# Kerodon

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Corollary 8.4.6.7. Let $\operatorname{\mathcal{D}}$ be an $\infty$-category. The following conditions are equivalent:

$(a)$

There exists an essentially small $\infty$-category $\operatorname{\mathcal{C}}$ and a functor $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a cocompletion of $\operatorname{\mathcal{C}}$.

$(b)$

The $\infty$-category $\operatorname{\mathcal{D}}$ is locally small and cocomplete. Moreover, it contains a small collection of atomic objects $\{ X_ i \} _{i \in I}$ which is weakly dense.

Proof. We first show that $(a)$ implies $(b)$. Without loss of generality, we may assume that $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, where $\operatorname{\mathcal{C}}$ is a small $\infty$-category. Since the $\infty$-category $\operatorname{\mathcal{S}}$ is cocomplete (Corollary 7.4.5.6), the $\infty$-category $\operatorname{\mathcal{D}}$ is also cocomplete (Remark 7.6.7.5). Corollary 4.7.8.9 guarantees that $\operatorname{\mathcal{D}}$ is locally small. For each object $X \in \operatorname{\mathcal{C}}$, let $h_{X} \in \operatorname{\mathcal{D}}$ be a functor represented by $X$. The collection of objects $\{ h_ X \} _{X \in \operatorname{\mathcal{C}}}$ span a full subcategory of $\operatorname{\mathcal{D}}$ which is dense (Corollary 8.4.2.2), and therefore weakly dense (Example 8.4.6.5). We conclude by observing that each of the representable functors $h_{X}$ is a atomic object of $\operatorname{\mathcal{D}}$ (Example 8.4.6.2).

We now show that $(b)$ implies $(a)$. Assume that $\operatorname{\mathcal{D}}$ is locally small and cocomplete. Let $\{ X_ i \} _{i \in I}$ be a small collection of atomic objects of $\operatorname{\mathcal{D}}$, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the full subcategory that they span. It follows from Proposition 4.7.8.7 that the $\infty$-category $\operatorname{\mathcal{D}}_0$ is essentially small. If $\operatorname{\mathcal{D}}_0$ is weakly dense in $\operatorname{\mathcal{D}}$, then the inclusion map $\operatorname{\mathcal{D}}_0 \hookrightarrow \operatorname{\mathcal{D}}$ satisfies the hypotheses of Proposition 8.4.6.6, and therefore exhibits $\operatorname{\mathcal{D}}$ as a cocompletion of $\operatorname{\mathcal{D}}_0$. $\square$