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7.4.7 Transitivity of Kan Extensions

Let $\overline{\operatorname{\mathcal{C}}}$ be an $\infty $-category equipped with full subcategories $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$. Our goal in this section is to show that a functor of $\infty $-categories $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is left Kan extended from $\operatorname{\mathcal{C}}$ and $\overline{F}_{\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ (Corollary 7.4.7.8). We begin by analyzing the case special case where the $\infty $-category $\overline{\operatorname{\mathcal{C}}}$ has the form $\operatorname{\mathcal{C}}^{\triangleright }$.

Proposition 7.4.7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be another functor of $\infty $-categories. Assume that $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $\overline{F}$ is a $U$-colimit diagram if and only if the composite map

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

is a $U$-colimit diagram.

Proof. For each object $D \in \operatorname{\mathcal{D}}$, let $\underline{D} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{D}})$ denote the constant functor taking the value $D$. By virtue of Proposition 7.1.7.12, the functor $\overline{F}$ is a $U$-colimit diagram if and only if each of the diagrams

7.45
\begin{equation} \begin{gathered}\label{equation:Kan-extension-relative-colimit} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{D}})}( \overline{F}, \underline{D} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, \underline{D}|_{ \operatorname{\mathcal{C}}} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{E}})}( U \circ \overline{F}, U \circ \underline{D} ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( U \circ F, U \circ \underline{D}|_{\operatorname{\mathcal{C}}}) } \end{gathered} \end{equation}

is a homotopy pullback square. To simplify the notation, let $\operatorname{\mathcal{C}}^{1}$ denote the cone $(\operatorname{\mathcal{C}}^0)^{\triangleright }$, which we regard as a full subcategory of $\operatorname{\mathcal{C}}^{\triangleright }$. Our assumption that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ guarantees also that $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{1}$ (Example 7.4.3.7). Applying Proposition 7.4.6.7, we see that the restriction map from (7.45) to the diagram

7.46
\begin{equation} \begin{gathered}\label{equation:Kan-extension-relative-colimit2} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^1, \operatorname{\mathcal{D}})}( \overline{F}|_{\operatorname{\mathcal{C}}^1} , \underline{D}|_{\operatorname{\mathcal{C}}^{1}} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{0}}, \underline{D}|_{ \operatorname{\mathcal{C}}^0} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^1, \operatorname{\mathcal{E}})}( U \circ \overline{F}|_{\operatorname{\mathcal{C}}^1}, U \circ \underline{D}|_{\operatorname{\mathcal{C}}^{1}} ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{E}}) }( U \circ F, U \circ \underline{D}|_{\operatorname{\mathcal{C}}^0}) } \end{gathered} \end{equation}

is a levelwise homotopy equivalence. In particular, (7.45) is a homotopy pullback square if and only if (7.46) is a homotopy pullback square (Corollary 3.4.1.10). By virtue of Proposition 7.1.7.12, this is satisfied for every object $D \in \operatorname{\mathcal{D}}$ if and only if $F^{1}$ is a $U$-colimit diagram. $\square$

Corollary 7.4.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $\overline{F}$ is a colimit diagram if and only if the composite map

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

is a $U$-colimit diagram.

Proof. Apply Proposition 7.4.7.1 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Proposition 7.4.7.3. 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 we are given a right fibration of $\infty $-categories $V: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{B}}^{0} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{B}}$. Then, for every object $B \in \operatorname{\mathcal{B}}$, the functor $F \circ V$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$ at $B$ if and only if $F$ is $U$-left extended from $\operatorname{\mathcal{C}}^{0}$ at $V(B)$.

Proof. Set $C = V(B)$, and let $F_{C}$ denote the composite map

\[ (\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C})^{\triangleright } (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]

We wish to show that $F_{C}$ is a $U$-colimit diagram if the composite map

\[ (\operatorname{\mathcal{B}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}_{/B})^{\triangleright } \rightarrow (\operatorname{\mathcal{B}}_{/B})^{\triangleright } \rightarrow \operatorname{\mathcal{B}}\xrightarrow {V} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]

is a $U$-colimit diagram. By virtue of Corollary 7.2.2.2, it will suffice to show that the natural map

\[ \theta : \operatorname{\mathcal{B}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}_{/B} \rightarrow \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \]

is right cofinal. By construction, $\theta $ is a pullback of the map $V_{/B}: \operatorname{\mathcal{B}}_{/B} \rightarrow \operatorname{\mathcal{C}}_{ / V(B) }$. Our assumption that $V$ is a right fibration guarantees that $V_{/B}$ is a trivial Kan fibration (Corollary 4.3.7.13). It follows that $\theta $ is also a trivial Kan fibration, and therefore right cofinal by virtue of Corollary 7.2.1.11. $\square$

Corollary 7.4.7.4. 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 we are given a right fibration of $\infty $-categories $V: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{B}}^{0} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{B}}$. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F \circ V$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$. The converse holds if every fiber of $V$ is nonempty.

