Kerodon

Proof. Assume that $\operatorname{ev}_0(e)$ is $U$-cocartesian; we will show that $e$ is $V$-cocartesian (the second assertion follows by a similar argument). Let $n \geq 2$; we wish to show that every lifting problem

admits a solution, provided that the composite map

coincides with $e$. Let $X(0)$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ n$ given by the union of $\operatorname{\partial \Delta }^1 \times \Delta ^ n$ with $\Delta ^1 \times \Lambda ^{n}_0$. Unwinding the definitions, we can rewrite (5.34) as a lifting problem

Choose a filtration

satisfying the requirements of Lemma 4.4.4.7. We will complete the proof by showing that, for each $s \leq t$, the morphism $\tau _0$ admits an extension $\tau _ s: X(s) \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ \tau _{s} = \overline{\tau }|_{ X(s) }$. The proof proceeds by induction on $s$, the case $s = 0$ being vacuous. Let us therefore assume that $0 < s \leq t$ and that $\tau _0$ has already been extended to a morphism of simplicial sets $\tau _{s-1}: X(s-1) \rightarrow \operatorname{\mathcal{C}}$. By construction, we have a pushout diagram of simplicial sets

for some $q \geq 2$ and $0 \leq p < q$. Consequently, to prove the existence of $\tau _ s$, it will suffice to show that $\tau _{s-1} \circ \varphi _0$ can be extended to a $q$-simplex of $\operatorname{\mathcal{C}}$ lying over the simplex $\overline{\tau } \circ \varphi : \Delta ^ q \rightarrow \operatorname{\mathcal{D}}$. For $p \neq 0$, the existence of this extension follows from our assumption that $U$ is an inner fibration. To handle the case $p= 0$, we observe that the morphism $\varphi $ carries the initial edge of $\Delta ^{q}$ to the edge $(0,0) \rightarrow (0,1)$ of $\Delta ^{1} \times \Delta ^ n$, so that $\tau _{s-1} \circ \varphi _0$ carries the initial edge of $\Delta ^ q$ to the edge $\operatorname{ev}_0(e)$ of $\operatorname{\mathcal{C}}$, which is $U$-cocartesian by assumption. $\square$

Proof. Let us say that an edge $e$ of $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is special if it satisfies condition $(\ast )$. We first show that if $e$ is a special edge of $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, then $e$ is $V$-cocartesian. Let $e_0$ and $e_1$ denote the images of $e$ in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively. Note that $V$ factors as a composition

Here $V'$ is a pullback of the projection map $\overline{V}': \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) = \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. Since $F( e_0 )$ is $U$-cocartesian, Lemma 5.3.7.1 implies that $e$ is $V'$-cocartesian. Moreover, $V''$ is a pullback of the product map $(U_0 \times U_1): \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}_0 \times \operatorname{\mathcal{D}}_1$. By assumption, $e_0$ is $U_0$-cocartesian and $e_1$ is $U_1$-cocartesian. It follows that $V'(e)$ is $V''$-cocartesian, so that $e$ is $V$-cocartesian by virtue of Remark 5.1.1.6.

