Kerodon

Proof. We first prove $(1)$. Assume that $U$ and $V$ are inner fibration and that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$. We wish to show that $F$ is a categorical equivalence of simplicial sets. Fix an $\infty $-category $\operatorname{\mathcal{K}}$, and let $\theta _{F}: \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{K}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{K}})^{\simeq } )$ be the map given by precomposition with $F$. We wish to show that $\theta _{F}$ is a bijection. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$, so that precomposition with $G$ determines a map $\theta _{G}: \pi _0( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{K}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{K}})^{\simeq } )$. We claim that $\theta _{G}$ is an inverse of $\theta _{F}$. We will show that $\theta _{G}$ is a left inverse of $\theta _{F}$; a similar argument will show that $\theta _{G}$ is a right inverse of $\theta _{F}$. Fix a morphism $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{K}}$; we wish to show that $H$ is isomorphic to $H \circ G \circ F$ as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{K}})$. This is clear, since postcomposition with $H$ determines a functor of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{K}})$.

We now prove $(2)$. Let $Q$ be a contractible Kan complex containing a pair of distinct vertices $x$ and $y$, and form a pushout diagram of simplicial sets

Since the vertical maps are monomorphisms, this diagram is also a categorical pushout square (Proposition 4.5.4.11). In particular, if $F$ is a categorical equivalence, then the map $Q \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{M}}$ is also a categorical equivalence (Proposition 4.5.4.10). Since $Q$ is contractible, the inclusion $\{ y\} \times \operatorname{\mathcal{C}}\hookrightarrow Q \times \operatorname{\mathcal{C}}$ is a categorical equivalence (Remark 4.5.3.7), so the inclusion $\{ y\} \times \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{M}}$ is also a categorical equivalence. If $U$ is an isofibration, then the lifting problem

admits a solution, which we can identify with a pair of morphisms $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $u: Q \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ satisfying $u(x) = G \circ F$ and $u(y) = \operatorname{id}_{\operatorname{\mathcal{C}}}$. It follows that $G \circ F$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as an object of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{C}})$.

Repeating the above argument with $F$ replaced by $G$, we conclude that there exists a morphism $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ in $(\operatorname{Set_{\Delta }})_{/S}$ such that $H \circ G$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as an object of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{D}})$. Then $F$ and $H$ are both isomorphic to $H \circ G \circ F$ as objects of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, and are therefore isomorphic to each other. We may therefore assume without loss of generality that $H = F$, so that $G$ is a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$. In particular, $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$. $\square$

Proof. Our assumption that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex guarantees that the inner fibrations $U$ and $V$ are isofibrations (Example 4.4.1.6), so the desired result follows from Proposition 5.1.7.5. $\square$

Proof. We first show that $(1)$ implies $(2)$. Assume that $(1)$ is satisfied and let $B \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets; we wish to show that the induced map $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories. By virtue of Theorem 4.5.7.1, it will suffice to show that for every simplicial set $A$, the induced map

is a homotopy equivalence of Kan complexes. This follows by applying $(1)$ to the composite map $B' \times B \rightarrow B \rightarrow \operatorname{\mathcal{E}}$.

We now prove that $(2)$ implies $(3)$. Assume that condition $(2)$ is satisfied. Setting $B=\operatorname{\mathcal{D}}$, we deduce that composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{D}})$. In particular, there exists a morphism $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ in $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$ such that $F \circ G$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as an object of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}},\operatorname{\mathcal{D}})$. It follows that $F \circ G \circ F$ is isomorphic to $F$ as an object of the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$. Applying condition $(2)$ in the case $B = \operatorname{\mathcal{C}}$, we see that postcomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})$, so that $G \circ F$ is isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as an object of $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. It follows that $G$ is a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$. In particular, $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$.

