Proposition 8.4.1.22. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is locally $\lambda $-small, and let
\[ h_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \quad \quad Y \mapsto h_{Y} \]
be a covariant Yoneda embedding for $\operatorname{\mathcal{D}}$ (Definition 8.3.3.9). If $\operatorname{\mathcal{C}}$ is a simplicial set, then a diagram $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is dense if and only if the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) \xrightarrow { \circ F^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) \]
is fully faithful.
Proof.
Choose an uncountable cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and $\operatorname{\mathcal{D}}$ is locally $\kappa $-small. Enlarging $\lambda $ if necessary, we may assume that the exponential cofinality of $\lambda $ is $\geq \kappa $ (see Remark 4.7.3.19). For each object $Y \in \operatorname{\mathcal{D}}$, let $h_{Y}^{\circ }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ denote the composite functor $h_{Y} \circ F^{\operatorname{op}}$, given on objects by the construction $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), Y )$. By virtue of Remark 8.4.1.21, it will suffice to show that the following conditions are equivalent:
- $(1_ Y)$
The identity transformation $\operatorname{id}: h_{Y} \circ F^{\operatorname{op}} \rightarrow h_{Y}$ exhibits $h_{Y}$ as a right Kan extension of $h_{Y}^{\circ }$ along the functor $F^{\operatorname{op}}$.
- $(2_ Y)$
For each object $X \in \operatorname{\mathcal{C}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) }( h_{X}, h_{Y} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) }( h_{X}^{\circ }, h_{Y}^{\circ } ) \]
is a homotopy equivalence of Kan complexes.
Since the covariant Yoneda embedding $X \mapsto h_{X}$ is fully faithful (Theorem 8.3.3.13), we can reformulate $(2_ Y)$ as follows:
- $(2'_ Y)$
The restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\lambda }) }( h_{X}, h_{Y} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\lambda }) }( h_{X}^{\circ }, h_{Y}^{\circ } ) \]
is a homotopy equivalence of Kan complexes.
The inequality $\kappa \leq \mathrm{ecf}(\lambda )$ guarantees that the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa $-small limits (Corollary 7.4.1.13). Using Proposition 7.6.7.13, we can choose a functor $\mathscr {G}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{<\lambda }$ and a natural transformation $\alpha : \mathscr {G} \circ F^{\operatorname{op}} \rightarrow h_{Y}^{\circ }$ which exhibits $\mathscr {G}$ as a right Kan extension of $h_{Y}^{\circ }$ along the functor $F^{\operatorname{op}}$. Invoking the universal mapping property of $\mathscr {G}$ (Proposition 7.3.6.1), we see that there exists a natural transformation $\beta : h_{Y} \rightarrow \mathscr {G}$ and a commutative diagram
8.60
\begin{equation} \begin{gathered}\label{equation:density-via-Yoneda2} \xymatrix@R =50pt@C=50pt{ & \mathscr {G} \circ F^{\operatorname{op}} \ar [dr]^{\alpha } & \\ h_{Y} \circ F^{\operatorname{op}} \ar [rr]^{ \operatorname{id}} \ar [ur]^{\beta } & & h_{Y}^{\circ } } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$. Using Remark 7.3.1.12, we see that condition $(1_ Y)$ is satisfied if and only if the natural transformation $\beta $ is an isomorphism: that is, it induces a homotopy equivalence of Kan complexes $\beta _{X}: h_{Y}(X) \rightarrow \mathscr {G}(X)$ for each object $X \in \operatorname{\mathcal{D}}$ (Theorem 4.4.4.4). Combining this observation with Proposition 8.3.1.3, we can reformulate $(1_ Y)$ as follows:
- $(1'_ Y)$
For each object $X \in \operatorname{\mathcal{C}}$, precomposition with $\beta $ induces a homotopy equivalence
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\lambda }) }( h_{X}, h_{Y} ) \xrightarrow {\circ [\beta ]} \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) }( h_{X}, \mathscr {G} ). \]
Using the commutativity of (8.60), we see that the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=30pt{ & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) }( h_{X}, \mathscr {G} ) \ar [dr] & \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) }( h_{X}, h_{Y} ) \ar [rr] \ar [ur]^{ \circ [\beta ] } & & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) }( h_{X}^{\circ }, h_{Y}^{\circ } ) } \]
commutes up to homotopy, where the diagonal map on the right is the homotopy equivalence of Proposition 7.3.6.1. It follows that conditions $(1'_ Y)$ and $(2'_ Y)$ are equivalent.
$\square$