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7.4.3 Relative Kan Extensions

For many applications, it will be convenient to work with a generalization of Definition 7.4.2.1. In what follows, we assume that the reader is familiar with the theory of relative (co)limit diagrams introduced in ยง7.1.7.

Definition 7.4.3.1 (Relative Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. For each object $C \in \operatorname{\mathcal{C}}$, we will say that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$ if the composite map

\[ (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \xrightarrow {c} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]

is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. We say that $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$ if the composite map

\[ (\operatorname{\mathcal{C}}^{0}_{C/})^{\triangleleft } \hookrightarrow (\operatorname{\mathcal{C}}_{C/})^{\triangleleft } \xrightarrow {c'} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]

is a $U$-limit diagram in $\operatorname{\mathcal{D}}$. Here $c$ and $c'$ denote the slice and coslice contraction morphisms of Construction 4.3.5.12. We say that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$. We say that $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$.

Remark 7.4.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ (in the sense of Definition 7.4.2.1) if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ (in the sense of Definition 7.4.3.1), where $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ is the projection map. Similarly, $F$ is right Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$. See Example 7.1.7.3.

Remark 7.4.3.3. In the situation of Definition 7.4.3.1, the morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is $U^{\operatorname{op}}$-left Kan extended from $(\operatorname{\mathcal{C}}^0)^{\operatorname{op}}$.

Remark 7.4.3.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. Suppose that $X$ belongs to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, and that the morphism $e$ exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$ (see Definition 6.2.2.1). Then $e$ is final when viewed as an object the $\infty $-category $\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$. It follows that the functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $Y$ if and only if $F(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$ (see Corollary 7.2.3.7). More generally, if $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is an inner fibration of $\infty $-categories, then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $Y$ if and only if $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$ (Corollary 7.2.3.6).

Example 7.4.3.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. If $U$ is fully faithful, then $F$ is $U$-left Kan extended and $U$-right Kan extended from any full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ (see Example 7.1.7.4).

To verify the Kan extension conditions of Definition 7.4.3.1, it suffices to consider objects $C$ which do not belong to the full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$.

Proposition 7.4.3.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory and let $C \in \operatorname{\mathcal{C}}$ be an object which is isomorphic to an object of $\operatorname{\mathcal{C}}^{0}$. Then $F$ is both $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ and $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$.

Proof. We will show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$; the analogous statement for the right Kan extension condition follows by a similar argument. Let $c: (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the slice contraction morphism; we wish to show that the composition $(F \circ c): (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram. Choose an object $C' \in \operatorname{\mathcal{C}}^{0}$ and an isomorphism $u: C' \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $u$ is an isomorphism guarantees that it is final when viewed as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/C}$ (Proposition 7.1.3.8), and therefore also when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/C}^{0}$. The desired result now follows from Corollary 7.2.3.9, since $F(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. $\square$

Example 7.4.3.7. Let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and set $F = \overline{F}|_{\operatorname{\mathcal{C}}}$. Then $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ if and only if $\overline{F}$ is $U$-left Kan extended from the cone $(\operatorname{\mathcal{C}}^0)^{\triangleright } \subseteq \operatorname{\mathcal{C}}^{\triangleright }$. To prove this, it suffices (by virtue of Proposition 7.4.3.6) to show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $C \in \operatorname{\mathcal{C}}$ if and only if $\overline{F}$ is $U$-left Kan extended from $( \operatorname{\mathcal{C}}^{0} )^{\triangleright }$ at $C$, which follows immediately from the definition.

Example 7.4.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories. It follows from Proposition 7.4.3.6 that a functor $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram (in the sense of Definition 7.1.7.1) if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}$.

We now record some basic stability properties enjoyed by the class of relative Kan extensions, which follow easily from the analogous stability properties of relative (co)limit diagrams.

Remark 7.4.3.9. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [r]^-{G} \ar [d]^{U} & \operatorname{\mathcal{D}}' \ar [d]^{U'} \\ \operatorname{\mathcal{E}}\ar [r] & \operatorname{\mathcal{E}}', } \]

where the horizontal functors are equivalence of $\infty $-categories. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G \circ F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Remark 7.1.7.6). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G \circ F$ is $U'$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Remark 7.4.3.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor which is isomorphic to $U$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$). Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $V$-left Kan extended from $\operatorname{\mathcal{C}}^0$ (see Remark 7.1.7.7). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $V$-right Kan extended from $\operatorname{\mathcal{C}}^0$.

Remark 7.4.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which is isomorphic to $F$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Proposition 7.1.7.13). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Remark 7.4.3.12 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that $U \circ F$ is $V$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $(V \circ U)$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Proposition 7.1.7.14). Similarly, if $U \circ F$ is $V$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $(V \circ U)$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Remark 7.4.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that $U \circ F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Then $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$; this follows by applying Remark 7.4.3.12 in the special case $\operatorname{\mathcal{E}}' = \Delta ^0$. Similarly, if $U \circ F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proposition 7.4.3.14 (Base Change). Suppose we are given a commutative diagram of $\infty $-categories

7.26
\begin{equation} \begin{gathered}\label{equation:base-change-relative-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [rr]^{H'} \ar [dr] \ar [dd]^{G} & & \operatorname{\mathcal{E}}' \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{B}}' \ar [dd] & \\ \operatorname{\mathcal{D}}\ar [dr]_{ U } \ar [rr]^(.4){H} & & \operatorname{\mathcal{E}}\ar [dl]^{V} \\ & \operatorname{\mathcal{B}}& } \end{gathered} \end{equation}

where each square is a pullback and the diagonal maps are inner fibrations. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be a functor of $\infty $-categories and $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then:

$(1)$

If $G \circ F$ is $H$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $H'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

$(2)$

Assume that $U$ and $V$ are cartesian fibrations and that the functor $G$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{E}}$. If $F$ is $H'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $G \circ F$ is an $H$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. Use Proposition 7.1.7.19. $\square$

Corollary 7.4.3.15. Suppose we are given a pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [d]^{U'} \ar [r]^-{G} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{E}}' \ar [r] & \operatorname{\mathcal{E}}, } \]

where the vertical maps are inner fibrations. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be a functor of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $G \circ F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. The converse holds if $U$ is a cartesian fibration.

Proof. Apply Proposition 7.4.3.14 in the special case $\operatorname{\mathcal{B}}= \operatorname{\mathcal{E}}$. $\square$

Corollary 7.4.3.16. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories, let $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ be the fiber of $U$ over an object $E \in \operatorname{\mathcal{E}}$, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_{E}$ be a functor of $\infty $-categories, and $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (when regarded as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$), then it is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (when regarded as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}_{E}$). The converse holds if $U$ is a cartesian fibration.

Proof. Apply Corollary 7.4.3.15 in the special case $\operatorname{\mathcal{E}}' = \{ E\} $. $\square$