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7.4.1 Kan Extensions along General Functors

We begin by introducing some notation.

Notation 7.4.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. For each object $C \in \operatorname{\mathcal{C}}$, we let $K_{/C}$ denote the fiber product $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the slice diagonal of Construction 4.6.4.12 determines a map $K_{/C} \rightarrow K \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $, which we can identify with a natural transformation of diagrams $\gamma : \delta |_{ K_{/C} } \rightarrow \underline{C}$; here $\delta |_{K_{/C} }$ denotes the composition $K_{/C} \rightarrow K \xrightarrow {\delta } \operatorname{\mathcal{C}}$, while $\underline{C}$ denotes the constant diagram $K_{/C} \rightarrow \operatorname{\mathcal{C}}$ taking the value $C$. Similarly, we let $K_{C/}$ denote the fiber product $\operatorname{\mathcal{C}}_{C/} \times _{\operatorname{\mathcal{C}}} K$, so that the coslice diagonal of Construction 4.6.4.12 determines a natural transformation $\gamma ': \underline{C} \rightarrow \delta |_{ K_{C/} }$.

Definition 7.4.1.2 (Right Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose we are given a simplicial set $K$ together with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\alpha : F \circ \delta \rightarrow F_0$, as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}. } \]

We will say that $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, the following condition is satisfied:

$(\ast _ C)$

Let $\alpha _{C}$ denote a composition of the natural transformations

\[ \underline{ F(C) } \xrightarrow { F( \gamma ' ) } (F \circ \delta )|_{ K_{C/} } \xrightarrow { \alpha } F_0|_{ K_{C/} } \]

(formed in the $\infty $-category $\operatorname{Fun}( K_{C/}, \operatorname{\mathcal{D}})$), where $\gamma ': \underline{C} \rightarrow \delta |_{ K_{C/} }$ is defined in Notation 7.4.1.1. Then $\alpha _{C}$ exhibits $F(C)$ as a limit of the diagram

\[ K_{C/} = \operatorname{\mathcal{C}}_{C/} \times _{\operatorname{\mathcal{C}}} K \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}, \]

in the sense of Definition 7.1.1.1.

Remark 7.4.1.3. Stated more informally, a diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}. } \]

exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, we can calculate the value $F(C) \in \operatorname{\mathcal{D}}$ as a limit of the diagram

\[ K_{C/} = \operatorname{\mathcal{C}}_{C/} \times _{\operatorname{\mathcal{C}}} K \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}. \]

Note that this requirement characterizes the object $F(C) \in \operatorname{\mathcal{D}}$ up to isomorphism (see Proposition 7.1.1.11). We will later prove a stronger assertion: if the diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ are fixed, then a right Kan extension of $F_0$ along $\delta $ is uniquely determined (up to isomorphism) as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Remark 7.4.6.6).

Warning 7.4.1.4. In the situation of Definition 7.4.1.2, the natural transformation $\alpha _{C}$ appearing in condition $(\ast _ C)$ is defined as a composition of morphisms in the $\infty $-category $\operatorname{Fun}( K_{/C}, \operatorname{\mathcal{D}})$, which is only well-defined up to homotopy. However, the condition that $\beta _{C}$ exhibits $F(C)$ as a colimit of the diagram $F_0|_{ K_{/C} }$ depends only on the homotopy class $[\beta _ C]$ (Remark 7.1.1.6).

Variant 7.4.1.5 (Left Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose we are given a simplicial set $K$ together with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : F_0 \rightarrow F \circ \delta $, as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\beta } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}. } \]

We will say that $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, the following condition is satisfied:

$(\ast _ C)$

Let $\beta _{C}$ denote a composition of the natural transformations

\[ F_0|_{ K_{/C} } \xrightarrow {\beta } (F \circ \delta )|_{ K_{/C} } \xrightarrow { F(\gamma ) } \underline{ F(C) } \]

(formed in the $\infty $-category $\operatorname{Fun}( K_{/C}, \operatorname{\mathcal{D}})$), where $\gamma : \delta |_{ K_{/C} } \rightarrow \underline{C}$ is defined in Notation 7.4.1.1. Then $\beta _{C}$ exhibits $F(C)$ as a colimit of the diagram

\[ K_{/C} = K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}, \]

in the sense of Definition 7.1.1.1.

