Kerodon

Proof. Let $S$ be the collection of all morphisms of simplicial sets $f: A \rightarrow B$ having the property that, for every morphism $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the map $\theta _{g}$ is inner anodyne. It follows immediately from the definitions that $S$ is weakly saturated (in the sense of Definition 1.5.4.12). Consequently, to show that every inner anodyne morphism belongs to $S$, it will suffice to prove that $S$ contained every inner horn inclusion $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. Using Remark 5.3.2.3, we can reduce to the case where $\operatorname{\mathcal{C}}= [n]$ and $g: \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the identity map. In this case, Remark 5.3.2.14 shows that $\theta _{g}$ is a pushout of the inclusion map $\Lambda ^{n}_{i} \times \mathscr {F}(0) \hookrightarrow \Delta ^ n \times \mathscr {F}(0)$, which is inner anodyne by virtue of Lemma 1.5.7.5. $\square$

Proof. By virtue of Corollary 4.5.7.3, we may assume without loss of generality that $S = \Delta ^ n$ is a standard simplex. Replacing $\operatorname{\mathcal{C}}$ by the category $[n] = \{ 0 < 1 < \cdots < n \} $, we are reduced to proving that $\lambda $ is a categorical equivalence, which follows from Theorem 5.3.5.7. $\square$

Proof. We will give the proof under the assumption that $U$ is a cocartesian fibration; the proof when $U$ is a cartesian fibration is similar. Let $S$ be the collection of all monomorphisms of simplicial sets $f: A \hookrightarrow B$ with the following property: for every morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$, the induced map $A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a categorical equivalence. To complete the proof, it will suffice to show that the morphism $\overline{F}: \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ belongs to $S$. In fact, we claim that every inner anodyne morphism of simplicial sets belongs to $S$. Using Remark 4.5.3.6, Remark 4.5.3.5, Corollary 4.5.7.2, and Remark 4.5.4.13, we see that $S$ is weakly saturated (see Definition 1.5.4.12). It will therefore suffice to show that $S$ contains every inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. In particular, we are reduced to proving Lemma 5.3.6.5 in the special case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is the nerve of a category $\operatorname{\mathcal{C}}_0$. Applying Corollary 5.3.4.9, we deduce that there exists a diagram of $\infty $-categories $\mathscr {G}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \rightarrow \operatorname{\mathcal{E}}$. We then have a commutative diagram of simplicial sets

where the vertical maps are categorical equivalences (Lemma 5.3.6.4). Consequently, to show that $F$ is a categorical equivalence, it will suffice to show that $\widetilde{F}$ is a categorical equivalence, which follows from Lemma 5.3.6.3. $\square$

Proof of Proposition 5.3.6.1. Without loss of generality we may assume that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of $\infty $-categories. Suppose we are given a commutative diagram of simplicial sets

where both squares are pullbacks and $\overline{F}$ is a categorical equivalence. We wish to show that $F$ is also a categorical equivalence. By virtue of Proposition 4.1.3.2, the morphism $\overline{G}$ factors as a composition $\operatorname{\mathcal{C}}' \xrightarrow {\overline{G}'} \operatorname{\mathcal{B}}\xrightarrow { \overline{G}'' } \operatorname{\mathcal{C}}$, where $\overline{G}'$ is inner anodyne and $\overline{G}''$ is an inner fibration. Note that the projection map $V: \operatorname{\mathcal{B}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{B}}$ is a cocartesian fibration of $\infty $-categories. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{B}}$ and thereby reduce to the special case where $\overline{G}$ is inner anodyne. In this case, the morphism $G: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets (Lemma 5.3.6.5). Consequently, to show that $F$ is a categorical equivalence of simplicial sets, it will suffice to show that the composite map $(G \circ F): \operatorname{\mathcal{E}}'' \rightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets (Remark 4.5.3.5).

