Kerodon

Proof. Combine Propositions 7.3.6.1 and 6.2.4.1. $\square$

Proof. Combine Corollaries 7.3.6.3 and 7.3.5.2. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.1 and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3) \Rightarrow (1)$. By assumption, there exists a functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta ': F_0 \rightarrow F' \circ \delta $ which exhibits $F'$ as a left Kan extension of $F$ along $\delta $. Applying Proposition 7.3.6.1, we deduce that there exists a natural transformation $\gamma : F' \rightarrow F$ for which $\beta $ is a composition of $\beta '$ with the induced transformation $\gamma |_{K}: (F' \circ \delta ) \rightarrow (F \circ \delta )$. For each object $G \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, we have a commutative diagram

where the right vertical map is bijective. If condition $(3)$ is satisfied, then the left vertical map is also bijective. Allowing the functor $G$ to vary, it follows that the homotopy class $[\gamma ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }$, so that $\gamma $ is an isomorphism in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Invoking Remark 7.3.1.13, we conclude that $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $. $\square$

Proof of Proposition 7.3.6.7. We will assume that the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (the proof in the case where $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ is similar). Using Corollary 4.5.2.23, we can factor the functor $U$ as a composition $\operatorname{\mathcal{D}}\xrightarrow {T} \operatorname{\mathcal{D}}' \xrightarrow {U'} \operatorname{\mathcal{E}}$, where $U'$ is an isofibration and $T$ is an equivalence of $\infty $-categories. Note that the functor $T \circ F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.15), and that the natural maps

are homotopy equivalences. Consequently, we can replace $\operatorname{\mathcal{D}}$ by $\operatorname{\mathcal{D}}'$ and thereby reduce to proving Proposition 7.3.6.7 in the special case where the functor $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is an isofibration of $\infty $-categories.

Let $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be as in Remark 7.3.6.8; we wish to show that $F$ is a $V$-initial object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Note that $V$ is also an isofibration (Proposition 4.4.5.1). By virtue of Corollary 7.1.5.17, it will suffice to show that every lifting problem

has a solution, provided that $n \geq 0$ and $\sigma _0(0) = F$. Unwinding the definitions, we can rewrite (7.29) as a lifting problem

Note that $V'$ is also an isofibration of $\infty $-categories (Proposition 4.4.5.1).

We will complete the proof by showing that the lifting problem (7.30) admits a solution $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\Delta ^ n, \operatorname{\mathcal{D}})$ which is $V'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. By virtue of Proposition 7.3.5.5, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the induced lifting problem

admits a solution $Q: (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$ which is a $V'$-colimit diagram. Our assumption that $\sigma _0(0) = F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ guarantees that the composite map

is a $U$-colimit diagram. Applying Corollary 7.1.7.9, we conclude that the lifting problem (7.31) admits a solution $Q$, and Proposition 7.1.7.12 guarantees that $Q$ is automatically a $V'$-colimit diagram. $\square$

Proof. Let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories and let $\operatorname{\mathcal{C}}^{0}$ be a full subcategory of $\operatorname{\mathcal{C}}$. Assume either that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ or that $G$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$. Applying Proposition 7.3.6.7 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$, we deduce that the restriction map

is a homotopy equivalence of Kan complexes. Since the restriction map $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}})$ is an inner fibration of $\infty $-categories (Corollary 4.1.4.2), the map $\theta $ is also a Kan fibration (Proposition 4.6.1.21), and therefore a trivial Kan fibration (Proposition 3.3.7.6). $\square$

Proof. Let $\operatorname{\mathcal{C}}^{\mathrm{init}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its initial objects. The morphism $\theta $ then factors as a composition

Our assumption guarantees that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{\mathrm{init}}$ (Corollary 7.3.3.13), so that the morphism $\theta '$ is a homotopy equivalence (Proposition 7.3.6.7). Since $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex (Corollary 4.6.7.14), the inclusion map $\{ C\} \hookrightarrow \operatorname{\mathcal{C}}^{\mathrm{init}}$ is a categorical equivalence of simplicial sets, which implies that $\theta ''$ is also a homotopy equivalence. $\square$

Proof. It follows from Corollary 7.3.6.10 that any object of $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ which satisfies conditions $(1)$ and $(2)$ is initial. It will therefore suffice to show that there exists an object $F \in \operatorname{Fun}_{ /\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $(1)$ and $(2)$ (any other initial object of $\operatorname{Fun}_{ /\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ will be isomorphic to $F$, and will therefore also satisfy $(1)$ and $(2)$).

