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8.3.1 Dense Functors

To study the behavior of a (large) category $\operatorname{\mathcal{D}}$, it is often useful to approximate $\operatorname{\mathcal{D}}$ by well-chosen (small) subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$. The following condition guarantees that, for some purposes, passage from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ does not lose too much information:

Definition 8.3.1.1. Let $\operatorname{\mathcal{D}}$ be a (locally small) category. We say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if the functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \bullet , Y ) \]

is fully faithful.

Remark 8.3.1.2. Definition 8.3.1.1 was introduced by Isbell in [MR175954]. Beware that Isbell uses the term left adequate subcategory for what we refer to as a dense subcategory.

Example 8.3.1.3. Let $\operatorname{Cat}$ denote the ordinary category whose objects are small categories and whose morphisms are functors, and let $\operatorname{{\bf \Delta }}\subset \operatorname{Cat}$ be the simplex category. Proposition 1.2.2.1 asserts that the restricted Yoneda embedding

\[ \operatorname{Cat}\rightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set}) = \operatorname{Set_{\Delta }}\quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \]

is fully faithful, so that $\operatorname{{\bf \Delta }}$ is a dense subcategory of $\operatorname{Cat}$.

Exercise 8.3.1.4. Let $\operatorname{\mathcal{C}}$ denote the category of partially ordered sets, and let $\operatorname{{\bf \Delta }}_{\leq 1}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects $[0]$ and $[1]$. Show that $\operatorname{{\bf \Delta }}_{\leq 1}$ is a dense subcategory of $\operatorname{\mathcal{C}}$.

We now introduce an $\infty $-categorical counterpart of Definition 8.3.1.1.

Definition 8.3.1.5. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. We will say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if, for every object $X \in \operatorname{\mathcal{D}}$, the composition

\[ ( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X} )^{\triangleright } \hookrightarrow \operatorname{\mathcal{D}}_{/X}^{\triangleright } \rightarrow \operatorname{\mathcal{D}} \]

is a colimit diagram.

Remark 8.3.1.6. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if and only if the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ is left Kan extended from $\operatorname{\mathcal{C}}$.

Example 8.3.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is a dense full subcategory of itself (see Example 7.3.3.6).

In the situation of Definition 8.3.1.5, suppose that the $\infty $-category $\operatorname{\mathcal{D}}$ is locally small, and let $h_{\bullet }: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding for $\operatorname{\mathcal{D}}$ (see Definition 8.2.5.1). Composing with the restriction functor $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, we obtain a functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto h_{Y}^{\circ } \]

which we will refer to as the restricted Yoneda embedding.

Proposition 8.3.1.8. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty $-category. A full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if and only if the restricted Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful.

We will deduce Proposition 8.3.1.8 from a more general result (Proposition 8.3.1.23), which we prove at the end of this section.

Corollary 8.3.1.9. Let $\operatorname{\mathcal{D}}$ be a locally small category. Then a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense (in the sense of Definition 8.3.1.1) if and only if $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a dense subcategory of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ (in the sense of Definition 8.3.1.5).

Warning 8.3.1.10. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Consider the following conditions on a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense, in the sense of Definition 8.3.1.5.

$(2)$

Every object $X \in \operatorname{\mathcal{D}}$ can be realized as the colimit of a diagram taking values in the full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$.

$(3)$

The $\infty $-category $\operatorname{\mathcal{D}}$ is generated by $\operatorname{\mathcal{C}}$ under colimits. That is, if $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is a full subcategory which contains $\operatorname{\mathcal{C}}$ and is closed under the formation of colimits in $\operatorname{\mathcal{D}}$, then $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$.

It follows immediately from the definitions that $(1) \Rightarrow (2) \Rightarrow (3)$. Beware that neither of this implications is reversible. See Exercises 8.3.1.11 and 8.3.1.12.

Exercise 8.3.1.11. Let $\operatorname{\mathcal{D}}$ denote the category of free abelian groups, and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ denote the full subcategory spanned by object $\operatorname{\mathbf{Z}}$. Show that $\operatorname{\mathcal{C}}$ is not a dense subcategory of $\operatorname{\mathcal{D}}$. Consequently, the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \subset \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ satisfies condition $(2)$ of Warning 8.3.1.10, but does not satisfy condition $(1)$.

Exercise 8.3.1.12. Let $\operatorname{Cat}$ denote the (ordinary) category of small categories, and let $\operatorname{{\bf \Delta }}_{\leq 1} \subset \operatorname{Cat}$ denote the full subcategory spanned by the objects $[0]$ and $[1]$. Show that:

  • The full subcategory $\operatorname{{\bf \Delta }}_{\leq 1}$ generates $\operatorname{Cat}$ under colimits.