Corollary 7.4.7.5. 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. Suppose we are given a right fibration of $\infty $-categories $V: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{B}}^{0} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{B}}$. If $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F \circ V$ is left Kan extended from $\operatorname{\mathcal{B}}^{0}$. The converse holds if every fiber of $V$ is nonempty.

Proof. Apply Corollary 7.4.7.4 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Proposition 7.4.7.6 (Transitivity for Kan Extensions). Let $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be full subcategories. Then $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it satisfies the following pair of conditions:

$(1)$

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

$(2)$

The restriction $\overline{F}|_{\operatorname{\mathcal{C}}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Remark 7.4.7.7. In the special case $\overline{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}^{\triangleright }$, Proposition 7.4.7.6 is essentially a restatement of Proposition 7.4.7.1 (see Example 7.4.3.8).

Proof of Proposition 7.4.7.6. It follows immediately from the definitions that if $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$, then the functor $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ has the same property. We may therefore assume that condition $(2)$ is satisfied. Fix an object $X \in \overline{\operatorname{\mathcal{C}}}$. We will complete the proof by showing that $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at $X$ if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}$ at $X$. Let $\overline{F}_{X}$ denote the composite map

\[ (\operatorname{\mathcal{C}}\times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X})^{\triangleright } (\overline{\operatorname{\mathcal{C}}}_{/X})^{\triangleright } \rightarrow \overline{\operatorname{\mathcal{C}}} \xrightarrow { \overline{F} } \operatorname{\mathcal{D}}. \]

We wish to show that $\overline{F}_{X}$ is a $U$-colimit diagram if and only if its restriction to $(\operatorname{\mathcal{C}}^{0} \times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X})^{\triangleright }$ is a $U$-colimit diagram. Let $F_{X}$ denote the restriction of $\overline{F}_{X}$ to $\operatorname{\mathcal{C}}\times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X}$. By virtue of Proposition 7.4.7.1, it will suffice to show that $F_{X}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X}$. This follows by applying Corollary 7.4.7.4 to the right fibration $\operatorname{\mathcal{C}}\times _{ \overline{\operatorname{\mathcal{C}}} } \overline{\operatorname{\mathcal{C}}}_{/X} \rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 7.4.7.8. Let $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be full subcategories. Then $\overline{F}$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it satisfies the following pair of conditions:

$(1)$

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

$(2)$

The restriction $\overline{F}|_{\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. Apply Proposition 7.4.7.6 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Corollary 7.4.7.9. 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 $C,C' \in \operatorname{\mathcal{C}}$ be objects which are isomorphic. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$, then it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C'$.

Proof. Let $\operatorname{\mathcal{C}}^{1} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the objects of $\operatorname{\mathcal{C}}^{0}$ together with the object $C$, and let $\operatorname{\mathcal{C}}^{2} \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the objects of $\operatorname{\mathcal{C}}$ together with the objects $C$ and $C'$. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$, then the functor $F|_{ \operatorname{\mathcal{C}}^{1}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Since every object of $\operatorname{\mathcal{C}}^{2}$ is isomorphic to an object of $\operatorname{\mathcal{C}}^{1}$. the functor $F|_{ \operatorname{\mathcal{C}}^{2} }$ is automatically $U$-left Kan extended from $\operatorname{\mathcal{C}}^{1}$ (Proposition 7.4.3.6). Applying Proposition 7.4.7.6, we see that $F|_{ \operatorname{\mathcal{C}}^{2} }$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. In particular, $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at the object $C' \in \operatorname{\mathcal{C}}^{2}$. $\square$

We now prove a variant of Proposition 7.4.7.6, which gives a criterion for the existence of relative Kan extensions.