Let $\operatorname{\mathcal{C}}^{\mathrm{init}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its initial objects, and let $D$ be an initial object of the $\infty $-category $\operatorname{\mathcal{D}}_{E}$. Since $\operatorname{\mathcal{C}}^{\mathrm{init}}$ is a contractible Kan complex, we can lift $\overline{F}|_{ \operatorname{\mathcal{C}}^{\mathrm{init}}}$ to a functor $F_0: \operatorname{\mathcal{C}}^{\mathrm{init}} \rightarrow \operatorname{\mathcal{D}}$ satisfying $F_0(C) = D$. Corollary 7.3.5.11 then guarantees that $F_0$ admits a $U$-left Kan extension $F \in \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, which satisfies condition $(2)$ by virtue of Corollary 7.3.3.13. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.7. To prove the converse, let $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a solution to the lifting problem (7.32) which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, and let $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ be as in Remark 7.3.6.8. If condition $(2)$ is satisfied, then $F$ and $F'$ are both $V$-initial objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $V(F) = V(F')$. Applying Corollary 7.1.5.12, we see that $F$ and $F'$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, so that $F$ is also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.17). $\square$

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows by applying Corollary 7.3.6.12 in the special case $\operatorname{\mathcal{E}}= \Delta ^{0}$. The equivalence $(2) \Leftrightarrow (3)$ is a special case of Proposition 3.3.7.6, since the morphism $\theta $ is automatically a Kan fibration (see Corollary 4.1.4.2 and Proposition 4.6.1.21). $\square$

Proof of Theorem 7.3.6.14. Note that the functor $V$ is an isofibration of $\infty $-categories (Proposition 4.4.5.1). It follows from Proposition 7.3.5.5 that $\operatorname{\mathcal{B}}$ is the essential image of the functor $V|_{ \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}$, and from Proposition 7.3.6.7 (together with Remark 7.3.6.8) that every object of $\operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is $V$-initial when regarded as an object of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Applying Corollary 7.1.5.18, we see that the functor $V|_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }: \operatorname{Fun}'(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is a trivial Kan fibration. $\square$

Proof. Apply Theorem 7.3.6.14 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$

Proof of Proposition 7.3.6.1. Let $F, G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories. Suppose we are given a simplicial set $K$ equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$, together with a natural transformation $\beta : F_0 \rightarrow F \circ \delta $ which exhibits $F$ as a left Kan extension of $F_0$ along $\delta $. Let $\theta $ denote the composite map

We wish to show that $\theta $ is a homotopy equivalence.

It follows from Corollary 4.1.3.3 that there exists an inner anodyne morphism $K \hookrightarrow \operatorname{\mathcal{K}}$, 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 $\delta ': \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ and $F'_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, respectively (Proposition 1.5.6.7). Moreover, the restriction functor $\operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{D}})$ is a trivial Kan fibration (Proposition 1.5.7.6). We can therefore extend $\beta $ to a natural transformation $\beta ': F'_0 \rightarrow F \circ \delta '$, which induces a map of Kan complexes $\theta ': \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F'_0, G \circ \delta ' )$. By construction, the map $\theta $ is obtained (up to homotopy) by composing $\theta '$ with the restriction map $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F'_0, G \circ \delta ' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F_0, G \circ \delta )$, which is a trivial Kan fibration. Consequently, to show that $\theta $ is a homotopy equivalence, it will suffice to show that $\theta '$ is a homotopy equivalence. We may therefore replace $K$ by $\operatorname{\mathcal{K}}$ and thereby reduce to proving Proposition 7.3.6.1 in the special case where $K = \operatorname{\mathcal{K}}$ is an $\infty $-category.

