# Kerodon

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### 8.4.6 Recognition of Cocompletions

Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty$-category, let $\widehat{\operatorname{\mathcal{C}}}$ denote the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Then:

$(1)$

The functor $h$ is fully faithful (Theorem 8.3.3.13).

$(2)$

For each object $X \in \operatorname{\mathcal{C}}$, the functor

$\operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( h(X), \bullet ): \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}$

preserves small colimits (see Example 8.4.6.2 below).

$(3)$

The $\infty$-category $\widehat{\operatorname{\mathcal{C}}}$ is generated (under small colimits) by the essential image of $h$ (in fact, the functor $h$ is dense: see Theorem 8.4.2.1).

Our goal in this section is to prove the converse: if $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ is any functor which satisfies conditions $(1)$ through $(3)$, then $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ (Proposition 8.4.6.6) and is therefore equivalent to the covariant Yoneda embedding of $\operatorname{\mathcal{C}}$. First, let us introduce a bit of terminology.

Definition 8.4.6.1. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty$-category which admits small colimits. We say that an object $X \in \operatorname{\mathcal{D}}$ is atomic if the corepresentable functor

$h^{X}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}\quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y)$

preserves small colimits.

Example 8.4.6.2 (Representable Functors are Atomic). Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty$-category. Then every representable functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is atomic when regarded as an object of the $\infty$-category $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. To see this, suppose that $\mathscr {F}$ is representable by an object $C \in \operatorname{\mathcal{C}}$. Using Remark 8.3.1.5, we see that $\mathscr {F}$ corepresents the evaluation functor

$\operatorname{ev}_{C}: \widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {G} \mapsto \mathscr {G}(C),$

and therefore preserves small colimits by virtue of Proposition 7.1.6.1.

Definition 8.4.6.3. Let $\widehat{\operatorname{\mathcal{C}}}$ be an $\infty$-category. We say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \widehat{\operatorname{\mathcal{C}}}$ is weakly dense if the following condition is satisfied:

• Let $f: Y \rightarrow Z$ be a morphism of $\widehat{\operatorname{\mathcal{C}}}$ such that, for every object $X \in \operatorname{\mathcal{C}}$, the induced map

$\operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}( X, Y) \xrightarrow { [f] \circ } \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(X, Z)$

is a homotopy equivalence of Kan complexes. Then $f$ is an isomorphism.

We say that a collection of objects $\{ X_ i \} _{i \in I}$ of $\widehat{\operatorname{\mathcal{C}}}$ is weakly dense if it spans a weakly dense full subcategory of $\widehat{\operatorname{\mathcal{C}}}$.

Remark 8.4.6.4. Let $\widehat{\operatorname{\mathcal{C}}}$ be a locally small $\infty$-category. A full subcategory $\operatorname{\mathcal{C}}\subseteq \widehat{\operatorname{\mathcal{C}}}$ is weakly dense if and only if the restricted Yoneda embedding $\widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is conservative.

Example 8.4.6.5. Let $\widehat{\operatorname{\mathcal{C}}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}\subseteq \widehat{\operatorname{\mathcal{C}}}$ be a full subcategory which generates $\widehat{\operatorname{\mathcal{C}}}$ under colimits (see Warning 8.4.1.10). Then the full subcategory $\operatorname{\mathcal{C}}$ is weakly dense. In particular, every dense subcategory of $\widehat{\operatorname{\mathcal{C}}}$ is weakly dense.

Proposition 8.4.6.6. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, where $\operatorname{\mathcal{C}}$ is essentially small. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.4.0.1) if and only if the following conditions are satisfied:

$(0)$

The $\infty$-category $\operatorname{\mathcal{D}}$ is locally small and cocomplete.

$(1)$

The functor $f$ is fully faithful.

$(2)$

For each object $C \in \operatorname{\mathcal{C}}$, the image $f(C) \in \operatorname{\mathcal{D}}$ is atomic.

$(3)$

The collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$.

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 $\operatorname{\mathcal{D}}_0$. $\square$

Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, where $\operatorname{\mathcal{C}}$ is essentially small and $\operatorname{\mathcal{D}}$ is cocomplete. Using Theorem 8.4.0.3, we see that $f$ admits an essentially unique factorization as a composition

$\operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {F} \operatorname{\mathcal{D}},$

where the functor $F$ preserves small colimits. As a first step towards the proof of Proposition 8.4.6.6, we study conditions which guarantee that $F$ is fully faithful. With an eye towards future applications, we consider a slightly more general situation.

Lemma 8.4.6.8. Let $\mathbb {K}$ be a collection of simplicial sets, let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty$-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$, and let $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ a functor which satisfies the following conditions:

$(0)$

The $\infty$-category $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete and the functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.

$(1)$

The functor $f = F \circ h$ is fully faithful.

$(2)$

Let $\kappa$ be an uncountable regular cardinal such that $\operatorname{\mathcal{D}}$ is locally $\kappa$-small and each $K \in \mathbf{K}$ is essentially $\kappa$-small. Then, for each $C \in \operatorname{\mathcal{C}}$, the corepresentable functor

$\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D )$

preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.

Then $F$ is fully faithful.

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

$\theta _{\mathscr {G}, \mathscr {G}'}: \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \mathscr {G}, \mathscr {G}' ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F( \mathscr {G}), F( \mathscr {G}' ) ).$

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

$\theta : \widehat{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}^{< \kappa } ).$

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 7.4.5.17). 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

$\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}^{< \lambda } ) \quad \quad \mathscr {G}' \mapsto \theta _{ \mathscr {G}, \mathscr {G}'}$

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

$\xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C') \ar [dl] \ar [dr] & \\ \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \mathscr {G}, \mathscr {G}' ) \ar [rr]^{ \theta _{\mathscr {G}, \mathscr {G}'} } & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(\mathscr {G}), F( \mathscr {G}') ), }$

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

$\operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {F} \operatorname{\mathcal{D}},$

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 6.2.2.19, 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. $\square$

Proposition 8.4.6.6 has a counterpart for more general cocompletions:

Variant 8.4.6.9. Let $\mathbb {K}$ be a collection of simplicial sets and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.4.5.1) if and only if the following conditions are satisfied:

$(0)$

The $\infty$-category $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete.

$(1)$

The functor $f$ is fully faithful.

$(2)$

Let $\kappa$ be an uncountable regular cardinal such that $\operatorname{\mathcal{D}}$ is locally $\kappa$-small and each $K \in \mathbf{K}$ is essentially $\kappa$-small. Then, for each $C \in \operatorname{\mathcal{C}}$, the corepresentable functor

$\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D )$

preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.

$(3)$

The $\infty$-category $\operatorname{\mathcal{D}}$ is generated by the objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ under the formation of $K$-indexed colimits for $K \in \mathbb {K}$.

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 $(3)$. $\square$

Corollary 8.4.6.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories which exhibits $\operatorname{\mathcal{D}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be the smallest full subcategory which contains the essential image of $f_0: f|_{\operatorname{\mathcal{C}}_0}$ and is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$. Then the functor $f_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}_0$ exhibits $\operatorname{\mathcal{D}}_0$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}_0$.