  • A small category $\operatorname{\mathcal{C}}$ realized as the colimit (in $\operatorname{Cat}$) of a diagram $\mathcal{K} \rightarrow \operatorname{{\bf \Delta }}_{\leq 1}$ if and only if the category $\operatorname{\mathcal{C}}$ is free, in the sense of Definition 1.2.6.7.

In particular, the inclusion $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\leq 1} ) \subset \operatorname{N}_{\bullet }( \operatorname{Cat})$ satisfies condition $(3)$ of Warning 8.3.1.10, but does not satisfy condition $(2)$.

Warning 8.3.1.13 (Failure of Transitivity). Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{E}}$ be full subcategories. Suppose that $\operatorname{\mathcal{C}}$ is a dense subcategory of $\operatorname{\mathcal{E}}$. Then $\operatorname{\mathcal{C}}$ is also a dense subcategory of $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{D}}$ is a dense subcategory of $\operatorname{\mathcal{E}}$ (see Corollary 7.3.7.8). Beware that the converse is false (Example 8.3.1.14).

Example 8.3.1.14. Let $\operatorname{Cat}$ denote the (ordinary) category of small categories. Then the simplex category $\operatorname{{\bf \Delta }}$ is a dense full subcategory of $\operatorname{Cat}$ (Example 8.3.1.3), and $\operatorname{{\bf \Delta }}_{\leq 1}$ is a dense full subcategory of $\operatorname{{\bf \Delta }}$ (Exercise 8.3.1.4). However, $\operatorname{{\bf \Delta }}_{\leq 1}$ is not a dense full subcategory of $\operatorname{Cat}$ (Exercise 8.3.1.12).

For some applications, it will be useful to consider the following generalization of Definition 8.3.1.5.

Definition 8.3.1.15. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. We say that $F$ is dense if the identity transformation $\operatorname{id}_{F}: F \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}} \circ F$ exhibits the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as a left Kan extension of $F$ along $F$ (see Variant 7.3.1.5).

Example 8.3.1.16. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense (in the sense of Definition 8.3.1.5) if and only if the inclusion functor $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is dense (in the sense of Definition 8.3.1.15). See Proposition 7.3.2.5.

Remark 8.3.1.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ denote the relative join of Construction 5.2.3.1. Then $F$ is dense if and only if the projection map $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}$. See Proposition 7.3.2.10.

Remark 8.3.1.18 (Homotopy Invariance). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $C$ be a simplicial set, and let $F,F' \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be diagrams which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then $F$ is dense if and only if $F'$ is dense. This follows by combining Remarks 7.3.1.10 and 7.3.1.11.

Remark 8.3.1.19 (Change of Source). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $G: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ be a categorical equivalence of simplicial sets. Then $F$ is dense if and only if $F \circ G$ is dense. See Proposition 7.3.1.14.

Remark 8.3.1.20 (Change of Target). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then:

  • If $G$ is fully faithful and $G \circ F$ is dense, then $F$ is dense.

  • If $G$ is an equivalence of $\infty $-categories and $F$ is dense, then $G \circ F$ is dense.

See Remark 7.3.1.13.

Remark 8.3.1.21. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $f$ is dense if and only if, for every object $Y \in \operatorname{\mathcal{D}}$, the composite map

\[ (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{/Y}^{\triangleright } \rightarrow \operatorname{\mathcal{D}} \]

is a colimit diagram in $\operatorname{\mathcal{D}}$.

Remark 8.3.1.22. Let $\kappa $ be an uncountable regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and that $\operatorname{\mathcal{D}}$ is a locally $\kappa $-small $\infty $-category. Then, for each object $X \in \operatorname{\mathcal{D}}$, the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ is also essentially $\kappa $-small (Corollary 5.4.8.11). Let $\lambda $ be an infinite cardinal satisfying $\mathrm{ecf}(\lambda ) \geq \kappa $ (see Definition 5.4.3.16). Then $F$ is dense if and only if, for every representable functor $h_{Y}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the identity transformation $\operatorname{id}: h_{Y} \circ F^{\operatorname{op}} \rightarrow h_{Y} \circ F^{\operatorname{op}}$ exhibits the functor $h_{Y}$ as a right Kan extension of $h_{Y} \circ F^{\operatorname{op}}$ along $F^{\operatorname{op}}$. This follows by combining Remark 8.3.1.21 with Proposition 7.4.5.13 (together with Remark 7.4.5.15).