Proposition 7.4.7.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, and suppose that $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Set $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$. Then the restriction map

\[ \theta : \operatorname{\mathcal{D}}_{ F / } \rightarrow \operatorname{\mathcal{D}}_{ F_0 / } \times _{ \operatorname{\mathcal{E}}_{ (U \circ F_0) / } } \operatorname{\mathcal{E}}_{ (U \circ F) / } \]

is an equivalence of $\infty $-categories.

Proof. Note that the restriction maps

\[ \operatorname{\mathcal{D}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{F_0 / } \quad \quad \operatorname{\mathcal{D}}_{F_0 / } \rightarrow \operatorname{\mathcal{D}}\quad \quad \operatorname{\mathcal{E}}_{ (U \circ F) / } \rightarrow \operatorname{\mathcal{E}}_{ (U \circ F_0) / } \]

are left fibrations of simplicial sets (Corollary 4.3.6.11). It follows that we can regard $\theta $ as a functor of $\infty $-categories which are left-fibered over $\operatorname{\mathcal{D}}$. Consequently, to show that $\theta $ is an equivalence of $\infty $-categories, it will suffice to show that for every object $D \in \operatorname{\mathcal{D}}$, the commutative diagram

7.47
\begin{equation} \begin{gathered}\label{equation:coslice-over-Kan-extension} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}_{F/} \times _{\operatorname{\mathcal{D}}} \{ D\} \ar [r] \ar [d] & \operatorname{\mathcal{D}}_{F_0 /} \times _{\operatorname{\mathcal{D}}} \{ D\} \ar [d] \\ \operatorname{\mathcal{E}}_{ (U \circ F) / } \times _{\operatorname{\mathcal{E}}} \{ U(D) \} \ar [r] & \{ U(D) \} \times _{\operatorname{\mathcal{E}}_{ (U \circ F_0) / }} \times _{ \operatorname{\mathcal{E}}} \{ U(D) \} } \end{gathered} \end{equation}

induces a homotopy equivalence of Kan complexes

\[ \operatorname{\mathcal{D}}_{F/} \times _{\operatorname{\mathcal{D}}} \{ D\} \rightarrow (\operatorname{\mathcal{D}}_{ F_0 / } \times _{ \operatorname{\mathcal{E}}_{ (U \circ F_0) / } } \operatorname{\mathcal{E}}_{ (U \circ F) / }) \times _{\operatorname{\mathcal{D}}} \{ D \} . \]

Note that the horizontal maps in the diagram (7.47) are left fibrations between Kan complexes (Corollary 4.3.6.11), and therefore Kan fibrations (Corollary 4.4.3.8). We are therefore reduced to showing that the diagram (7.47) is a homotopy pullback square (Example 3.4.1.5).

Let $\underline{D} \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the constant functor taking the value $D$. Using Theorem 3.4.1.5, we obtain a (termwise) homotopy equivalence from (7.47) to the diagram of morphism spaces

7.48
\begin{equation} \begin{gathered}\label{equation:coslice-over-Kan-extension2} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, \underline{D} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( F_0, \underline{D}|_{ \operatorname{\mathcal{C}}^{0} } ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})}( U \circ F, U \circ \underline{D} ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}})}( U \circ F_0, U \circ \underline{D}|_{ \operatorname{\mathcal{C}}^{0} } ). } \end{gathered} \end{equation}

Using Corollary 3.4.1.10, we are reduced to showing that the diagram (7.48) is a homotopy pullback square, which is a special case of Proposition 7.4.6.7. $\square$

Corollary 7.4.7.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories, where $U$ is an inner fibration and $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Set $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$. Then the restriction map

\[ \theta : \operatorname{\mathcal{D}}_{ F / } \rightarrow \operatorname{\mathcal{D}}_{ F_0 / } \times _{ \operatorname{\mathcal{E}}_{ (U \circ F_0) / } } \operatorname{\mathcal{E}}_{ (U \circ F) / } \]

is a trivial Kan fibration.

Proof. It follows from Proposition 4.3.6.8 that $\theta $ is a left fibration, and therefore an isofibration (Example 4.4.1.10). By virtue of Proposition 4.5.7.16, it will suffice to show that $\theta $ is an equivalence of $\infty $-categories, which follows from Proposition 7.4.7.10. $\square$

Corollary 7.4.7.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which is Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, and set $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$. Then the restriction functor $\theta : \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}}_{F_0 / }$ is a trivial Kan fibration.

