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:
The functor $h$ is fully faithful (Theorem 8.3.3.13).
For each object $X \in \operatorname{\mathcal{C}}$, the functor
preserves small colimits (see Example 8.4.5.2 below).
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 show that these properties characterize the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ (and the functor $h$) up to equivalence. First, let us introduce a bit of terminology.
Definition 8.4.5.1. 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 preserves small colimits.
\[ h^{X}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}\quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y) \]
Example 8.4.5.2 (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 and therefore preserves small colimits by virtue of Proposition 7.1.6.1.
\[ \operatorname{ev}_{C}: \widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {G} \mapsto \mathscr {G}(C), \]
Definition 8.4.5.3. 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
is a homotopy equivalence of Kan complexes. Then $f$ is an isomorphism.
\[ \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}( X, Y) \xrightarrow { [f] \circ } \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(X, Z) \]
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 8.4.5.4. 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 8.4.5.5. 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.
Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. In what follows, we will say that another functor $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is equivalent to $h$ if there is a diagram of $\infty $-categories
which commutes up to isomorphism, where $F$ is an equivalence of $\infty $-categories. We will be primarily interested in the special case where $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ and $h$ is a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$.
Proposition 8.4.5.6. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially small. Then $f$ is equivalent to the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ if and only if it satisfies the following conditions:
The $\infty $-category $\operatorname{\mathcal{D}}$ is locally small and admits small colimits.
The functor $f$ is fully faithful.
For each object $C \in \operatorname{\mathcal{C}}$, the image $f(C) \in \operatorname{\mathcal{D}}$ is atomic.
The collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$.
Corollary 8.4.5.7. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. The following conditions are equivalent:
There exists an essentially small $\infty $-category $\operatorname{\mathcal{C}}$ and an equivalence of $\infty $-categories $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$.
The $\infty $-category $\operatorname{\mathcal{D}}$ is locally small and admits small colimits. 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 $\operatorname{\mathcal{S}}$ admits small colimits (Corollary 7.4.5.6), the $\infty $-category $\operatorname{\mathcal{D}}$ also admits small colimits (Proposition 7.1.6.1). Moreover, Corollary 5.4.8.10 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 $X$. Then 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 8.4.5.5). We conclude by observing that each of the representable functors $h_{X}$ is a atomic object of $\operatorname{\mathcal{D}}$ (Example 8.4.5.2).
We now show that $(b)$ implies $(a)$. Assume that $\operatorname{\mathcal{D}}$ is locally small and admits small colimits. 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 5.4.8.8 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 8.4.5.6 and therefore induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$. $\square$
Following the convention of Remark 5.4.0.5, we will deduce Proposition 8.4.5.6 from the following more precise assertion.
Variant 8.4.5.8. Let $\kappa $ be an uncountable regular cardinal and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Then $f$ is equivalent to the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })$ if and only if it satisfies the following conditions:
The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits.
The functor $f$ is fully faithful.
Let $C$ be an object of $\operatorname{\mathcal{C}}$. Then the image $f(C) \in \operatorname{\mathcal{D}}$ corepresents a functor
which preserves $\kappa $-small colimits.
The collection of objects $\{ f(X) \} _{X \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$.
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), Y) \]
Warning 8.4.5.9. In the situation of Variant 8.4.5.8, it is not necessarily true that the $\infty $-category $\operatorname{\mathcal{D}}$ is locally $\kappa $-small. However, condition $(2)$ requires that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( X, Y)$ is essentially $\kappa $-small whenever the object $X$ belongs to the essential image of the functor $f$.
Proof of Proposition 8.4.5.6. By virtue of Variant 8.4.5.8, the only nontrivial point is to verify that if $\operatorname{\mathcal{C}}$ is an essentially small $\infty $-category, then the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is locally small. This follows from Corollary 5.4.8.10. $\square$
We first prove a weak version of Variant 8.4.5.8.
Lemma 8.4.5.10. Let $\kappa $ be an uncountable regular cardinal and let $\lambda $ be an uncountable cardinal of exponential cofinality $\geq \kappa $. Let $\operatorname{\mathcal{C}}$ be an essentially $\kappa $-small $\infty $-category, let $\operatorname{\mathcal{D}}$ be a locally $\lambda $-small $\infty $-category, and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which satisfies the following conditions:
The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits.
The functor $f$ is fully faithful.
Let $C$ be an object of $\operatorname{\mathcal{C}}$. Then the image $f(C) \in \operatorname{\mathcal{D}}$ corepresents a functor $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{<\lambda }$ which preserves $\kappa $-small colimits.
Then $f$ is isomorphic to the composition $F \circ h_{\bullet }$, for some fully faithful functor
which preserves $\kappa $-small colimits. Moreover, the functor $F$ is uniquely determined up to isomorphism.
Proof. It follows from Variant 8.4.3.6 that $f$ is isomorphic to a composition $F \circ h_{\bullet }$, for some functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ which preserves $\kappa $-small colimits (and that $F$ is uniquely determined up to isomorphism). To complete the proof, it will suffice to show that the functor $F$ is fully faithful. Since $\lambda $ has exponential cofinality $\geq \kappa $, the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small (Corollary 5.4.8.9). For every pair of functors $\mathscr {G}, \mathscr {G}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, 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 functors $\mathscr {G}, \mathscr {G}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the morphism $\theta _{\mathscr {G}, \mathscr {G}'}$ is a homotopy equivalence. Let us first regard the functor $\mathscr {G}'$ as fixed. Since $F$ preserves $\kappa $-small colimits, it follows from Remark 7.4.5.15 that the functor
preserves $\kappa $-small limits. By virtue of Corollary 8.4.3.12, it will suffice to show that $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence in the special case where $\mathscr {G} = h_{C}$ for some object $C \in \operatorname{\mathcal{C}}$. Let us now regard $\mathscr {G} = h_ C$ as fixed. Combining Example 8.4.5.2 with assumption $(2)$, we deduce that the functor
preserves $\kappa $-small colimits. Invoking Corollary 8.4.3.12 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). It will therefore suffice to show that the right vertical map is a homtoopy equivalence. Since $F \circ h_{\bullet }$ is isomorphic to $f$, this is equivalent to the assertion that the natural map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C') \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), f(C') )$ is a homotopy equivalence, which follows from assumption $(1)$. $\square$
Proof of Variant 8.4.5.8. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an essentially $\kappa $-small $\infty $-category, and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which satisfies the hypotheses of Variant 8.4.5.8. By virtue of Lemma 8.4.5.10, there exists a fully faithful functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ which preserves $\kappa $-small colimits such that $f$ is isomorphic to the composite functor
To complete the proof, it will suffice to show that $F$ is an equivalence of $\infty $-categories. Using Variant 8.4.4.2, we see that the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$, given on objects by the formula $G(D)(C) = \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D)$. By virtue of Corollary 6.3.3.14, it will suffice to show that the functor $G$ is conservative. This follows from our assumption that the collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$ (Remark 8.4.5.4). $\square$