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Proposition 9.5.1.11. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete and $(\kappa ,\lambda )$-compactly generated. Assume that $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small, where $\operatorname{\mathcal{C}}_{< \kappa }$ denotes the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects. Let $\mu $ be an uncountable regular cardinal. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is representable by an object of $\operatorname{\mathcal{C}}$ if and only if it satisfies the following conditions:
- $(1)$
The functor $\mathscr {F}$ is $\lambda $-continuous: that is, it carries $\lambda $-small colimit diagrams in $\operatorname{\mathcal{C}}$ to limit diagrams in $\operatorname{\mathcal{S}}_{< \mu }$.
- $(2)$
For every $(\kappa ,\lambda )$-compact object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is essentially $\lambda $-small.
Proof.
The necessity of condition $(1)$ follows from Corollary 7.4.1.23. To prove the necessity of condition $(2)$, we must show that if $C$ and $D$ are objects of $\operatorname{\mathcal{C}}$ where $C$ is $(\kappa ,\lambda )$-compact, then the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ is essentially $\lambda $-small. Let us regard the object $C \in \operatorname{\mathcal{C}}$ as fixed, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $D \in \operatorname{\mathcal{C}}$ for which $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, D)$ is essentially $\lambda $-small. Since $C$ is $(\kappa ,\lambda )$-compact, the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is closed under $\lambda $-small $\kappa $-filtered colimits (Corollary 7.4.3.8). Consequently, to show that $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}$, it will suffice to show that it contains every $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. This follows from our assumption that $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small.
We now prove the converse. Assume that conditions $(1)$ and $(2)$ are satisfied; we wish to show that the functor $\mathscr {F}$ is representable. By virtue of Proposition 8.6.8.3, we may assume that $\mathscr {F}$ is the contravariant transport representation of a right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. It follows from assumption $(1)$ that the $\infty $-category $\operatorname{\mathcal{E}}$ is $\lambda $-cocomplete and that the functor $U$ is $\lambda $-cocontinuous (Corollary 7.4.1.21). Let $\operatorname{\mathcal{E}}_{< \kappa } \subseteq \operatorname{\mathcal{E}}$ denote the inverse image of $\operatorname{\mathcal{C}}_{< \kappa }$, so that $\operatorname{\mathcal{E}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{E}}_{< \kappa }$ (Proposition 9.3.1.16). The $\infty $-category $\operatorname{\mathcal{E}}_{< \kappa }$ is closed under $\kappa $-small colimits in $\operatorname{\mathcal{E}}$; in particular, it is $\kappa $-filtered (Example 9.1.1.7). Assumption $(2)$ guarantees that the right fibration $U_{< \kappa }: \operatorname{\mathcal{E}}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small, so that the $\infty $-category $\operatorname{\mathcal{E}}_{< \kappa }$ is also essentially $\lambda $-small (Corollary 4.9.8.12). Applying Proposition 9.3.7.2, we deduce that the $\infty $-category $\operatorname{\mathcal{E}}$ has a final object (given by the colimit of the inclusion map $\operatorname{\mathcal{E}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{E}}$), so that the functor $\mathscr {F}$ is representable as desired (Remark 8.6.8.9).
$\square$