Since $\overline{F}$ is a categorical equivalence and $\overline{G}$ is inner anodyne, it follows that the composite map $\overline{G} \circ \overline{F}: \operatorname{\mathcal{C}}'' \rightarrow \operatorname{\mathcal{C}}$ is also a categorical equivalence. Applying Proposition 4.1.3.2, we can factor $\overline{G} \circ \overline{F}$ as a composition $\operatorname{\mathcal{C}}'' \xrightarrow {\overline{F}_0} \operatorname{\mathcal{C}}'_0 \xrightarrow {\overline{G}_0} \operatorname{\mathcal{C}}$, where $\overline{F}_0$ is inner anodyne and $\overline{G}_0$ is an inner fibration. Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, it follows that $\operatorname{\mathcal{C}}'_0$ is also an $\infty $-category (Remark 4.1.1.9). Set $\overline{\operatorname{\mathcal{E}}}'_0 = \operatorname{\mathcal{C}}'_{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, so that we have a commutative diagram

satisfying $G \circ F = G_0 \circ F_0$. Since $U$ is an isofibration (Proposition 5.1.4.9) and $\overline{G}_0$ is an equivalence of $\infty $-categories, it follows that $G_0$ is an equivalence $\infty $-categories (Corollary 4.5.2.29). Applying Lemma 5.3.6.5 to the square on the left, we see that $F_0$ is a categorical equivalence of simplicial sets. Invoking Remark 4.5.3.5, we deduce that $G \circ F = G_0 \circ F_0$ is also a categorical equivalence, as desired. $\square$

Proof. Apply Proposition 5.3.6.6 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (and use Example 5.1.1.4). $\square$

Proof. Replacing $\operatorname{\mathcal{C}}$ by the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$, we can assume without loss of generality that $\operatorname{\mathcal{B}}= \Delta ^1$ and $f$ is the nondegenerate edge of $\operatorname{\mathcal{B}}$. By virtue of Corollary 5.2.4.4, we can choose a functor $R: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{B'}$ such that $R|_{ \operatorname{\mathcal{C}}_{B} } = f_{!}$, $R|_{ \operatorname{\mathcal{C}}_{B'} } = \operatorname{id}$, and $R$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{C}}_{B'}$. Precomposition with functor $(U,R): \operatorname{\mathcal{C}}\rightarrow \Delta ^1 \times \operatorname{\mathcal{C}}_{B'}$ then determines a functor

Unwinding the definitions, we see that $H|_{ \{ 0\} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{B'}, \operatorname{\mathcal{D}}) }$ is the functor given by precomposition with $f_{!}$, and that $H|_{ \{ 1\} \times \operatorname{Fun}( \operatorname{\mathcal{C}}_{B'}, \operatorname{\mathcal{D}}) }$ is the identity functor from $\operatorname{Fun}( \operatorname{\mathcal{C}}_{B'}, \operatorname{\mathcal{D}})$ to itself. It will therefore suffice to show that, for each object $F \in \operatorname{Fun}( \operatorname{\mathcal{C}}_{B'}, \operatorname{\mathcal{D}})$, the restriction $H|_{ \Delta ^1 \times \{ F\} }$ is a $\pi $-cartesian morphism of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$. By virtue of Corollary 5.3.6.8, this is equivalent to the assertion that the composite functor $\operatorname{\mathcal{C}}\xrightarrow {R} \operatorname{\mathcal{C}}_{B'} \xrightarrow {F} \operatorname{\mathcal{D}}$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$. This follows from our assumption on $R$, since the functor $F$ carries isomorphisms of $\operatorname{\mathcal{C}}_{B'}$ to isomorphisms of $\operatorname{\mathcal{D}}$. $\square$

Proof. Let $n \geq 2$ and suppose we are given a lifting problem

where $\sigma _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n}$ to $e$; we wish to show that this lifting problem admits a solution. Replacing $U$ with the projection map $\Delta ^{n} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$, we can assume without loss of generality that $\operatorname{\mathcal{B}}= \Delta ^ n$ is a standard simplex, so that $\overline{\sigma }$ corresponds to a morphism of simplicial sets $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$. Invoking the universal property of the simplicial sets $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ (Proposition 4.5.9.5), we can rewrite (5.30) as a lifting problem

Note that since the edge $e$ satisfies condition $(\ast )$, the morphism $F_0$ satisfies the following condition:

• If $u$ is a $U$-cocartesian edge of $\operatorname{\mathcal{C}}$ lying over the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Delta ^ n$, then $F_0(u)$ is a $V$-cartesian edge of $\operatorname{\mathcal{D}}$.