The implication $(3) \Rightarrow (4)$ follows by combining Remark 5.1.7.4 with Corollary 5.1.7.8. We will complete the proof by showing that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied, and let $B$ be a simplicial set equipped with a morphism $B \rightarrow \operatorname{\mathcal{E}}$. We wish to show that composition with $F$ induces a homotopy equivalence of Kan complexes $\theta _{B}: \operatorname{Fun}_{/\operatorname{\mathcal{E}}}( B, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})^{\simeq }$. Assume first that the simplicial set $B$ has dimension $\leq n$, for some integer $n \geq -1$. Our proof proceeds by induction on $n$. If $n=-1$, then $B$ is empty and there is nothing to prove. We may therefore assume without loss of generality that $n \geq 0$. Let $A$ be the $(n-1)$-skeleton of $B$. Our inductive hypothesis guarantees that $\theta _{A}$ is a homotopy equivalence. By virtue of Proposition 3.2.8.1, it will suffice to verify the following:

$(\ast )$

The restriction maps

\[ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}}) \quad \quad \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}}) \]

are isofibrations of $\infty $-categories, and therefore induce Kan fibrations

\[ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})^{\simeq } \quad \quad \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})^{\simeq } \rightarrow \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}})^{\simeq }; \]

see Proposition 4.4.3.7.

$(\ast ')$

For every object $T \in \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})$, the induced map of fibers

\[ \{ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}}) \rightarrow \{ F \circ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}})} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}}) \]

is an equivalence of $\infty $-categories, and therefore induces a homotopy equivalence of Kan complexes

\[ \{ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{C}})^{\simeq }} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \{ F \circ T\} \times _{ \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(A,\operatorname{\mathcal{D}})^{\simeq }} \operatorname{Fun}_{/\operatorname{\mathcal{E}}}(B,\operatorname{\mathcal{D}})^{\simeq } \]

(see Remark 4.5.1.19).

Let $J$ denote the set of all nondegenerate $n$-simplices of $B$. Proposition 1.1.4.12 supplies a pushout diagram of simplicial sets

Consequently, to verify $(\ast )$ and $(\ast ')$, we can assume without loss of generality that $B = \Delta ^ n$ is a standard simplex and that $A = \operatorname{\partial \Delta }^ n$ is its boundary. Replacing $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ by the fiber products $\Delta ^ n \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}$ and $\Delta ^ n \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, we can reduce further to the case where $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex. Applying Example 4.4.1.6, we deduce that $U$ and $V$ are isofibrations, so that assertion $(\ast )$ follows from Proposition 4.4.5.1. Invoking assumption $(4)$, we deduce that $F$ is an equivalence of $\infty $-categories, and therefore induces equivalences

Assertion $(\ast ')$ now follows from Corollary 4.5.2.32.

We now treat the case where $B$ is a general simplicial set. For each $n \geq 0$, let $\operatorname{sk}_{n}(B)$ denote the $n$-skeleton of $B$ (Construction 1.1.4.1). Using $(\ast )$ and Corollary 4.5.6.22, we see that $\theta _{B}$ can be realized as the inverse limit of a tower

where each of the transition maps is a Kan fibration. The preceding arguments show that each of the vertical maps $\theta _{ \operatorname{sk}_ n(B) }$ is a homotopy equivalence of Kan complexes. Invoking Example 4.5.6.18, we deduce that $\theta _{B}$ is a homotopy equivalence of Kan complexes. $\square$

Proof of Proposition 5.1.7.10. Since $F$ and $G$ are isomorphic as objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, there exists a contractible Kan complex $X$ containing vertices $f$ and $g$ and a functor $H: X \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ satisfying $H(f) = F$ and $H(g) = G$. Let us identify $H$ with a morphism of simplicial sets $X \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, and let $\widetilde{\operatorname{\mathcal{C}}}$ denote the fiber product $(X \times \operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$. We will show that the inclusion maps

are equivalences of inner fibrations over $\operatorname{\mathcal{C}}$. To prove this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex (Proposition 5.1.7.9); in this case, we wish to show that both inclusion maps are equivalences of $\infty $-categories (Corollary 5.1.7.8). This follows by applying Corollary 4.5.2.29 to the diagram of pullback squares

here the vertical maps are isofibrations (since they are pullbacks of $V$) and the lower horizontal maps are equivalences of $\infty $-categories (since $X$ is a contractible Kan complex). $\square$

Proof. We first show that $(2)$ implies $(1)$. Suppose that there exists a morphism $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $G \circ F = \operatorname{id}_{\operatorname{\mathcal{C}}}$. Then $F$ and $G$ exhibit $\operatorname{\mathcal{C}}$ as a retract of $\operatorname{\mathcal{D}}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$. Since $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is an isofibration, it follows that $(U \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is an isofibration (Remark 4.5.5.10). In particular, $U \circ F$ is an inner fibration (Remark 4.5.5.7). If there exists an isomorphism $u: \operatorname{id}_{\operatorname{\mathcal{D}}} \rightarrow F \circ G$ in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, then $G$ is a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$, so that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$.