Let $\overline{\operatorname{\mathcal{C}}}$ denote the relative join $\operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$. Note that the definition of $\theta $ (as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$) depends only on the homotopy class of $\beta $. We may therefore assume without loss of generality that there exists a functor $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ for which $\overline{F}|_{\operatorname{\mathcal{K}}} = F_0$, $\overline{F}|_{\operatorname{\mathcal{C}}} = F$, and the natural transformation $\beta $ is given by the composition

Let $\overline{G}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ denote the functor given by the composition

Our assumption on $\beta $ guarantees that $\overline{F}$ is left Kan extended from the full subcategory $\operatorname{\mathcal{K}}\subseteq \overline{\operatorname{\mathcal{C}}}$ (Proposition 7.3.2.11). Applying Corollary 7.3.6.9, we deduce that precomposition with the inclusion $\operatorname{\mathcal{K}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$ determines a trivial Kan fibration

We claim that $\overline{G}$ is right Kan extended from the full subcategory $\operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$. To prove this, it will suffice to show that for every object $X \in \operatorname{\mathcal{K}}$, the functor $\overline{G}$ is right Kan extended from $\operatorname{\mathcal{C}}$ at $X$ (see Proposition 7.3.3.7). Let $e_{X}: X \rightarrow \delta (X)$ denote the morphism in $\overline{\operatorname{\mathcal{C}}}$ given by the edge

Note that $e_{X}$ is cocartesian with respect to the projection map $\overline{\operatorname{\mathcal{C}}} \rightarrow \Delta ^1$ (Proposition 5.2.3.15), and therefore exhibits $\delta (X)$ as a $\operatorname{\mathcal{C}}$-reflection of $X$ in the $\infty $-category $\overline{\operatorname{\mathcal{C}}}$ (Lemma 6.2.3.1). It will therefore suffice to show that $\overline{G}$ carries $e_{X}$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$, which is clear (by construction, $\overline{G}(e_ X)$ is the identity morphism $\operatorname{id}_{D}$ for $D = G( \delta (X) )$). Applying Corollary 7.3.6.9 again, we deduce that precomposition with the inclusion map $\operatorname{\mathcal{C}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$ determines a trivial Kan fibration

Let $\varphi _{\pm }: \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}})}( F \circ \delta , G \circ \delta )$ be given by precomposition with the functor $\operatorname{\mathcal{K}}\xrightarrow {\delta } \operatorname{\mathcal{C}}\hookrightarrow \overline{\operatorname{\mathcal{C}}}$. Consider the diagram of Kan complexes

Note that the diagonal maps are homotopy equivalences, and the triangle on the left is commutative. Consequently, to show that $\theta $ is a homotopy equivalence, it will suffice to show that the triangle on the right commutes up to homotopy.

Let $\operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta )$ be the Kan complex introduced in Notation 4.6.9.1. To verify the homotopy commutativity of the right triangle in the diagram (7.33), it will suffice to show that there is exists map of Kan complexes $\rho : \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta )$ satisfying the following conditions:

• The composition

  \[ \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta ) \]

  is the constant map taking the value $\beta $.

• The composition

  \[ \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, G \circ \delta ) \]

  is equal to $\varphi _{-}$.

• The composition

  \[ \operatorname{Hom}_{ \operatorname{Fun}( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) }( \overline{F}, \overline{G} ) \xrightarrow {\rho } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F_0, F \circ \delta , G \circ \delta ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta ) \]

  is equal to $\varphi _{\pm }$.

Let $\sigma $ denote the $2$-simplex of $\Delta ^1 \times \Delta ^1$ given on vertices by the formulae

and let $T: \Delta ^2 \times \operatorname{\mathcal{K}}\rightarrow \Delta ^1 \times \overline{\operatorname{\mathcal{C}}}$ be the functor given by the composition

More concretely, the functor $T$ is given on objects by the formulae

We conclude by observing that precomposition with $T$ induces a map of Kan complexes

having the desired properties. $\square$