Kerodon

Proof. For each object $C \in \operatorname{\mathcal{C}}$, the inclusion map

is right anodyne (Example 7.2.3.11), and therefore right cofinal. The desired result now follows by combining Proposition 7.3.5.1 with Corollary 7.2.2.10. $\square$

Proof. Assume that condition $(\ast _ C)$ is satisfied for every object $C \in \operatorname{\mathcal{C}}$; we will show that the lifting problem (7.19) admits a solution $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (the converse follows immediately from the definitions). Let $\operatorname{\mathcal{K}}$ denote the oriented fiber product $\operatorname{\mathcal{C}}^{0} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$: that is, the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by those morphisms $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ such that $X$ belongs to the subcategory $\operatorname{\mathcal{C}}^{0}$. Let $\pi : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}^{0}$ and $\pi ': \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be the evaluation maps, given on objects by $\pi (e) = X$ and $\pi '(e) = Y$, respectively. We then have a natural transformation $\alpha : \pi \rightarrow \pi '$ (which carries each morphism $e: X \rightarrow Y$ to itself). Regarding $\operatorname{\mathcal{K}}$ as an object of $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ via the functor $\pi '$, let $\operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ denote the relative join of Construction 5.2.3.1. We will write $\iota _{\operatorname{\mathcal{K}}}: \operatorname{\mathcal{K}}\hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ and $\iota _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ for the inclusion maps, and $\iota _{\operatorname{\mathcal{C}}^{0} }$ for the restriction of $\iota _{\operatorname{\mathcal{C}}}$ to the full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. The natural transformation $\alpha $ then determines a functor $S: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $S \circ \iota _{\operatorname{\mathcal{K}}} = \pi $ and $S \circ \iota _{\operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$. Consider the lifting problem

For each object $C \in \operatorname{\mathcal{C}}$, write $\operatorname{\mathcal{K}}_{C}$ for the fiber $\pi '^{-1} \{ C\} $, so that (7.21) restricts to a lifting problem

Note that $\operatorname{\mathcal{K}}_{C}$ can be identified with the oriented fiber product $\operatorname{\mathcal{C}}^{0} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $. Moreover, after precomposing with the slice diagonal equivalence $\operatorname{\mathcal{C}}^{0}_{/C} \rightarrow \operatorname{\mathcal{C}}^{0} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $ of Theorem 4.6.4.17, (7.22) recovers the lifting problem (7.20). Combining assumption $(\ast _ C)$ with Proposition 7.2.2.9, we deduce that the lifting problem (7.22) admits a solution $\overline{F}_{C}: \operatorname{\mathcal{K}}_{C}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram. Since $\pi ': \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration (Corollary 5.3.7.3), Proposition 7.3.4.7 guarantees that the lifting problem (7.21) admits a solution $\overline{F}: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{K}}$.

Note that the diagonal inclusion $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ restricts to a map $\delta : \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{K}}$. Let $\beta $ denote the composite map

which we regard as a natural transformation from $\iota _{\operatorname{\mathcal{K}}} \circ \delta $ to $\iota _{\operatorname{\mathcal{C}}^{0}}$. This natural transformation carries each object $X \in \operatorname{\mathcal{C}}^{0}$ to a morphism $\beta _{X}: \iota _{\operatorname{\mathcal{K}}}( \operatorname{id}_ X ) \rightarrow \iota _{\operatorname{\mathcal{C}}}(X)$ in the $\infty $-category $\operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$. Since $\operatorname{id}_ X$ is a final object of the $\infty $-category $\operatorname{\mathcal{K}}_{X} \simeq \operatorname{\mathcal{C}}^{0} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}^{0} } \{ X\} $ (Proposition 4.6.7.22) and $\overline{F}|_{ \operatorname{\mathcal{K}}_{X}^{\triangleright } }$ is a $U$-colimit diagram (Proposition 7.3.4.3), the image $\overline{F}( \beta _ X )$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.5). Since $U( \overline{F}(\beta _ X) ) = \operatorname{id}_{ G(X) }$ is an isomorphism in $\operatorname{\mathcal{E}}$, we conclude that $\overline{F}( \beta _ X )$ is an isomorphism in $\operatorname{\mathcal{D}}$. Applying Corollary 4.4.5.9, we deduce that $\overline{F}( \beta )$ can be lifted to an isomorphism $F \rightarrow \overline{F} \circ \iota _{\operatorname{\mathcal{C}}}$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a solution to the lifting problem (7.19). We will show $\overline{F} \circ \iota _{\operatorname{\mathcal{C}}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, so that $F$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.17).