Proof. Apply Corollary 7.4.7.11 in the special case $\operatorname{\mathcal{E}}= \Delta ^{0}$. $\square$

Corollary 7.4.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories, and suppose we are given a lifting problem

7.49
\begin{equation} \begin{gathered}\label{equation:relative-colimit-extension-criterion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d] \ar [r]^-{F} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F} } & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}

Assume that $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The lifting problem (7.49) admits a solution $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.

$(2)$

The induced lifting problem

7.50
\begin{equation} \begin{gathered}\label{equation:relative-colimit-extension-criterion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^0 \ar [d] \ar [r]^-{F|_{\operatorname{\mathcal{C}}^{0}}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ (\operatorname{\mathcal{C}}^0)^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F}_0 } & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}

admits a solution $\overline{F}_0: ( \operatorname{\mathcal{C}}^{0} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.

Proof. The implication $(1) \Rightarrow (2)$ follows immediately from Proposition 7.4.7.1. For the converse, suppose that $\overline{F}_0: ( \operatorname{\mathcal{C}}^{0} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram which solves the lifting problem (7.50). Applying Corollary 7.4.7.11, we see that $\overline{F}_0$ can be extended to a functor $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which solves the lifting problem (7.49). It then follows from Proposition 7.4.7.1 that $\overline{F}$ is a $U$-colimit diagram. $\square$

Corollary 7.4.7.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $F$ has a colimit in $\operatorname{\mathcal{D}}$ if and only if the restriction $F|_{\operatorname{\mathcal{C}}^{0}}$ has a colimit in $\operatorname{\mathcal{D}}$.

Proof. Apply Corollary 7.4.7.13 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Proposition 7.4.7.15. Let $\overline{\operatorname{\mathcal{C}}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be a full subcategory, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories. Suppose we are given a lifting problem

7.51
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d] \ar [r]^-{F} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \overline{\operatorname{\mathcal{C}}} \ar [r] \ar@ {-->}[ur]^{ \overline{F} } & \operatorname{\mathcal{E}}, } \end{gathered} \end{equation}

where $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The lifting problem (7.51) admits a solution $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}$.

$(2)$

The induced lifting problem

7.52
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0} \ar [d] \ar [r]^-{F|_{\operatorname{\mathcal{C}}^{0}}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \overline{\operatorname{\mathcal{C}}} \ar [r] \ar@ {-->}[ur]^{ \overline{F} } & \operatorname{\mathcal{E}}, } \end{gathered} \end{equation}

admits a solution $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. The implication $(1) \Rightarrow (2)$ follows immediately from Proposition 7.4.7.6. For the converse, assume that $(2)$ is satisfied. To prove $(1)$, it will suffice to show that for each object $C \in \overline{\operatorname{\mathcal{C}}}$, the induced lifting problem

7.53
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/C} \ar [d] \ar [r]^-{F_{C}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F}_{C} } & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}

admits a solution $\overline{F}_{C}: ( \operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram (Proposition 7.4.5.3). Arguing as in the proof of Proposition 7.4.7.6, we see that $F_{C}$ is $U$-left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{0}_{/C} \subseteq \operatorname{\mathcal{C}}_{/C}$. Let $F_{C}^{0}$ denote the restriction of $F_{C}$ to the subcategory $\operatorname{\mathcal{C}}^{0}_{/C} \subseteq \operatorname{\mathcal{C}}_{/C}$. By virtue of Corollary 7.4.7.13, it will suffice to show that the induced lifting problem

7.54
\begin{equation} \begin{gathered}\label{equation::relative-Kan-extension-existence-transitivity4} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}^{0}_{/C} \ar [d] \ar [r]^-{F_{C}^{0}} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \ar [r] \ar@ {-->}[ur]^{ \overline{F}^{0}_{C} } & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}

has a solution $\overline{F}^{0}_{C}: (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram, which follows immediately from assumption $(2)$. $\square$

Corollary 7.4.7.16. Let $\overline{\operatorname{\mathcal{C}}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be a full subcategory, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $F$ admits a left Kan extension $\overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ if and only if the restriction $F|_{\operatorname{\mathcal{C}}^{0}}$ admits a left Kan extension $\overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$.

Proof. Apply Proposition 7.4.7.15 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

We close this section by establishing counterparts of Corollaries 7.4.7.8 and 7.4.7.14 for Kan extensions along more general functors.