Proposition 8.3.1.23. 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 8.2.5.1). 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 5.4.3.19). 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 8.3.1.22, 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 8.2.5.5), 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 7.4.1.12). Using Proposition 7.6.7.12, 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

8.21
\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 8.2.1.3, 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 (8.21), 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. $\square$

Proof of Proposition 8.3.1.8. Let $\operatorname{\mathcal{D}}$ be a locally small $\infty $-category and let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ be a full subcategory. By virtue of Example 8.3.1.16, it will suffice to show that the inclusion functor $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is dense if and only if the restricted Yoneda embedding $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is fully faithful. Following the convention of Remark 5.4.0.5, this is a special case of Proposition 8.3.1.23. $\square$

Proposition 8.3.1.24. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, and let $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ be a full subcategory which contains the image of $F$. If $F$ is dense, then the subcategory $\operatorname{\mathcal{D}}_0$ is dense.

Proof. Using Proposition 4.1.3.2, we can factor $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{C}}' \xrightarrow {F''} \operatorname{\mathcal{D}}$, where $F'$ is inner anodyne and $F''$ is an inner fibration. Using Remark 8.3.1.19, we see that the functor $F''$ is dense. Replacing $F$ by $F'$, we can reduce to proving Proposition 8.3.1.24 in the special case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.

Let $\gamma $ denote the identity map $\operatorname{id}_{F}$, which we regard as a natural transformation from $F$ to $\operatorname{id}_{\operatorname{\mathcal{D}}} \circ F$. Our assumption that $F$ is dense guarantees that $\gamma $ exhibits $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as a left Kan extension of $F$ along $F$. To avoid confusion, let us write $F_0$ to denote the functor $F$, regarded as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}_0$. Let $\iota : \operatorname{\mathcal{D}}_0 \hookrightarrow \operatorname{\mathcal{D}}$ denote the inclusion map, so that $F = \iota \circ F_0$. We can therefore also regard the identity map $\operatorname{id}_{F}$ as a natural transformation $\alpha : F \rightarrow \iota \circ F_0$. Our assumption that $F$ is dense also guarantees that $\alpha $ exhibits $\iota $ as a left Kan extension of $F$ along $F_0$. Note that $\gamma = \operatorname{id}_{F}$ is a composition of $\alpha = \operatorname{id}_{F}$ with $\beta |_{ \operatorname{\mathcal{C}}}$, where $\beta = \operatorname{id}_{ \iota }$ is the identity transformation from $\iota $ to itself. Invoking the transitivity of Kan extensions (Proposition 7.3.7.18), we deduce that $\beta $ exhibits the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as a left Kan extension of $\iota $ along itself: that is, the functor $\iota $ is dense. Applying Example 8.3.1.16, we conclude that $\operatorname{\mathcal{D}}_0$ is a dense subcategory of $\operatorname{\mathcal{D}}$. $\square$

Remark 8.3.1.25. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a dense diagram. Choose another diagram $q: K \rightarrow \operatorname{\mathcal{D}}$ and set $\widetilde{\operatorname{\mathcal{D}}} = \operatorname{\mathcal{D}}_{/q}$, so that the projection map $\pi : \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ is a right fibration. Then the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is dense. To prove this, it will suffice to show that for every object $Y \in \widetilde{\operatorname{\mathcal{D}}}$, the induced map

\[ \theta : (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}_{/Y} )^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{D}}}_{/Y}^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{D}}} \]

is a colimit diagram in $\widetilde{\operatorname{\mathcal{D}}}$ (Remark 8.3.1.21). By virtue of Proposition 7.1.3.19, this is equivalent to the requirement that $\pi \circ \theta $ is a colimit diagram in $\operatorname{\mathcal{D}}$. Unwinding the definitions, we see that $\pi \circ \theta $ is given by the composition

\[ (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}_{/Y} )^{\triangleright } \rightarrow (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \pi (Y) })^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }^{\triangleright } \rightarrow \operatorname{\mathcal{D}}. \]

Since $\pi $ is a right fibration, the map $\widetilde{\operatorname{\mathcal{D}}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }$ is a trivial Kan fibration (Proposition 4.3.7.12). Using Corollary 7.2.2.2, we are reduced to showing that the map $(\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \pi (Y) })^{\triangleright } \rightarrow \operatorname{\mathcal{D}}_{ / \pi (Y) }^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram, which follows from our assumption that $F$ is dense (Remark 8.3.1.21).