Fix an object $C \in \operatorname{\mathcal{C}}$, let $c: ( \operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the slice contraction map and set $T^{+} = \iota _{\operatorname{\mathcal{C}}} \circ c$; we wish to show that $T^{+}: (\operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram. Let $\psi : \operatorname{\mathcal{C}}_{/C} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} $ be the slice diagonal of Construction 4.6.4.13. Note that $\psi $ is an equivalence of right fibrations over $\operatorname{\mathcal{C}}$ (Theorem 4.6.4.17 and Proposition 5.1.7.5), and therefore restricts to an equivalence of full subcategories $\psi _{0}: \operatorname{\mathcal{C}}^{0}_{/C} \rightarrow \operatorname{\mathcal{C}}^{0} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ C\} = \operatorname{\mathcal{K}}_{C}$. Let $T^{-}$ denote the composite functor

Because $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{K}}$, the $\overline{F}|_{ \operatorname{\mathcal{K}}_{C}^{\triangleright } }$ is a $U$-colimit diagram in $\operatorname{\mathcal{D}}$ (Proposition 7.3.4.3). Since the functor $\psi _{0}$ is right cofinal (Corollary 7.2.1.13), the functor $\overline{F} \circ T^{-}$ is also a $U$-colimit diagram (Corollary 7.2.2.2). Beware that the functors $T^{-}, T^{+}: (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ are not isomorphic: if $\widetilde{X}$ is an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/C}$ given by a morphism $e: X \rightarrow C$ in $\operatorname{\mathcal{C}}$, then we have $T^{+}( \widetilde{X} ) = \iota _{\operatorname{\mathcal{C}}}(X)$ and $T^{-}( \widetilde{X} ) = \iota _{\operatorname{\mathcal{K}}}( e )$. However, we will show that the functors $\overline{F} \circ T^{-}$ and $\overline{F} \circ T^{+}$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( ( \operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright }, \operatorname{\mathcal{D}})$, so that $\overline{F} \circ G^{+}$ a $U$-colimit diagram by virtue of Proposition 7.1.6.15.

Let $b: (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \Delta ^1$ be the map carrying $\operatorname{\mathcal{C}}^{0}_{/C}$ to the vertex $0 \in \Delta ^1$ and the cone point of $(\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright }$ to the vertex $1 \in \Delta ^1$. Note that the map $( b, c ): (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \Delta ^1 \times \operatorname{\mathcal{C}}$ factors through the full subcategory

We let $T: (\operatorname{\mathcal{C}}^0_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ denote the composite functor

Concretely, the functor $T$ carries the cone point of $(\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright }$ to the object $\iota _{\operatorname{\mathcal{C}}}(C) \in \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, and carries an object $(e: X \rightarrow C)$ of $\operatorname{\mathcal{C}}^{0}_{/C}$ to the object $\iota _{\operatorname{\mathcal{K}}}( \operatorname{id}_ X) \in \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$. We will complete the proof by verifying the following:

$(a)$

There exists a natural transformation of functors $\gamma ^{+}: T \rightarrow T^{+}$, which carries the cone point of $( \operatorname{\mathcal{C}}_{/C}^{0} )^{\triangleright }$ to the identity morphism $\iota _{\operatorname{\mathcal{C}}}( \operatorname{id}_ C )$, and carries each object $(e: X \rightarrow C) \in \operatorname{\mathcal{C}}_{/C}^{0}$ to the morphism $\beta _{X}$.

$(b)$

There exists a natural transformation of functors $\gamma ^{-}: T \rightarrow T^{-}$, which carries the cone point of $( \operatorname{\mathcal{C}}_{/C}^{0} )^{\triangleright }$ to the identity morphism $\iota _{\operatorname{\mathcal{C}}}( \operatorname{id}_ C )$ and carries each object $(e: X \rightarrow C)$ to the morphism of $\operatorname{\mathcal{K}}\subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ given by a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{ \operatorname{id}_ X } \ar [d]^{\operatorname{id}_ X} & X \ar [d]^{e} \\ X \ar [r]^-{e} & C } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$.

Assuming this has been done, we observe that the natural transformations $\overline{F}( \gamma ^{-} )$ and $\overline{F}( \gamma ^{+} )$ carry each object of $( \operatorname{\mathcal{C}}_{/C }^{0} )^{\triangleright }$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$ and therefore supply isomorphisms $\overline{F} \circ T^{-} \xleftarrow {\sim } \overline{F} \circ T \xrightarrow {|sim} \overline{F} \circ T^{+}$ in the $\infty $-category $\operatorname{Fun}( ( \operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright }, \operatorname{\mathcal{D}})$.

We begin by constructing the natural transformation $\gamma ^{+}$. Let $b': ( \operatorname{\mathcal{C}}_{/C}^{0} )^{\triangleright } \rightarrow \Delta ^1$ be the constant map taking the value $1$, so that there is a unique natural transformation $\xi : b \rightarrow b'$. Note that $\xi $ induces a natural transformation from $(b,c)$ to $(b',c)$ in the $\infty $-category $\operatorname{Fun}( ( \operatorname{\mathcal{C}}_{/C}^{0} )^{\triangleright }, \operatorname{\mathcal{C}}^0 \star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})$. Composing with the map $( \delta \star \operatorname{id}): \operatorname{\mathcal{C}}^0 \star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, we obtain a natural transformation $\gamma ^{+}: T \rightarrow T^{+}$ satisfying the requirements of $(a)$.

