$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.5.5.19. Let $\kappa $ be a small regular cardinal. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-presentable if and only if it satisfies the following conditions:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small.
- $(b)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete.
- $(c_{\kappa })$
There exists a small collection of $\kappa $-compact objects $\{ C_{i} \} _{i \in I}$ which generates $\operatorname{\mathcal{C}}$ under small colimits.
Proof.
If $\operatorname{\mathcal{C}}$ is $\kappa $-presentable, then assertions $(b)$ and $(c_{\kappa })$ are immediate and $(a)$ follows from Remark 9.4.6.14. Conversely, suppose that conditions $(a)$, $(b)$, and $(c_{\kappa })$ are satisfied. Let $\{ C_ i \} _{i \in I}$ be a small collection of $\kappa $-compact objects which generates $\operatorname{\mathcal{C}}$ under small colimits, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the objects $C_ i$. It follows from assumption $(a)$ that $\operatorname{\mathcal{C}}_0$ is essentially small. Enlarging $\operatorname{\mathcal{C}}_0$ if necessary, we can assume that it is closed under $\kappa $-small colimits (Proposition 9.2.5.24), so that $\operatorname{\mathcal{C}}_0$ is $\kappa $-cocomplete and the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is $\kappa $-cocontinuous. Applying Lemma 9.5.5.17, we conclude that $\iota $ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}_0$, and is therefore $\kappa $-presentable.
$\square$