7.3.2 Kan Extensions along Inclusions
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. In §7.3.1, we introduced the notion of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ being a left Kan extension of another diagram $F_0: K \rightarrow \operatorname{\mathcal{D}}$ along $\delta $ (Variant 7.3.1.5). Beware that this terminology is potentially misleading: if $F$ is a left Kan extension of $F_0$ along $\delta $, then the composition $F \circ \delta $ need not be equal to $F_0$. Instead, it is equipped with a natural transformation $\beta : F_0 \rightarrow F \circ \delta $ satisfying a certain universal property. In this section, we specialize to the case where $K = \operatorname{\mathcal{C}}^{0}$ is a full subcategory of $\operatorname{\mathcal{C}}$ and $\delta : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ is the inclusion map. In this case, the natural transformation $\beta $ is necessarily an isomorphism (Corollary 7.3.1.17). Consequently, the Kan extension condition can be substantially simplified: it can be regarded as a property of the functor $F$, which can be formulated without reference to the diagram $F_0$ or the natural transformation $\beta $.
Definition 7.3.2.1. 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. Fix an object $C \in \operatorname{\mathcal{C}}$. We will say that $F$ is 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 colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. Here $\operatorname{\mathcal{C}}^{0}_{/C}$ denotes the fiber product $\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ (Notation 7.3.1.1), and $c$ is the slice contraction morphism of Construction 4.3.5.12. Similarly, we say that $F$ is 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 limit diagram in $\operatorname{\mathcal{D}}$. We say that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ if it is left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$. We say that $F$ is right Kan extended from $\operatorname{\mathcal{C}}^0$ if it is right Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$.
Exercise 7.3.2.3. 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. Show that, for every object $C \in \operatorname{\mathcal{C}}^{0}$, the functor $F$ is both left and right Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$. For a more general statement, see Proposition 7.3.3.7.
Example 7.3.2.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of ordinary 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 7.3.0.1) if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is left Kan extended from $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{0} )$ (in the sense of Definition 7.3.2.1).
We now show that Definition 7.3.2.1 can be regarded as a special case of the notions introduced in §7.3.1:
Proposition 7.3.2.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $F_0$ denote the restriction of $F$ to a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, and let $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ denote the inclusion functor. Then:
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1) if and only if the identity transformation $\operatorname{id}: F_0 \rightarrow F \circ \iota $ exhibits $F$ as a left Kan extension of $F_0$ along $\iota $ (in the sense of Variant 7.3.1.5).
The functor $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1) if and only if the identity transformation $\operatorname{id}_{F_0}: F \circ \iota \rightarrow F_0$ exhibits $F$ as a right Kan extension of $F_0$ along $\iota $ (in the sense of Definition 7.3.1.2).
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$. It follows from Remark 7.1.3.6 that the composition
\[ (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]
is a colimit diagram in $\operatorname{\mathcal{D}}$ if and only if the natural transformation $\operatorname{id}_{ F_0 }$ satisfies condition $(\ast _ C)$ of Variant 7.3.1.5. The first assertion follows by allowing the object $C$ to vary, and the second follows by a similar argument.
$\square$
Corollary 7.3.2.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $\beta : F_0 \rightarrow F|_{ \operatorname{\mathcal{C}}^0 }$ be a natural transformation of functors from $\operatorname{\mathcal{C}}^{0}$ to $\operatorname{\mathcal{D}}$. Then $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along the inclusion map $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ (in the sense of Variant 7.3.1.5) if and only if the following pair of conditions is satisfied:
- $(1)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 7.3.2.1).
- $(2)$
The natural transformation $\beta $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$.
Proof.
By virtue of Corollary 7.3.1.17, we may assume that condition $(2)$ is satisfied. Using Remark 7.3.1.11, we can reduce further to the special case where $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$ and $\beta $ is the identity transformation, in which case the desired result is a restatement of Proposition 7.3.2.6.
$\square$
Corollary 7.3.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
- $(1)$
There exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : F_0 \rightarrow F|_{ \operatorname{\mathcal{C}}^{0} }$ which exhibits $F$ as a left Kan extension of $F_0$ along the inclusion functor $\operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$.
- $(2)$
There exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F_0 = F|_{ \operatorname{\mathcal{C}}^{0} }$.
Proof.