Remark 7.4.1.6. In the situation of Variant 7.4.1.5, the natural transformation $\beta : F_0 \rightarrow F \circ \delta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ if and only if it exhibits $F^{\operatorname{op}}$ as a right Kan extension of $F_0^{\operatorname{op}}$ along $\delta ^{\operatorname{op}}$, when regarded as a morphism in the $\infty $-category $\operatorname{Fun}(K^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}}) \simeq \operatorname{Fun}(K, \operatorname{\mathcal{D}})^{\operatorname{op}}$.

Example 7.4.1.7. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category, let $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be a diagram. Let $\delta : K \rightarrow \Delta ^0$ be the projection map and let $F: \Delta ^0 \rightarrow \operatorname{\mathcal{D}}$ be the functor corresponding to an object $Y \in \operatorname{\mathcal{D}}$. Then:

  • A natural transformation $\alpha : \underline{Y} = (F \circ \delta ) \rightarrow F_0$ exhibits $Y$ as a limit of $F_0$ (in the sense of Definition 7.1.1.1) if and only if it exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ (in the sense of Definition 7.4.1.2).

  • A natural transformation $\beta : F_0 \rightarrow (F \circ \delta ) = \underline{Y}$ exhibits $Y$ as a colimit of $F_0$ (in the sense of Definition 7.1.1.1) if and only if it exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ (in the sense of Variant 7.4.1.5).

Example 7.4.1.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $\alpha : F \rightarrow G$ be a morphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. The following conditions are equivalent:

$(1)$

The natural transformation $\alpha $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

$(2)$

The natural transformation $\alpha $ exhibits $F$ as a right Kan extension of $G$ along the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

$(3)$

The natural transformation $\alpha $ exhibits $G$ as a left Kan extension of $F$ along the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

To prove the equivalence of $(1)$ and $(2)$, fix an object $C \in \operatorname{\mathcal{C}}$. Since the identity morphism $\operatorname{id}_{C}$ is an initial object of the $\infty $-category $\operatorname{\mathcal{C}}_{C/}$ (Proposition 7.1.3.8), the natural transformation $\alpha $ satisfies condition $(\ast _ C)$ of Definition 7.4.1.2 if and only if the induced map $\alpha _ C: F(C) \rightarrow G(C)$ is an isomorphism in $\operatorname{\mathcal{D}}$ (Corollary 7.2.3.7). The equivalence $(1) \Leftrightarrow (2)$ now follows from the criterion of Theorem 4.4.4.4. The equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument.

Remark 7.4.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be diagrams. Then:

  • The condition that a natural transformation $\alpha : F \circ \delta \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ depends only on the homotopy class $[\alpha ]$ (as a morphism in the $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{D}})$).

  • The condition that a natural transformation $\beta : F_0 \rightarrow F \circ \delta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ depends only on the homotopy class $[\beta ]$ (as a morphism in the $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{D}})$).

See Remark 7.1.1.6.

Remark 7.4.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, and let $\rho : F_0 \rightarrow F'_0$ be an isomorphism in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$. Then:

  • A natural transformation $\alpha : F \circ \delta \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if and only if the composite natural transformation

    \[ F \circ \delta \xrightarrow { \alpha } F_0 \xrightarrow {\rho } F'_0 \]

    exhibits $F$ as a right Kan extension of $F'_0$ along $\delta $ (note that this condition is independent of the composition chosen, by virtue of Remark 7.4.1.9).