We now construct the natural transformation $\gamma ^{-}$. Note that $T$ and $T_{-}$ both carry $\operatorname{\mathcal{C}}^{0}_{/C}$ into $\operatorname{\mathcal{K}}$ and the cone point of $(\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright }$ to the object $\iota _{\operatorname{\mathcal{C}}}(C)$ and can therefore be identified with functors $T_{0}, T^{-}_{0}: \operatorname{\mathcal{C}}^{0}_{/C} \rightarrow \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Let $\sigma $ be an $n$-simplex of the product $\Delta ^1 \times \operatorname{\mathcal{C}}^{0}_{/C}$, which we identify with a pair $(\epsilon , \tau )$ where $\epsilon : [n] \rightarrow [1]$ is a nondecreasing function and $\tau : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ has the property that $\tau |_{\Delta ^ n}$ factors through $\operatorname{\mathcal{C}}^{0}$ and $\tau ( n+1 ) = C$. Let $\rho : \Delta ^1 \times \Delta ^{n} \rightarrow \Delta ^{n+1}$ and $\rho ': \Delta ^{n+1} \rightarrow \Delta ^{n+1}$ denote the maps given on vertices by the formulae

Then $(\rho \circ \tau ): \Delta ^1 \times \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ can be identified with an $n$-simplex of the simplicial set $\operatorname{\mathcal{K}}\subseteq \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$, so that $( \rho \circ \tau , \rho ' \circ \tau )$ is an $n$-simplex of $\operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. The construction $\sigma \mapsto ( \rho \circ \tau , \rho ' \circ \tau )$ depends functorially on $[n]$, and therefore determines a morphism of simplicial sets

We can identify this map with a natural transformation $\gamma ^{-}_{0}: T_0 \rightarrow T^{-}_{0}$, which then determines a natural transformation $\gamma ^{-}: T \rightarrow T^{-}$ satisfying the requirements of $(b)$. $\square$

Proof. Since $U$ is a cartesian fibration, an object $D \in \operatorname{\mathcal{D}}$ is $U$-initial if and only if it is initial when viewed as an object of the $\infty $-category $\operatorname{\mathcal{D}}_{C}$ for $C = U(D)$ (Corollary 7.1.5.21). It follows from Example 7.3.5.6 that there exists a functor $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ which satisfies condition $(\ast )$. Proposition 7.1.7.12 then guarantees that $F$ is an initial object of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Any other initial object of $\operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is isomorphic to $F$, and therefore also satisfies condition $(\ast )$. $\square$

Proof. The first assertion follows by applying the criterion of Proposition 7.3.5.5 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$, and the second assertion follows by a similar argument. $\square$

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$, and choose a morphism $u: C' \rightarrow C$ which exhibits $C'$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $\operatorname{\mathcal{C}}$. By virtue of Proposition 7.3.5.5, it will suffice to show that the lifting problem

admits a solution which is a $U$-colimit diagram if and only if $G(u)$ can be lifted to a $U$-cocartesian morphism $F_0(C') \rightarrow D$. This follows from Corollary 7.2.2.14, since $u$ is final when viewed as an object of the $\infty $-category $\operatorname{\mathcal{C}}^{0}_{/C}$. $\square$

Proof. Combine Corollary 7.3.5.10 with Example 6.2.2.5. $\square$

Proof of Proposition 7.3.5.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and suppose we are given diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, the composite map

has a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$. We wish to show that $F_0$ has a left Kan extension along $\delta $ (the converse assertion is immediate from the definitions, and the analogous assertion for right Kan extensions will follow by a similar argument). Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $\iota : K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. Since $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, we can extend $\delta $ and $F_0$ to functors $\overline{\delta }: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ and $\overline{F}_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, respectively (Proposition 4.1.3.1). For every object $C \in \operatorname{\mathcal{C}}$, the induced map $K \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} \hookrightarrow \operatorname{\mathcal{K}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is a categorical equivalence (Corollary 5.6.7.6), and therefore right cofinal (Corollary 7.2.1.13). Applying Proposition 7.2.2.9, we deduce that the composite map

has a colimit in $\operatorname{\mathcal{D}}$. Corollary 7.3.5.8 now guarantees that the functor $\overline{F}_0$ admits a left Kan extension $\overline{F}: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Set $F = \overline{F}|_{\operatorname{\mathcal{C}}}$. Applying Proposition 7.3.2.11, we obtain a natural transformation $\overline{\beta }: \overline{F}_0 \rightarrow F \circ \overline{\delta }$ which exhibits $F$ as a left Kan extension of $\overline{F}_0$ along $\overline{\delta }$. Since $\iota $ is a categorical equivalence, it follows that $\overline{\beta }$ restricts to a natural transformation $F_0 \rightarrow F \circ \delta $ which exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ (Proposition 7.3.1.15). $\square$