Kerodon

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 7.4.3.3), the $\infty $-category $\operatorname{\mathcal{D}}$ is also cocomplete (Remark 7.6.6.5). 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 $\operatorname{\mathcal{D}}_0$. $\square$

Proof. By virtue of Proposition 8.4.5.7, we may assume without loss of generality that $\widehat{\operatorname{\mathcal{C}}}$ is the smallest replete full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ which contains all representable functors and is closed under the formation of $K$-indexed colimits for $K \in \mathbb {K}$ (see Construction 8.4.5.5). For every pair of objects $\mathscr {G}, \mathscr {G}' \in \widehat{\operatorname{\mathcal{C}}}$, the functor $F$ induces a morphism of Kan complexes

By virtue of Corollary 8.3.5.8 (and Remark 8.3.5.9), we can promote the construction $(\mathscr {G}, \mathscr {G}') \mapsto \theta _{ \mathscr {G}, \mathscr {G}' }$ to a functor of $\infty $-categories

We wish to show that, for every pair of objects $\mathscr {G}, \mathscr {G}' \in \widehat{\operatorname{\mathcal{C}}}$, the morphism $\theta _{\mathscr {G}, \mathscr {G}'}$ is a homotopy equivalence. Let us first regard the functor $\mathscr {G}'$ as fixed. Let $\widehat{\operatorname{\mathcal{C}}}'$ denote the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by those objects $\mathscr {G}$ for which $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence. For each $K \in \mathbb {K}$, our assumption that $F$ preserves $K$-indexed colimits guarantees that the functor $\mathscr {G} \mapsto \theta _{ \mathscr {G}, \mathscr {G}' }$ preserves $K^{\operatorname{op}}$-indexed limits (Proposition 7.4.1.18). Consequently, the full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under the formation of $K$-indexed colimits. It will therefore suffice to show that $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence in the special case where $\mathscr {G} = h_{C}$ is the functor represented by some object $C \in \operatorname{\mathcal{C}}$.

Let us now regard $\mathscr {G} = h_ C$ as fixed. Combining Example 8.4.6.2 with assumption $(2)$, we deduce that the functor

preserves $K$-indexed colimits, for each $K \in \mathbb {K}$. Invoking Corollary 8.4.3.9 again, we are reduced to proving that $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence in the special case where $\mathscr {G}' = h_{C'}$ for some object $C' \in \operatorname{\mathcal{C}}$. In this case, we have a commutative diagram of Kan complexes

where the left vertical map is a homotopy equivalence by virtue of Yoneda's lemma (Theorem 8.3.3.13) and the right vertical map is a homotopy equivalence by virtue of assumption $(1)$. It follows that lower horizontal map is also a homotopy equivalence. $\square$

Proof of Proposition 8.4.6.6. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category, let $\operatorname{\mathcal{D}}$ be a cocomplete $\infty $-category, and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. By virtue of Theorem 8.4.0.3, the functor $f$ admits an essentially unique factorization as a composition

where $h_{\bullet }$ is the covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ and the functor $F$ preserves small colimits. Moreover, $f$ exhibits $\operatorname{\mathcal{D}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ if and only if the functor $F$ is an equivalence of $\infty $-categories. If this condition is satisfied, then $\operatorname{\mathcal{D}}$ is locally small (Corollary 4.7.8.9), the functor $f$ is fully faithful (Theorem 8.3.3.13), the essential image of $f$ consists of atomic objects of $\operatorname{\mathcal{D}}$ (Example 8.4.6.2). We may therefore assume without loss of generality that $f$ satisfies conditions $(0)$, $(1)$, and $(2)$ of Proposition 8.4.6.6, so that $F$ is fully faithful (Lemma 8.4.6.8). Using Proposition 8.4.4.1, we see that the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, given on objects by the formula $G(D)(C) = \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D)$. By virtue of Corollary 6.2.2.21, the functor $F$ is an equivalence if and only if the functor $G$ is conservative: that is, if and only if the collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense. $\square$

Proof. Let $\kappa $ be as in $(2)$, and let $\widehat{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the smallest replete full subcategory which contains all representable functors and is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$ (see Construction 8.4.5.5). By virtue of Proposition 8.4.5.7, the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Assume that $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-complete, so that $f$ factors (up to isomorphism) as a composition $\operatorname{\mathcal{C}}\xrightarrow {h_{\bullet }} \widehat{\operatorname{\mathcal{C}}} \xrightarrow {F} \operatorname{\mathcal{D}}$, where the functor $F$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. To complete the proof, it will suffice to show that if $f$ satisfies conditions $(1)$, $(2)$, and $(3)$, then the functor $F$ is an equivalence of $\infty $-categories (the reverse implication follows from Theorem 8.3.3.13 and Example 8.4.6.2). Applying Lemma 8.4.6.8, we see that the functor $F$ is fully faithful and therefore restricts to an equivalence of $\widehat{\operatorname{\mathcal{C}}}$ with a replete full subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$. For each $K \in \mathbb {K}$, our assumption that $F$ preserves $K$-indexed colimits guarantees that the subcategory $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is closed under the formation of $K$-indexed colimits. Since $\operatorname{\mathcal{D}}_0$ contains the essential image of the functor $f$, the equality $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$ follows from assumption $(3)$. $\square$