$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.5.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is 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.
- $(c)$
There exists a small collection of objects $\{ C_ i \} _{i \in I}$ which generates $\operatorname{\mathcal{C}}$ under small colimits in the following sense: if $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a full subcategory which contains each $C_ i$ and is closed under small colimits, then $\operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}$.
- $(d)$
For every object $C \in \operatorname{\mathcal{C}}$, there exists a small regular cardinal $\kappa $ such that $C$ is $\kappa $-compact.
Proof of Proposition 9.5.5.15.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which satisfies the hypotheses of Proposition 9.5.5.15; we wish to show that $\operatorname{\mathcal{C}}$ is presentable (the converse follows from Remark 9.4.6.14 and 9.4.6.8). By assumption, $\operatorname{\mathcal{C}}$ is generated under small colimits by a small collection of objects $\{ C_ i \} _{i \in I}$. Moreover, each $C_ i$ is $\kappa _ i$-compact for some small regular cardinal $\kappa _ i$. Choose a small regular cardinal $\kappa $ which is an upper bound for the collection $\{ \kappa _ i \} _{i \in I}$. Then each of the objects $C_ i$ is $\kappa $-compact. Applying Proposition 9.5.5.19, we conclude that $\operatorname{\mathcal{C}}$ is $\kappa $-presentable. In particular, it is presentable.
$\square$