# Kerodon

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### 7.3.4 Kan Extensions along Fibrations

In this section, we study the formation of left Kan extension along cocartesian fibrations. We can state a preliminary version of our main result as follows:

Proposition 7.3.4.1. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories. Suppose we are given functors of $\infty$-categories $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ and $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : F_0 \rightarrow F \circ \delta$. The following conditions are equivalent:

$(1)$

The natural transformation $\beta$ exhibits $F$ as a left Kan extension of $F_0$ along $\delta$.

$(2)$

For each object $C \in \operatorname{\mathcal{C}}$, the restriction of $\beta$ to the fiber $\operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}$ determines a natural transformation $F_0|_{ \operatorname{\mathcal{K}}_{C} } \rightarrow \underline{ F(C) }$ which exhibits $F(C)$ as a colimit of the diagram $F_0|_{ \operatorname{\mathcal{K}}_{C} }$ in the $\infty$-category $\operatorname{\mathcal{D}}$.

Proof. By virtue of Corollary 7.2.2.7, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the tautological map

$\operatorname{\mathcal{K}}_{C} = \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \{ C\} \hookrightarrow \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$

is right cofinal. Since $\delta$ is a cocartesian fibration, it will suffice to show that the inclusion map $\{ \operatorname{id}_ C \} \hookrightarrow \operatorname{\mathcal{C}}_{/C}$ is right cofinal (Proposition 7.2.3.13). This follows from Corollary 4.6.6.25, since $\operatorname{id}_{C}$ is a final object of the $\infty$-category $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.6.23). $\square$

Corollary 7.3.4.2. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories and let $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $F$ is left Kan extended from $\operatorname{\mathcal{K}}$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the functor

$F_{C}: \operatorname{\mathcal{K}}_{C}^{\triangleright } \simeq \operatorname{\mathcal{K}}_{C} \star _{ \{ C\} } \{ C\} \hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a colimit diagram.

Corollary 7.3.4.2 generalizes to the setting of relative Kan extensions:

Proposition 7.3.4.3. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories and let $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors. The following conditions are equivalent:

$(1)$

The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{K}}$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the functor

$F_{C}: \operatorname{\mathcal{K}}_{C}^{\triangleright } \simeq \operatorname{\mathcal{K}}_{C} \star _{ \{ C\} } \{ C\} \hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a $U$-colimit diagram.

Proof. By virtue of Proposition 7.3.3.7, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the following conditions are equivalent:

$(1_ C)$

The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{K}}$ at $C$.

$(2_ C)$

The functor $F_{C}$ is a $U$-colimit diagram.

This follows from Corollary 7.2.2.3, since the tautological map

$\operatorname{\mathcal{K}}_{C} \simeq \{ \operatorname{id}_ C \} \times _{ \operatorname{\mathcal{C}}_{/C} } \operatorname{\mathcal{K}}_{/C} \hookrightarrow \operatorname{\mathcal{K}}_{/C}$

is right cofinal (as noted in the proof of Proposition 7.3.4.1). $\square$

Our next goal is establish a companion to Proposition 7.3.4.1, which provides necessary and sufficient conditions for the existence of a left Kan extension.

Proposition 7.3.4.4. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories and let $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $F_0$ admits a left Kan extension along $\delta$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the induced diagram

$\operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}}$

has a colimit in the $\infty$-category $\operatorname{\mathcal{D}}$.

Note that the implication $(1) \Rightarrow (2)$ of Proposition 7.3.4.4 follows immediately from Proposition 7.3.4.1. To prove the converse, it will be convenient to again translate to a question about the inclusion map $\operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, which we will address in a more general form. First, we need a variant of Corollary 7.1.6.6.

Lemma 7.3.4.5. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset which contains every vertex of $\operatorname{\mathcal{C}}$, and set $\operatorname{\mathcal{K}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}$. Suppose we are given a lifting problem

7.14
$$\begin{gathered}\label{equation:fiberwise-relative-Kan-general} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\coprod _{\operatorname{\mathcal{K}}_0} (\operatorname{\mathcal{K}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0) \ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \end{gathered}$$

which satisfies the following condition:

$(\ast )$

Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex which is not contained in $\operatorname{\mathcal{C}}_0$ and set $C = \sigma (0)$. Then the composite map

$\operatorname{\mathcal{K}}_ C^{\triangleright } \simeq \operatorname{\mathcal{K}}_{C} \star _{ \{ C\} } \{ C\} \hookrightarrow \operatorname{\mathcal{K}}_0 \star _{ \operatorname{\mathcal{C}}_0 } \operatorname{\mathcal{C}}_0 \xrightarrow {F_0} \operatorname{\mathcal{D}}$

is a $U$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$.

Then the lifting problem (7.14) admits a solution.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty$-category (working one simplex at a time, we could even assume that $\operatorname{\mathcal{C}}= \Delta ^ m$ is a standard simplex and that $\operatorname{\mathcal{C}}_0 = \operatorname{\partial \Delta }^ m$ is its boundary). Set $\overline{\operatorname{\mathcal{K}}} = \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, so that $\delta$ extends to a map

$\overline{\delta }: \overline{\operatorname{\mathcal{K}}} = \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\simeq \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}.$

Since $\delta$ is a cocartesian fibration, Lemma 5.2.3.17 guarantees that $\overline{\delta }$ is also a cocartesian fibration. Applying Proposition 5.3.6.6, we obtain a commutative diagram of $\infty$-categories

