal{C}}} )$. Fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{Fun}(K, \widehat{\operatorname{\mathcal{C}}} )$ is locally $\mu $-small, so that the functor
\[ Q: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}_{< \mu } \quad \quad Q(X) = \operatorname{Hom}_{ \operatorname{Fun}(K, \widehat{\operatorname{\mathcal{C}}} ) }( F, \underline{X} ) \]
is $(\kappa ,\lambda )$-finitary. Since $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa $-filtered, there exists a natural transformation from $F$ to a constant diagram in $\widehat{\operatorname{\mathcal{C}}}$: that is, the Kan complex $Q(X)$ is nonempty for some $X \in \widehat{\operatorname{\mathcal{C}}}$. Invoking the universal property of $\widehat{\operatorname{\mathcal{C}}}$, we see that the restriction functor $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ is an equivalence of $\infty $-categories. It follows that $Q(X)$ must be also be nonempty for some $X \in \operatorname{\mathcal{C}}$, as desired.
$\square$