We will show that $(1)$ implies $(2)$; the converse is an immediate consequence of Proposition 7.3.2.6. Let $\beta : F_0 \rightarrow F'|_{ \operatorname{\mathcal{C}}^{0} }$ exhibit $F'$ as a left Kan extension of $F_0$ along the inclusion functor $\operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$. Then $\beta $ is an isomorphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$ (Corollary 7.3.1.17). Using Corollary 4.4.5.9, we can lift $\beta $ to an isomorphism $\widetilde{\beta }: F \rightarrow F'$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, where $F$ satisfies $F|_{ \operatorname{\mathcal{C}}^{0} } = F_0$. Applying Remark 7.3.1.13, we deduce that the identity transformation $\operatorname{id}_{F_0}$ exhibits $F$ as a left Kan extension of $F_0$ along the inclusion map $\operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$. Invoking Proposition 7.3.2.6, we conclude that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
$\square$
Definition 7.3.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and suppose we are given functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$. We will say that $F$ is a left Kan extension of $F_0$ if $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{\operatorname{\mathcal{C}}^{0}} = F_0$. We will say that $F$ is a right Kan extension of $F_0$ if $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{\operatorname{\mathcal{C}}^{0}} = F_0$.
Warning 7.3.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\iota : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion of a full subcategory, and let $F_0: \operatorname{\mathcal{C}}^0 \rightarrow \operatorname{\mathcal{D}}$. be a functor. We have given two definitions for the notion of Kan extension:
- $(a)$
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a left Kan extension of $F_0$ if it is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{\operatorname{\mathcal{C}}^{0}} = F_0$ (Definition 7.3.2.9).
- $(b)$
A functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a left Kan extension of $F_0$ along $\iota $ if there exists a natural transformation $\beta : F_0 \rightarrow F|_{ \operatorname{\mathcal{C}}^{0} }$ which exhibits $F$ as a left Kan extension of $F_0$ along $\iota $, in the sense of Variant 7.3.1.5.
These definitions are not quite equivalent. By virtue of Proposition 7.3.2.6, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfies condition $(a)$ if and only if it satisfies a stronger version of condition $(b)$, where $\beta $ is required to be an identity natural transformation. In particular, condition $(a)$ implies condition $(b)$. However, the converse is false: if $F$ is a left Kan extension of $F_0$ along $\iota $, then the restriction $F|_{ \operatorname{\mathcal{C}}^{0} }$ need not be equal to $F_0$. However, it is necessarily isomorphic to $F_0$, by virtue of Corollary 7.3.2.7.
Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. The preceding results show that, if $\delta $ is an isomorphism from $\operatorname{\mathcal{K}}$ to a full subcategory of $\operatorname{\mathcal{C}}$, then the theory of Kan extensions along $\delta $ (in the sense of §7.3.1) can be reformulated in terms of Definition 7.3.2.1. We now extend this observation to the case of a general functor, by identifying $\operatorname{\mathcal{K}}$ with a full subcategory of the relative join $\operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ of Construction 5.2.3.1.
Proposition 7.3.2.11. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be another functor having restrictions $F_0 = F|_{ \operatorname{\mathcal{K}}}$ and $F_{1} = F|_{\operatorname{\mathcal{C}}}$, so that the composition
\[ \Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]
determines a natural transformation $\beta : F_0 \rightarrow F_1 \circ \delta $. The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from the full subcategory $\operatorname{\mathcal{K}}\subseteq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ (in the sense of Definition 7.3.2.1).
- $(2)$
The natural transformation $\beta $ exhibits $F_1$ as a left Kan extension of $F_0$ along $\delta $ (in the sense of Variant 7.3.1.5).
Proof.
By virtue of Exercise 7.3.2.3, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the following conditions are equivalent:
- $(1_ C)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{K}}$ at $C$ (in the sense of Definition 7.3.2.1).
- $(2_ C)$
The natural transformation $\beta $ satisfies condition $(\ast _ C)$ of Variant 7.3.1.5.