7.15
$$\begin{gathered}\label{equation:diagram-of-Res} \xymatrix@R =50pt@C=50pt{ \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}} / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}}) \ar [rr] \ar [dd]^{U \circ } \ar [dr]^{T} & & \operatorname{Res}_{\operatorname{\mathcal{K}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \ar [dd]^{U \circ } \ar [dl] \\ & \operatorname{\mathcal{C}}& \\ \operatorname{Res}_{ \overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) \ar [rr] \ar [ur] & & \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}), \ar [ul] } \end{gathered}$$

where the diagonal arrows are cartesian fibrations and the morphisms on the outside of the diagram preserve cartesian morphisms. Applying Proposition 5.1.4.20, we see that the induced map

$T': \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \times _{ \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}) } \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) \rightarrow \operatorname{\mathcal{C}}$

is also a cartesian fibration, and that the outer square of the diagram (7.15) determines a functor

$V: \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}}) \rightarrow \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \times _{ \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}) } \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) \rightarrow \operatorname{\mathcal{C}}$

which carries $T$-cartesian morphisms to $T'$-cartesian morphisms. Moreover, the functor $V$ is an isofibration (Proposition 4.5.9.17).

Unwinding the definitions, we can rewrite (7.14) as a lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r]^-{ G_0} \ar [d] & \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}}) \ar [d]^{V} & \\ \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur] & \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \times _{ \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}) } \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) . }$

By virtue of Corollary 7.1.6.6, to show that this lifting problem admits a solution, it will suffice to verify the following:

$(\ast ')$

Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex which is not contained in $\operatorname{\mathcal{C}}_0$ and set $C = \sigma (0)$. Then $G_0(C)$ is a $V$-initial object of the $\infty$-category $\operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}})$.

Unwinding the definitions, we see that the functor $T^{-1} \{ C\} \rightarrow T'^{-1} \{ C\}$ induced by $V$ can be identified with the restriction map

$V_ C: \operatorname{Fun}( \operatorname{\mathcal{K}}_{C}^{\triangleright }, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{K}}_ C, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{K}}_ C, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{K}}_ C^{\triangleright }, \operatorname{\mathcal{E}}).$

Combining assumption $(\ast )$ with Proposition 7.1.6.3, we see that $G_0( C )$ is a $V_ C$-initial object of the $\infty$-category $\operatorname{Fun}( K_{C}^{\triangleright }, \operatorname{\mathcal{D}})$. Proposition 7.1.4.19 then guarantees that $G_0(C)$ is also $V$-initial when regarded as an object of the $\infty$-category $\operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}})$. $\square$

Lemma 7.3.4.6. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories, and suppose we are given a lifting problem

7.16
$$\begin{gathered}\label{equation:fiberwise-relative-Kan-general2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur]^{F} \ar [r]^-{G} & \operatorname{\mathcal{E}}} \end{gathered}$$

with the following property:

$(\ast )$

For each vertex $C \in \operatorname{\mathcal{C}}$, the induced lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}_ C \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}_ C \star _{ \{ C\} } \{ C\} \ar@ {-->}[ur]^{ F_ C } \ar [r] & \operatorname{\mathcal{E}}}$

admits a solution $F_ C: \operatorname{\mathcal{K}}_{C}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.

Then (7.16) admits a solution $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $F|_{ X_{C}^{\triangleright } } = F_ C$ for each vertex $C \in \operatorname{\mathcal{C}}$.

Proof. Let $\operatorname{\mathcal{C}}_0 = \operatorname{sk}_0(\operatorname{\mathcal{C}})$ be the $0$-skeleton of $\operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{K}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}= \coprod _{C \in \operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}_ C$, so that we can amalgamate $F_0$ with the morphisms $\{ F_ C \} _{C \in \operatorname{\mathcal{C}}}$ to obtain a map $F_1: \operatorname{\mathcal{K}}\coprod _{ \operatorname{\mathcal{K}}_0 } ( \operatorname{\mathcal{K}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 ) \rightarrow \operatorname{\mathcal{D}}$. To prove Lemma 7.3.4.6, we must show that the lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\coprod _{ \operatorname{\mathcal{K}}_0 } (\operatorname{\mathcal{K}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 ) \ar [r]^-{F_1} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur] \ar [r]^-{G} & \operatorname{\mathcal{E}}}$

has a solution, which is a special case of Lemma 7.3.4.5. $\square$

Proposition 7.3.4.7. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories, and suppose we are given a lifting problem

7.17
$$\begin{gathered}\label{equation:existence-Kan-fiberwise} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\ar [r]^-{ F_0 } \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur]^{F} \ar [r] & \operatorname{\mathcal{E}}. } \end{gathered}$$

The following conditions are equivalent:

$(1)$

The lifting problem (7.17) has a solution $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{K}}$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the associated lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}_{C} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}_{C}^{\triangleright } \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}}$

has a solution $\operatorname{\mathcal{K}}_{C}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.

Corollary 7.3.4.8. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories and let $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

There exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{K}}$ and satisfies $F|_{\operatorname{\mathcal{K}}} = F_0$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the diagram

$\operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}}$

admits a colimit in the $\infty$-category $\operatorname{\mathcal{D}}$.

Proof of Proposition 7.3.4.4. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories and let $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the diagram

$\operatorname{\mathcal{K}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\xrightarrow {F_0} \operatorname{\mathcal{D}}$

has a colimit in the $\infty$-category $\operatorname{\mathcal{D}}$. Applying Corollary 7.3.4.8, we deduce that there exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{K}}$ and satisfies $F|_{\operatorname{\mathcal{K}}} = F_0$. Applying Proposition 7.3.1.15, we see that the restriction $F|_{\operatorname{\mathcal{C}}}$ is a left Kan extension of $F_0$ along $\delta$. $\square$