We now show that $(1)$ implies $(2)$. Assume that $U \circ F$ is an inner fibration and that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$. Let $G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$, so that there exists an isomorphism $e: \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G' \circ F$ in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. Applying Proposition 4.4.5.8, we can lift $e$ to an isomorphism $\widetilde{e}: G \rightarrow G'$ in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$, where $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ satisfies $G \circ F = \operatorname{id}_{\operatorname{\mathcal{C}}}$. Note that $F$ is a categorical equivalence of simplicial sets (Proposition 5.1.7.5), and therefore induces a categorical equivalence

Since $U$ is an isofibration, every lifting problem

admits a solution. In particular, there exists a morphism $u: \operatorname{id}_{\operatorname{\mathcal{D}}} \rightarrow F \circ G$ in the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$ whose image in $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is the identity map $\operatorname{id}_{F}$. We will complete the proof by showing that $u$ is an isomorphism in the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$. Using the criterion of Proposition 4.4.4.9, we are reduced to checking that, for each vertex $D \in \operatorname{\mathcal{D}}$ having image $E = U(D) \in \operatorname{\mathcal{E}}$, the induced map $u_{D}: D \rightarrow (F \circ G)(D)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$. This is clear, since $D$ is isomorphic (in the $\infty $-category $\operatorname{\mathcal{D}}_{E}$) to an object of the form $F(C)$ for $C \in \operatorname{\mathcal{C}}_{E}$, and the morphism $u_{F(C)}$ is equal to the identity $\operatorname{id}_{F(C)}$. $\square$

Proof. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$. We first prove $(1)$. Assume that $V$ is an isofibration; we will show that $U$ is also an isofibration. Choose a monomorphism of simplicial sets $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{Q}}$, where $\operatorname{\mathcal{Q}}$ is a contractible Kan complex (Exercise 3.1.7.11). Replacing $\operatorname{\mathcal{D}}$ by the product $\operatorname{\mathcal{D}}\times \operatorname{\mathcal{Q}}$, we can assume that $F$ is a monomorphism of simplicial sets. In this case, Lemma 5.1.7.12 guarantees that $F$ exhibits $\operatorname{\mathcal{C}}$ as a retract of $\operatorname{\mathcal{D}}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$, so that $U$ is an isofibration by virtue of Remark 4.5.5.10.

To prove $(2)$, we may assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex (Proposition 5.1.4.7). In this case, $U$ and $V$ are isofibrations (Example 4.4.1.6) and $F$ is an equivalence of $\infty $-categories (Corollary 5.1.7.8). It follows from Corollary 5.1.6.2 that $U$ is a cartesian fibration if and only if $V$ is a cartesian fibration.

To prove $(3)$, suppose that $U$ is a right fibration; we will show that $V$ is a right fibration. It follows from $(2)$ that $V$ is a cartesian fibration. It will therefore suffice to show that, for each vertex $E \in \operatorname{\mathcal{E}}$, the $\infty $-category $\{ E \} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is a Kan complex (Proposition 5.1.4.14). By virtue of Remark 5.1.7.4, the morphism $F$ induces an equivalence of $\infty $-categories $F_{E}: \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$. It will therefore suffice to show that $\{ E \} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}$ is a Kan complex (Remark 4.5.1.21), which follows from our assumption that $U$ is a right fibration.

Assertions $(4)$ and $(5)$ follow by similar arguments. Assertion $(6)$ follows by combining $(3)$ and $(5)$ (see Example 4.2.1.5). $\square$

Proof. By virtue of Proposition 5.1.7.9, we may assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex, so that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ if and only if it is an equivalence of $\infty $-categories (Corollary 5.1.7.8). Since $U$ and $V$ are isofibrations (Example 4.4.1.6), the desired result follows from Theorem 5.1.6.1. $\square$

Proof. Combine Proposition 5.1.7.14 with Proposition 5.1.4.14. $\square$