  • A natural transformation $\beta : F'_0 \rightarrow F \circ \delta $ exhibits $F$ as a left Kan extension of $F'_0$ along $\delta $ if and only if the composite natural transformation

    \[ F_0 \xrightarrow {\rho } F'_0 \xrightarrow {\beta } F \circ \delta \]

    exhibits $F$ as a left Kan extension of $F_0$ along $\delta $.

See Remark 7.1.1.7.

Remark 7.4.1.11. Suppose we are given a diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}} \]

as in Definition 7.4.1.2. Let $\rho : F' \rightarrow F$ be a morphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Then any two of the following conditions imply the third:

  • The natural transformation $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $.

  • The composite natural transformation

    \[ \delta \circ F' \xrightarrow { \rho } \delta \circ F \xrightarrow { \alpha } F_0 \]

    exhibits $F'$ as a right Kan extension of $F_0$ along $\delta $ (note that this condition does not depend on the composition chosen, by virtue of Remark 7.4.1.9).

  • The morphism $\rho $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

This follows by combining Remark 7.1.1.8 with Theorem 4.4.4.4.

Remark 7.4.1.12 (Change of Target). Suppose we are given a diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}} \]

as in Definition 7.4.1.2, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories. Then:

  • If $G$ is fully faithful and $G(\alpha ): (G \circ F) \circ \delta \rightarrow G \circ F_0$ exhibits $G \circ F$ as a right Kan extension of $G \circ F_0$ along $\delta $, then $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $.

  • If $G$ is an equivalence of $\infty $-categories and $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $, then $G(\alpha )$ exhibits $G \circ F$ as a right Kan extension of $G \circ F_0$ along $\delta $.

See Remark 7.1.1.9.

Proposition 7.4.1.13 (Change of Diagram). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be diagrams, and let $\epsilon : K' \rightarrow K$ be a categorical equivalence of simplicial sets. Then:

$(1)$

A natural transformation $\alpha : F \circ \delta \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if and only if the induced transformation $\alpha ': F \circ (\delta \circ \epsilon ) \rightarrow F_0 \circ \epsilon $ exhibits $F$ as a right Kan extension of $F_0 \circ \epsilon $ along $\delta \circ \epsilon $.

$(2)$

A natural transformation $\beta : F_0 \rightarrow F \circ \delta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ if and only if the induced transformation $\beta ': F_0 \circ \epsilon \rightarrow F \circ (\delta \circ \epsilon )$ exhibits $F$ as a left Kan extension of $F_0 \circ \epsilon $ along $\delta \circ \epsilon $.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Fix an object $C \in \operatorname{\mathcal{C}}$. Since $\epsilon $ is a categorical equivalence and the projection map $\operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration (Proposition 4.3.6.1), it follows that the induced map $\epsilon _{/C}: K' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/} \rightarrow K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is also a categorical equivalence of simplicial sets (Corollary 5.6.4.6). In particular, $\epsilon $ is left cofinal (Corollary 7.2.1.11). Applying Corollary 7.2.2.3, we see that the natural transformation $\alpha $ satisfies condition $(\ast _ C)$ of Definition 7.4.1.2 if and only if $\alpha '$ satisfies condition $(\ast _ C)$. The desired result now follows by allowing the object $C \in \operatorname{\mathcal{C}}$ to vary. $\square$

Proposition 7.4.1.14. Suppose we are given a diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}} \]

as in Definition 7.4.1.2, where $\delta $ factors as a composition

\[ K \xrightarrow { \delta ^{0} } \operatorname{\mathcal{C}}^{0} \xrightarrow {G} \operatorname{\mathcal{C}} \]

for some $\infty $-category $\operatorname{\mathcal{C}}^{0}$. Then:

$(1)$

If $G$ is fully faithful and $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $, then it also exhibits $F \circ G$ as a right Kan extension of $F_0$ along $\delta ^{0}$.