Proposition 7.4.7.17. Let $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, $\operatorname{\mathcal{C}}_2$, and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Suppose we are given functors $F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ for $0 \leq i \leq 2$, functors $G: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_1$ and $H: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_2$, and natural transformations

\[ \alpha : F_0 \rightarrow F_1 \circ G \quad \quad \beta : F_1 \rightarrow F_2 \circ H, \]

where $\alpha $ exhibits $F_1$ as a left Kan extension of $F_0$ along $G$. The following conditions are equivalent:

$(1)$

The natural transformation $\beta $ exhibits $F_2$ as a left Kan extension of $F_1$ along $H$.

$(2)$

Let $\gamma : F_0 \rightarrow F_2 \circ H \circ G$ be a composition of $\alpha $ with $\beta |_{\operatorname{\mathcal{C}}^{0}}$ (formed in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}})$). Then $\gamma $ exhibits $F_2$ as a left Kan extension of $F_0$ along $H \circ G$.

Proof. Let $\operatorname{\mathcal{C}}$ denote the iterated relative join $(\operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1) \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2$, so that we have a cocartesian fibration of $\infty $-categories $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^2$ having fibers $\pi ^{-1} \{ i\} = \operatorname{\mathcal{C}}_ i$ for $0 \leq i \leq 2$ (see Exercise 5.2.4.20). For $0 \leq i < j \leq 2$, let $\operatorname{\mathcal{C}}_{ij}$ denote the fiber product $\operatorname{N}_{\bullet }( \{ i < j \} ) \times _{\Delta ^2} \operatorname{\mathcal{C}}$, which we will identify with $\operatorname{\mathcal{C}}_{i} \star _{\operatorname{\mathcal{C}}_ j} \operatorname{\mathcal{C}}_ j$. By virtue of Remark 7.4.1.9, we are free to replace $\alpha $ and $\beta $ by homotopic natural transformations. We can therefore assume that there exist functors

\[ F_{01}: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{D}}\quad \quad F_{12}: \operatorname{\mathcal{C}}_{12} \rightarrow \operatorname{\mathcal{D}} \]

satisfying $F_{01}|_{\operatorname{\mathcal{C}}_0} = F_0$, $F_{01}|_{\operatorname{\mathcal{C}}_1} = F_1 = F_{12}|_{\operatorname{\mathcal{C}}_1}$, and $F_{12}|_{\operatorname{\mathcal{C}}_2} = F_2$, where $\alpha $ and $\beta $ are given by the composite maps

\[ \Delta ^1 \times \operatorname{\mathcal{C}}_0 \simeq \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1 \xrightarrow {F_{01}} \operatorname{\mathcal{D}} \]

\[ \Delta ^1 \times \operatorname{\mathcal{C}}_1 \simeq \operatorname{\mathcal{C}}_1 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_1 \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2 \xrightarrow {F_{12}} \operatorname{\mathcal{D}} \]

(see Warning 7.4.2.11). Note that $F_{01}$ and $F_{12}$ can be amalgamated to a morphism of simplicial sets $F': \Lambda ^{2}_{1} \times _{\Delta ^1} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Since $\pi $ is a cocartesian fibration, the inclusion map $\Lambda ^{2}_{1} \times _{\Delta ^1} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}$ is a categorical equivalence (Proposition 5.2.6.24). Applying Lemma 4.5.6.2, we can extend $F'$ to a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

Let $F_{02}$ denote the restriction of $F$ to $\operatorname{\mathcal{C}}_{02}$, and let $\gamma : F_0 \rightarrow F_2 \circ H \circ G$ denote the natural transformation given by the composite map

\[ \Delta ^1 \times \operatorname{\mathcal{C}}_0 \simeq \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2 \xrightarrow {F_{02}} \operatorname{\mathcal{D}}. \]

Note that the composite map

\[ \Delta ^2 \times \operatorname{\mathcal{C}}_0 \simeq (\operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0) \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow (\operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1) \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2 \xrightarrow {F} \operatorname{\mathcal{D}} \]

can be regarded as a $2$-simplex of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}})$, which witnesses $\gamma $ as a composition of $\alpha $ with $\beta |_{\operatorname{\mathcal{C}}_0}$. Applying Proposition 7.4.2.10, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

The functor $F_{12}: \operatorname{\mathcal{C}}_{12} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_1$.

$(2')$

The functor $F_{02}: \operatorname{\mathcal{C}}_{02} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$.