Using Corollary 5.3.4.9, we can choose a diagram of $\infty $-categories $\mathscr {F}: [n] \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{C}}$. Set $\operatorname{\mathcal{C}}' = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\operatorname{\mathcal{C}}'_0 = \Lambda ^{n}_{n} \times _{ \Delta ^ n } \operatorname{\mathcal{C}}'$, so that $\lambda $ restricts to a map $\lambda _0: \operatorname{\mathcal{C}}'_0 \rightarrow \operatorname{\mathcal{C}}_0$. We then have a commutative diagram of $\infty $-categories

Since $V$ is an isofibration (Proposition 5.1.4.9), the vertical maps in this diagram are isofibrations (Proposition 4.5.5.14). Since $\lambda $ and $\lambda _0$ are categorical equivalences of simplicial sets (Lemma 5.3.6.4), the horizontal maps are equivalences of $\infty $-categories. Applying Corollary 4.5.2.32, we deduce that the upper horizontal map in the diagram (5.32) restricts to an equivalence from each fiber of the left vertical map to the corresponding fiber of the right vertical map. Consequently, we can replace (5.31) with the lifting problem

Using Remark 5.3.2.12, we obtain a pushout square

Let us identify $F_0 \circ \lambda _0 \circ G_0$ with a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{n} \rightarrow \operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{D}})$, and $\overline{F} \circ \lambda \circ G$ with an $n$-simplex $\overline{\tau }$ of $\operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}})$, so that we can rewrite (5.33) again as a lifting problem

To show that this lifting problem admits a solution, it will suffice to show that $\tau _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} )$ of $\Lambda ^{n}_{n}$ to a $V''$-cocartesian edge of the simplicial set $\operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}})$. Since $\lambda $ is a scaffold, this follows by combining $(\ast ')$ with the criterion of Theorem 5.2.1.1. $\square$

Proof. As in the proof of Lemma 5.3.6.11, we may assume without loss of generality that $\operatorname{\mathcal{B}}= \Delta ^1$, so that $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\overline{e}$ corresponds to a morphism $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$. Replacing $\operatorname{\mathcal{D}}$ by the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, we can further reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{E}}$ and $T$ is the identity functor (so that $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration of $\infty $-categories). Let $\operatorname{\mathcal{C}}(0)$ and $\operatorname{\mathcal{C}}(1)$ denote the fibers of $\operatorname{\mathcal{C}}$ over the the vertices $0,1 \in \Delta ^1$, so that we can identify $Y$ with a functor $\operatorname{\mathcal{C}}(1) \rightarrow \operatorname{\mathcal{D}}$ such that $V \circ Y$ is the inclusion map $\operatorname{\mathcal{C}}(1) \hookrightarrow \operatorname{\mathcal{C}}$. Applying Proposition 5.2.2.8, we can choose a functor $F: \operatorname{\mathcal{C}}(0) \rightarrow \operatorname{\mathcal{C}}(1)$ and a diagram

which exhibits $F = H|_{ \{ 1\} \times \operatorname{\mathcal{C}}(0) }$ as given by covariant transport along the nondegenerate edge of $\operatorname{\mathcal{B}}= \Delta ^1$. Since $V$ is a cartesian fibration, Proposition 5.2.1.3 guarantees that the lifting problem

admits a solution with the property that, for every object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}(0)$, the restriction $G|_{ \Delta ^1 \times \{ C\} }$ is a $V$-cartesian morphism of $\operatorname{\mathcal{D}}$.

Let $\operatorname{\mathcal{C}}' = (\Delta ^1 \times \operatorname{\mathcal{C}}(0)) {\coprod }_{ (\{ 1\} \times \operatorname{\mathcal{C}}(0)) } \operatorname{\mathcal{C}}(1)$ denote the mapping cylinder of the functor $F$. Amalgamating $H$ with the inclusion map $\operatorname{\mathcal{C}}(1) \hookrightarrow \operatorname{\mathcal{C}}$, we obtain a morphism of simplicial sets $\overline{H}: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which is a categorical equivalence by virtue of Corollary 5.2.4.2. Amalgamating $G$ with $Y$, we obtain a diagram $\overline{G}: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$ satisfying $V \circ \overline{G}= \overline{H}$. We have a commutative diagram of simplicial sets