$(2)$

If $G$ is an equivalence of $\infty $-categories and $\alpha $ exhibits $F \circ G$ as a right Kan extension of $F_0$ along $\delta ^0$, then it exhibits $F$ as a right Kan extension of $F_0$ along $\delta $.

Proof. Assume that $G$ is fully faithful. Then, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}^{0}$, the induced map of left-pinched morphism spaces

\[ \operatorname{\mathcal{C}}^{0}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} = \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}^{0}}( X,Y) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( G(X), G(Y) ) = \operatorname{\mathcal{C}}_{ G(X) / } \times _{\operatorname{\mathcal{C}}} \{ G(Y) \} \]

is a homotopy equivalence. Allowing $Y$ to vary and applying Corollary 5.1.6.15, we see that the natural map $\operatorname{\mathcal{C}}^{0}_{X/} \rightarrow \operatorname{\mathcal{C}}_{ G(X)/ } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^0$ of left fibrations over $\operatorname{\mathcal{C}}^{0}$. It follows that the induced map

\[ \operatorname{\mathcal{C}}^{0}_{X/} \times _{ \operatorname{\mathcal{C}}^{0} } K \rightarrow \operatorname{\mathcal{C}}_{ G(X) / } \times _{ \operatorname{\mathcal{C}}} K \]

is an equivalence of left fibrations over $K$. In particular it is a categorical equivalence of simplicial sets (Proposition 5.1.6.5) and therefore left cofinal (Corollary 7.2.1.11). Applying Corollary 7.2.2.3, we see that the natural transformation $\alpha $ satisfies condition $(\ast _ X)$ of Definition 7.4.1.2 if and only if it satisfies condition $(\ast _{ G(X) })$. Assertion $(1)$ now follows by allowing the object $X \in \operatorname{\mathcal{C}}^{0}$ to vary.

We now prove $(2)$. Assume that $G$ is an equivalence of $\infty $-categories and that $\alpha $ exhibits $F \circ G$ as a right Kan extension of $F_0$ along $\delta ^0$; we wish to show that $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{0}$ be a homotopy inverse of $G$, so that there is an isomorphism $\rho : G \circ H \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. Let $\alpha '$ denote a composition of the natural transformations

\[ F \circ G \circ H \circ \delta \xrightarrow {\rho } F \circ \delta \xrightarrow {\alpha } F_0. \]

Using Remark 7.4.1.10, we see that $\alpha '$ exhibits $F \circ G$ as a right Kan extension of $F_0$ along $H \circ \delta $. Applying assertion $(1)$ to the fully faithful functor $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{0}$, we deduce that $\alpha '$ also exhibits $F \circ G \circ H$ as a right Kan extension of $F_0$ along $\delta $. Invoking Remark 7.4.1.11, we see that the original natural transformation $\alpha $ exhibits $F$ as a right Kan extension of $F_0$ along $\delta $, as desired. $\square$

Corollary 7.4.1.15. Let $G: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{C}}$, $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$, and $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $G$ is fully faithful. Then:

  • If $\alpha : F \circ G \rightarrow F_0$ is a natural transformation which exhibits $F$ as a right Kan extension of $F_0$ along $G$, then $\alpha $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$.

  • If $\beta : F_0 \rightarrow F \circ G$ is a natural transformation which exhibits $F$ as a left Kan extension of $F_0$ along $G$, then $\beta $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$.

Proof. Let $\alpha : F \circ G \rightarrow F_0$ be a natural transformation which exhibits $F$ as a right Kan extension of $F_0$ along $G$. Applying Proposition (in the special case where $K = \operatorname{\mathcal{C}}^0$), we deduce that $\alpha $ also exhibits $F \circ G$ as a right Kan extension of $F_0$ along the identity functor $\operatorname{id}_{ \operatorname{\mathcal{C}}^{0} }: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{C}}^{0}$. Invoking Example 7.4.1.8, we see that $\alpha $ is an isomorphism. This proves the first assertion; the second follows by a similar argument. $\square$