For the remainder of the proof, let us regard the object $C \in \operatorname{\mathcal{C}}$ as fixed, and set $\operatorname{\mathcal{K}}_{/C} = \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Let $\pi : \Delta ^2 \times \operatorname{\mathcal{K}}_{/C} \rightarrow ( \Delta ^1 \times \operatorname{\mathcal{K}}_{/C} )^{\triangleright }$ be the functor which is the identity on $\Delta ^1 \times \operatorname{\mathcal{K}}_{/C}$ and carries $\{ 2\} \times \operatorname{\mathcal{K}}_{/C}$ to the cone point of $( \Delta ^1 \times \operatorname{\mathcal{K}}_{/C} )^{\triangleright }$. Let $\sigma $ denote the composite map
\begin{eqnarray*} \Delta ^2 \times \operatorname{\mathcal{K}}_{/C} & \xrightarrow {\pi } & ( \Delta ^1 \times \operatorname{\mathcal{K}}_{/C} )^{\triangleright } \\ & \simeq & (( \operatorname{\mathcal{K}}\times \Delta ^1) \times _{ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}} ( \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})_{/C})^{\triangleright } \\ & \rightarrow & \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\\ & \xrightarrow {F} & \operatorname{\mathcal{D}}. \end{eqnarray*}
We will regard $\sigma $ as a $2$-simplex in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{K}}_{/C}, \operatorname{\mathcal{D}})$, which we display as a diagram
\[ \xymatrix@R =50pt@C=50pt{ & (F_{1} \circ \delta )|_{ \operatorname{\mathcal{K}}_{/C} } \ar [dr] & \\ F_0|_{ \operatorname{\mathcal{K}}_{/C} } \ar [ur] \ar [rr] & & \underline{ F_1(C) } } \]
which witnesses the bottom horizontal map as the natural transformation $\beta _{C}$ appearing in condition $(\ast _ C)$. By construction, this natural transformation $\beta _{C}$ is given by the composite map
\[ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \times \operatorname{\mathcal{K}}_{/C} \rightarrow (\operatorname{\mathcal{K}}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}, \]
so the equivalence $(1_ C) \Leftrightarrow (2_ C)$ is a special case of Remark 7.1.3.6.
$\square$
Warning 7.3.2.12. For a general diagram
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F_1} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\beta } & \\ \operatorname{\mathcal{K}}\ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}, } \]
we cannot always arrange that there exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying the requirements of Proposition 7.3.2.11. However, we can always find a functor $F': \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which satisfies $F'|_{\operatorname{\mathcal{K}}} = F_0$, $F'|_{\operatorname{\mathcal{C}}} = F_1$, and the map
\[ \Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}} \]
determines a natural transformation $\beta ': F_0 \rightarrow F_1 \circ \delta $ which is homotopic to $\beta $. To see this, set $M = ( \Delta ^1 \times \operatorname{\mathcal{K}}) {\coprod }_{ (\{ 1\} \times \operatorname{\mathcal{K}}) } \operatorname{\mathcal{C}}$, so that the pair $(\beta , F_1)$ determines a morphism of simplicial sets $f: M \rightarrow \operatorname{\mathcal{D}}$. Proposition 5.2.4.5 supplies a categorical equivalence of simplicial sets $\theta : M \rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, so the induced map
\[ \operatorname{Fun}_{ \operatorname{\mathcal{K}}\coprod \operatorname{\mathcal{C}}/}( \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \xrightarrow {\circ \theta } \operatorname{Fun}_{ \operatorname{\mathcal{K}}\coprod \operatorname{\mathcal{C}}/ }( M, \operatorname{\mathcal{D}}) \]
is an equivalence of $\infty $-categories (Corollary 4.5.4.5). It follows that there exists a functor $F': \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ such that $F'|_{\operatorname{\mathcal{K}}} = F_0$, $F'|_{\operatorname{\mathcal{C}}} = F_1$, and $F' \circ \theta $ is isomorphic to $f$ as an object of the $\infty $-category $\operatorname{Fun}_{ \operatorname{\mathcal{K}}\coprod \operatorname{\mathcal{C}}/ }( M, \operatorname{\mathcal{D}})$. The last requirement is a reformulation of the condition that $\beta ' = F'|_{ \Delta ^1 \times \operatorname{\mathcal{K}}}$ is homotopic to $\beta $.
Corollary 7.3.2.13. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, and $F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors 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 $\operatorname{\mathcal{K}}$ which satisfies $F_0 = F|_{\operatorname{\mathcal{K}}}$ and $F_1 = F|_{\operatorname{\mathcal{C}}}$.
- $(2)$
There exists a natural transformation $\beta : F_0 \rightarrow F_1 \circ \delta $ which exhibits $F_1$ as a left Kan extension of $F_0$ along $\delta $.
Proof.