where the horizontal maps are equivalences of $\infty $-categories. Since $V$ is an isofibration, the vertical maps in this diagram are isofibrations (Proposition 4.5.5.14). Applying Corollary 4.5.2.32, we deduce that the upper horizontal map in the diagram (5.32) restricts to an equivalence of the fibers of the vertical maps over the object $Y \in \operatorname{Fun}_{ /\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}(1), \operatorname{\mathcal{D}})$. It follows that there there exists a functor $E: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ such that $V \circ E = \operatorname{id}_{\operatorname{\mathcal{C}}}$, $E|_{\operatorname{\mathcal{C}}(1)} = Y$, and $E \circ \overline{H}$ is isomorphic to $\overline{G}$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}})$. By construction, we can identify $E$ with an edge $e: X \rightarrow Y$ of $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ satisfying $V'(e) = \overline{e}$. To complete the proof, it will suffice to show that $e$ satisfies condition $(\ast )$ of Proposition 5.3.6.6. Let $f: C \rightarrow C'$ be a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$; we wish to show that $E(f)$ is a $V$-cartesian morphism of $\operatorname{\mathcal{D}}$. Without loss of generality, we may assume that $U(f)$ is the nondegenerate edge of $\operatorname{\mathcal{B}}= \Delta ^1$ (otherwise, $f$ is an isomorphism and there is nothing to prove). By virtue of Remark 5.1.3.8, we can assume without loss of generality that $f: C \rightarrow F(C)$ is the $U$-cocartesian morphism given by the restriction $H|_{ \Delta ^1 \times \{ C\} }$. In this case, $E(f)$ is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$) to the $V$-cartesian morphism $G|_{ \Delta ^1 \times \{ C\} }$, and is therefore also $V$-cartesian (Corollary 5.1.2.5). $\square$

Proof of Proposition 5.3.6.6. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a cocartesian fibration of simplicial sets, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cartesian fibration of simplicial sets, and let $V': \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ be given by postcomposition with $V$. We first claim that $V'$ is an inner fibration. To prove this, we may assume without loss of generality that $\operatorname{\mathcal{B}}$ is a standard simplex, so that $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ is a cocartesian fibration of $\infty $-categories. Proposition 5.3.6.1 then guarantees that $U$ is exponentiable, so that $V'$ is an isofibration (Proposition 4.5.9.18) and therefore an inner fibration by virtue of Remark 4.5.5.7.

Let us say that an edge of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ is special if it satisfies condition $(\ast )$ appearing in the statement of Proposition 5.3.6.6. Lemma 5.3.6.11 guarantees that every special edge of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ is $V'$-cartesian. Moreover, if $Y$ is a vertex of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ and $\overline{e}: \overline{X} \rightarrow V'(Y)$ is an edge of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$, then Lemma 5.3.6.12 guarantees that there exists a special edge $e: X \rightarrow Y$ of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ satisfying $V'(e) = \overline{e}$. It follows that $V'$ is a cartesian fibration of simplicial sets.

To complete the proof of Proposition 5.3.6.6, we must show that every $V'$-cartesian edge $e: X \rightarrow Y$ of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ is special. Without loss of generality we may assume that $\operatorname{\mathcal{B}}= \Delta ^1$ and that $e$ lies over the nondegenerate edge of $\operatorname{\mathcal{B}}$, so that $e$ corresponds to a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Replacing $\operatorname{\mathcal{D}}$ by the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, we can assume that $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}$ and that $V \circ F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is the identity functor, so that $V$ is a cartesian fibration of $\infty $-categories. Using Lemma 5.3.6.12, we can choose a special edge $e': X' \rightarrow Y$ of $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}},\operatorname{\mathcal{D}})$ satisfying $V'(e') = V'(e)$, corresponding to another functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $V \circ F' = \operatorname{id}_{\operatorname{\mathcal{C}}}$. Since $e'$ is also $V'$-cartesian, it is isomorphic to $e$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \Delta ^1, \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}},\operatorname{\mathcal{D}}) )$. It follows that $F$ and $F'$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. If $u$ is a $U$-cocartesian edge of $\operatorname{\mathcal{C}}$, then $F(u)$ is isomorphic to the $V$-cartesian morphism $E'(u)$ (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$), and is therefore also $V$-cartesian (Corollary 5.1.2.5). $\square$