By assumption, the natural transformation $\alpha $ exhibits $F_1$ as a left Kan extension of $F_0$ along $G$. Applying Proposition 7.4.2.10, we see that the functor $F_{01}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$. In particular, $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object of the full subcategory $\operatorname{\mathcal{C}}_1 \subseteq \operatorname{\mathcal{C}}$. It follows that $(2')$ is equivalent to the following:

$(2'')$

The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$.

Using Corollary 7.4.7.8, we see that $(2'')$ is equivalent to the following:

$(1'')$

The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_{01}$.

To complete the proof, it will suffice to show that conditions $(1')$ and $(1'')$ are equivalent. We will prove something slightly more precise: for every object $X \in \operatorname{\mathcal{C}}_2$, the conditions are equivalent:

$(1'_ X)$

The functor $F_{12}: \operatorname{\mathcal{C}}_{12} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_1$ at $X$.

$(1''_{X})$

The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_{01}$ at $X$.

Let us regard the object $X$ as fixed, and let $F_{X}$ denote the composite map

\[ (\operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \hookrightarrow ( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]

We wish to show that $F_{X}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ if and only if its restriction to $( \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} )^{\triangleright }$ is a colimit diagram in $\operatorname{\mathcal{D}}$. By virtue of Corollary 7.2.2.3, it will suffice to show that the inclusion map $\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X}$ is right cofinal. This follows by applying Proposition 7.2.3.13 to the upper square of the pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \ar [r] \ar [d] & \{ 1\} \ar [d] \\ \operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \operatorname{\mathcal{C}}_{/X} \ar [r]^-{\pi '} & \Delta ^2, } \]

where $\pi '$ denotes the composite map $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}\rightarrow \Delta ^2$ (which is a cocartesian fibration by virtue of Proposition 5.1.4.19). $\square$

Proposition 7.4.7.18. Let $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, $\operatorname{\mathcal{C}}_2$, and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Suppose we are given functors $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$, $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}$, $G: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_1$, and $H: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_2$, where $F_1$ is a left Kan extension of $F_0$ along $G$. The following conditions are equivalent:

$(1)$

The functor $F_1$ admits a left Kan extension along $H$.

$(2)$

The functor $F_0$ admits a left Kan extension along $H \circ G$.

Proof. The implication $(1) \Rightarrow (2)$ is immediate from Proposition 7.4.7.17. To prove the converse, assume that $(2)$ is satisfied. Define $\operatorname{\mathcal{C}}$ as in the proof of Proposition 7.4.7.17. Using the criterion of Corollary 7.4.5.4, we see that $F_0$ admits a left Kan extension $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. It follows from Proposition 7.4.2.10 that $F|_{ \operatorname{\mathcal{C}}_1 }$ is a left Kan extension of $F_0$ along $G$, and is therefore isomorphic to $F_1$ (Remark 7.4.6.6). We may therefore assume without loss of generality that $F_1 = F|_{\operatorname{\mathcal{C}}_1}$ (Remark 7.4.1.10). We will complete the proof by showing that $F_{12} = F|_{\operatorname{\mathcal{C}}_{12}}$ is left Kan extended from $\operatorname{\mathcal{C}}_{1}$, and therefore exhibits $F|_{\operatorname{\mathcal{C}}_2}$ as a left Kan extension of $F_1$ along $H$ (Proposition 7.4.2.10).

Fix an object $X \in \operatorname{\mathcal{C}}_2$, and let $F_{X}$ denote the composite map

\[ (\operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \hookrightarrow ( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]

We wish to show that the composite map

\[ (\operatorname{\mathcal{C}}_{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \xrightarrow {F_ X} \operatorname{\mathcal{D}} \]

is a colimit diagram in $\operatorname{\mathcal{D}}$. As in the proof of Proposition 7.4.7.17, the inclusion map $\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X}$ is right cofinal. It will therefore suffice to show that $F_{X}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.3). This is clear: by construction, the functor $F$ is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$, and is therefore also left Kan extended from the larger subcategory $\operatorname{\mathcal{C}}_{01} \subseteq \operatorname{\mathcal{C}}$ (Proposition 7.4.7.6). $\square$

Exercise 7.4.7.19. Show that the conclusions of Propositions 7.4.7.17 and 7.4.7.18 hold if we drop the assumption that the simplicial set $\operatorname{\mathcal{C}}_0$ is an $\infty $-category.