The implication $(1) \Leftrightarrow (2)$ follows immediately from Proposition 7.3.2.11. Conversely, suppose that there exists a natural transformation $\beta : F_0 \rightarrow F_1 \circ \delta $ which exhibits $F_1$ as a left Kan extension of $F_0$ along $\delta $. By virtue of Remark 7.3.1.10, we can modify $\beta $ by a homotopy and thereby arrange that there exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $F|_{\operatorname{\mathcal{K}}} = F_0$, $F|_{\operatorname{\mathcal{K}}} = F_1$ and for which the induced map
\[ \Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]
coincides with $\beta $ (Warning 7.3.2.12). Applying Proposition 7.3.2.11, we see that $F$ is left Kan extended from $\operatorname{\mathcal{K}}$.
$\square$
For later use, we record a slightly more general version of Proposition 7.3.2.11.
Corollary 7.3.2.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ be a cocartesian fibration having fibers $\operatorname{\mathcal{C}}_{0} = \{ 0\} \times _{ \Delta ^1} \operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}_{1} = \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{C}}$. Choose a functor $G: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_1$ and a natural transformation $\beta : \operatorname{id}_{\operatorname{\mathcal{C}}_0} \rightarrow G$ which exhibits $G$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$ (see Definition 5.2.2.4). The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}_0$.
- $(2)$
The natural transformation $F(\beta ): F|_{ \operatorname{\mathcal{C}}_0} \rightarrow F|_{ \operatorname{\mathcal{C}}_1 } \circ G$ exhibits $F|_{ \operatorname{\mathcal{C}}_1 }$ as a left Kan extension of $F|_{ \operatorname{\mathcal{C}}_0 }$ along $G$.
Proof.
Let us regard the functor $G$ as fixed. Let $M = (\Delta ^1 \times \operatorname{\mathcal{C}}_0) {\coprod }_{ ( \{ 1\} \times \operatorname{\mathcal{C}}_0 )} \operatorname{\mathcal{C}}_{1}$ be the mapping cylinder of $G$, and let us abuse notation by identifying $\operatorname{\mathcal{C}}_0 \simeq \{ 0\} \times \operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ with (disjoint) simplicial subsets of $M$. We can then identify $\alpha $ with a morphism of simplicial sets $\mu : M \rightarrow \operatorname{\mathcal{C}}$ which is the identity when restricted to $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$.
Note that the tautological map
\[ \Delta ^1 \times \operatorname{\mathcal{C}}_0 \simeq \operatorname{\mathcal{C}}_{0} \star _{ \operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_{0} \star _{ \operatorname{\mathcal{C}}_1 } \operatorname{\mathcal{C}}_1 \]
extends to a morphism of simplicial sets $\lambda : M \rightarrow \operatorname{\mathcal{C}}_0 \star _{ \operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1$ which is the identity on $\operatorname{\mathcal{C}}_1$; moreover, $\lambda $ is a categorical equivalence (Proposition 5.2.4.5). It follows that precomposition with $\lambda $ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}_{ \operatorname{\mathcal{C}}_0 \coprod \operatorname{\mathcal{C}}_1 / }( \operatorname{\mathcal{C}}_0 \star _{ \operatorname{\mathcal{C}}_1 } \operatorname{\mathcal{C}}_1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ \operatorname{\mathcal{C}}_0 \coprod \operatorname{\mathcal{C}}_1 / }( M, \operatorname{\mathcal{C}}). \]
We can therefore choose a functor $G: \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ satisfying $G|_{ \operatorname{\mathcal{C}}_0 } = \operatorname{id}_{\operatorname{\mathcal{C}}_0}$ and $G|_{ \operatorname{\mathcal{C}}_1} = \operatorname{id}_{ \operatorname{\mathcal{C}}_1 }$, where $G \circ \lambda $ is isomorphic to $\mu $ as an object of the $\infty $-category $\operatorname{Fun}_{ \operatorname{\mathcal{C}}_0 \coprod \operatorname{\mathcal{C}}_1 / }( M, \operatorname{\mathcal{C}})$ Since condition $(2)$ depends only on the homotopy class of the natural transformation $\beta $ (Remark 7.3.1.10), we are free to modify $\beta $ and may therefore assume that $G \circ \lambda = \mu $. In this case, Proposition 7.3.2.11 allows us to reformulate condition $(2)$ as follows:
- $(2')$
The functor $(F \circ G): \operatorname{\mathcal{C}}_0 \star _{ \operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$.
Since $\lambda $ and $\mu $ are categorical equivalences of simplicial sets (Proposition 5.2.4.5), the functor $G$ is an equivalence of $\infty $-categories (Remark 4.5.3.5). The equivalence of $(1)$ and $(2')$ is now a special case of Proposition 7.3.3.18.
$\square$