Theorem 8.4.2.1. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding (Definition 8.3.3.9). Then $h_{\bullet }$ is a dense functor.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8.4.2 Density of Yoneda Embeddings
Our goal in this section is to prove the following result, which supplies an important source of examples of dense functors:
Since the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful, Theorem 8.4.2.1 can be reformulated as follows:
Corollary 8.4.2.2. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category and let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ denote the full subcategory spanned by the representable functors. Then $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is a dense subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
Proof. By virtue of Example 8.4.1.16, it will suffice to show that the inclusion map $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \hookrightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is a dense functor. Since the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is an equivalence of $\infty $-categories (Theorem 8.3.3.13), this is equivalent to the assertion that $h_{\bullet }$ is a dense functor from $\operatorname{\mathcal{C}}$ to $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (Remark 8.4.1.18), which follows from Theorem 8.4.2.1. $\square$
Example 8.4.2.3. Let $\operatorname{\mathcal{S}}_{ \mathrm{cont} }$ denote the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the contractible Kan complexes. Then $\operatorname{\mathcal{S}}_{ \mathrm{cont} }$ is a dense subcategory of $\operatorname{\mathcal{S}}$. This follows by applying Corollary 8.4.2.2 in the special case $\operatorname{\mathcal{C}}= \Delta ^0$. Moreover, the same assertion holds if we replace $\operatorname{\mathcal{S}}_{ \mathrm{cont} }$ by any nonempty subcategory of itself; for example, the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the standard $0$-simplex $\Delta ^0$.
By virtue of the convention of Remark 4.7.0.5, Theorem 8.4.2.1 can be regarded as a special case of the following:
Variant 8.4.2.4. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. Then the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is a dense functor.
We will deduce Variant 8.4.2.4 from a more precise result. Recall that, if $X$ is an object of a (locally small) $\infty $-category $\operatorname{\mathcal{C}}$, then the representable functor $h_{X} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ corepresents the evaluation functor $\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ (Remark 8.3.1.5). That is, for every functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$, there is a canonical homotopy equivalence
\[ \mathscr {F}(X) \xrightarrow {\sim } \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) }( h_{X}, \mathscr {F} ), \]
which depends functorially on $\mathscr {F}$. The following result guarantees that this homotopy equivalence can also be chosen to depend functorially on $X$:
Proposition 8.4.2.5. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. Then the profunctor is corepresentable by the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
\[ \operatorname{ev}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad (X, \mathscr {F}) \mapsto \mathscr {F}(X) \]
Proof. Let $\operatorname{Tw}(\operatorname{\mathcal{C}})$ denote the twisted arrow $\infty $-category of $\operatorname{\mathcal{C}}$, let $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ be the left fibration of Proposition 8.1.1.11, and let $\underline{ \Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) }$ denote the constant functor $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. Let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ denote the composition $\operatorname{ev}\circ (\operatorname{id}\times h_{\bullet })$, so that $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. We can therefore choose a natural transformation
\[ \alpha : \underline{ \Delta ^0 }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H} \circ \lambda = \operatorname{ev}\circ ( \operatorname{id}\times h_{\bullet } ) \circ \lambda \]
which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, in the sense of Definition 8.3.5.1. By virtue of Proposition 8.3.4.15, it will suffice to show that the natural transformation $\alpha $ also exhibits the profunctor $\operatorname{ev}$ as corepresented by the functor $h_{\bullet }$, in the sense of Variant 8.3.4.16. Fix an object $X \in \operatorname{\mathcal{C}}$, so that $\alpha $ carries the object $\operatorname{id}_{X} \in \operatorname{Tw}(\operatorname{\mathcal{C}})$ to a vertex $\eta \in \mathscr {H}(X,X) = \operatorname{ev}( X, h_ X)$. We wish to show that $\eta $ exhibits evaluation functor $\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ as corepresented by $h_{X}$. This follows from Proposition 8.3.1.3, since $\eta $ exhibits the functor $h_{X}$ as represented by $X$. $\square$
Example 8.4.2.6. Let $\operatorname{\mathcal{C}}$ be a locally small category. Then the evaluation profunctor is corepresentable by the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
\[ \operatorname{ev}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad (X, \mathscr {F}) \mapsto \mathscr {F}(X) \]
Proof of Variant 8.4.2.4. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small. We wish to show that the covariant Yoneda embedding $h^{\operatorname{\mathcal{C}}}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is dense. Choose a cardinal $\lambda \geq \kappa $ for which the $\infty $-category $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small, and let $h^{\operatorname{\mathcal{D}}}_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{D}}$. By virtue of Proposition 8.4.1.22, it will suffice to show that the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { h^{\operatorname{\mathcal{D}}}_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \xrightarrow { \circ h^{\operatorname{\mathcal{C}}}_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \]
is fully faithful. Applying Proposition 8.4.2.5, we see that this functor is isomorphic to the inclusion of $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ as a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$. $\square$
Using Proposition 8.4.2.5, we can also give an alternative characterization of the covariant transport representation associated to a left fibration of $\infty $-categories.
Corollary 8.4.2.7. Let $\operatorname{\mathcal{C}}$ be a locally $\kappa $-small $\infty $-category, let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a right fibration, and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor. The following conditions are equivalent:
The functor $\mathscr {F}$ is a covariant transport representation for the left fibration $U^{\operatorname{op}}: \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.
There exists a categorical pullback square
where $h_{\bullet }$ is a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$.
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [d]^{U} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })_{ / \mathscr {F} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h_{\bullet }} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa } ), } \]
Proof. Choose a cardinal $\lambda \geq \kappa $ such that $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small, and let $H: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ be a functor represented by $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. It follows from Proposition 5.6.6.21 that $H$ is a covariant transport representation for the left fibration $(\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F} })^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \kappa } )^{\operatorname{op}}$. Consequently, if condition $(2)$ is satisfied, then $H \circ h_{\bullet }^{\operatorname{op}}$ is a covariant transport representation for the left fibration $U^{\operatorname{op}}$. Proposition 8.4.2.5 implies that $H \circ h_{\bullet }^{\operatorname{op}}$ is isomorphic to $\mathscr {F}$. This proves the implication $(2) \Rightarrow (1)$, and the reverse implication follows from the fact that the equivalence class of a left fibration is determined by its covariant transport representation (Corollary 5.6.0.6). $\square$
Corollary 8.4.2.8. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Suppose we are given a diagram $f: K \rightarrow \operatorname{\mathcal{C}}$. Assume that $K$ is $\mathrm{ecf}(\kappa )$-small, so that $F = h_{\bullet } \circ f$ admits a limit $\widehat{C} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })$. Then $\widehat{C}$ is a covariant transport representation for the left fibration $( \operatorname{\mathcal{C}}_{ / f } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.
Proof. Since the functor $h_{\bullet }$ is fully faithful, it induces a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/f} \ar [d]^{U} \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })_{ / F} \ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r]^-{h_{\bullet }} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa } ). } \]
Our assumption that $\widehat{C}$ is a limit of $F$ guarantees that $U$ is equivalent to the right fibration $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \widehat{C} } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. The desired result now follows from Corollary 8.4.2.7. $\square$