Since $U_0$, $U_1$, and $U$ are inner fibrations, the morphisms $\overline{V}'$ and $(U_0 \times U_1)$ are also inner fibrations (see Proposition 4.1.4.1). It follows that $V'$ and $V''$ are inner fibrations (Remark 4.1.1.5), so that $V$ is an inner fibration (Remark 4.1.1.8). To show that $V$ is a cocartesian fibration, it will suffice to show that if $C$ is an object of $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ and $\overline{e}: V(C) \rightarrow \overline{C}'$ is an edge of $\operatorname{\mathcal{D}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$, then there exists a special edge $e: C \rightarrow C'$ satisfying $V(e) = \overline{e}$. Let us identify $C$ with a triple $(C_0, C_1, u)$ where $C_0$ is a vertex of $\operatorname{\mathcal{C}}_0$, $C_1$ is a vertex of $\operatorname{\mathcal{C}}_1$, and $u: F_0(C_0) \rightarrow F_1(C_1)$ is an edge of $\operatorname{\mathcal{C}}$. Similarly, we can identify $\overline{C}'$ with a triple $(\overline{C}'_0, \overline{C}'_1, \overline{u}' )$ where $\overline{C}_0$ is a vertex of $\operatorname{\mathcal{D}}_0$, $\overline{C}'_1$ is a vertex of $\operatorname{\mathcal{D}}_1$, and $\overline{u}': G_0( \overline{C}'_0 ) \rightarrow G_1( \overline{C}'_1 )$ is an edge of $\operatorname{\mathcal{D}}$. The edge $\overline{e}$ has images $\overline{e}_{0}: U_0(C_0) \rightarrow \overline{C}'_0$ an $\overline{e}_{1}: U_1(C_1) \rightarrow \overline{C}'_{1}$ in $\operatorname{\mathcal{D}}_0$ and $\operatorname{\mathcal{D}}_1$-respectively. Since $U_0$ is a cocartesian fibration, we can lift $\overline{e}_0$ to a $U_0$-cocartesian edge $e_0: C_0 \rightarrow C'_0$ of $\operatorname{\mathcal{C}}_0$. Similarly, we can lift $\overline{e}_1$ to a $U_1$-cocartesian edge $e_1: C_1 \rightarrow C'_1$ of $\operatorname{\mathcal{C}}_1$. The edge $\overline{e}$ also determines a map $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{D}}$, which we depict informally in the diagram

Using our assumption that $U$ is an inner fibration, we can lift the upper right triangle to a $2$-simplex $\sigma $:

of the simplicial set $\operatorname{\mathcal{C}}$. Using the fact that $F_0( e_0 )$ is $U$-cocartesian, we can lift the lower triangle to a $2$-simplex $\tau $

of $\operatorname{\mathcal{C}}$. Setting $C' = (C'_0, C'_1, w) \in \operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, we observe that the tuple $( e_0, e_1, \sigma , \tau )$ determines a special edge $e: C \rightarrow C'$ satisfying $V(e) = \overline{e}$.

We now complete the proof by showing that every $V$-cocartesian edge $f: C \rightarrow C''$ in $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is special. Using the preceding argument, we can choose a special edge $e: C \rightarrow C'$ satisfying $V(e) = V(f)$. Set $\overline{C}' = V( C' ) = V(C'' )$. Applying Remark 5.1.3.8, we deduce that there is a $2$-simplex $\rho :$

of the simplicial set $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, where $s$ is an isomorphism in the $\infty $-category $V^{-1}( \{ \overline{C}' \} )$. Applying Example 5.1.3.6, we deduce that the images of $s$ in $\operatorname{\mathcal{C}}_0$ is $U_0$-cocartesian, and the image of $s$ in $\operatorname{\mathcal{C}}_1$ is $U_1$-cocartesian. Since the collections of $U_0$-cocartesian and $U_1$-cocartesian edges are closed under composition (Corollary 5.1.2.4), we conclude that $f$ is also special. $\square$

Proof. Assertion $(1)$ follows by applying Proposition 5.3.7.2 in the special case $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}= \Delta ^0$ and $\operatorname{\mathcal{D}}_1 = \operatorname{\mathcal{C}}_1$. Assertion $(2)$ follows by a similar argument. $\square$

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $\Delta ^0 \diamond K$ denote the blunt join of Notation 4.5.8.3, and let $c: \Delta ^0 \diamond K \rightarrow \Delta ^{0} \star K = K^{\triangleleft }$ be the categorical equivalence of Theorem 4.5.8.8. We have a commutative diagram of $\infty $-categories

where the horizontal map is an equivalence of $\infty $-categories (Proposition 4.5.3.8) and the vertical maps are isofibrations (Corollary 4.4.5.3). Unwinding the definitions, we can identify $\operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}})$ with the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Under this identification, the functor $U'$ is given by projection onto the second factor, and is therefore a cocartesian fibration (Corollaryt 5.3.7.3). Applying Corollary 5.1.6.2, we deduce that $U$ is also a cocartesian fibration. Moreover, a morphism $e$ of $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}})$ is $U$-cocartesian if and only if its image in $\operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}})$ is $U'$-cocartesian (Proposition 5.1.6.6). Using the criterion of Corollary 5.3.7.3, we see that this is equivalent to the requirement that $e$ carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism in $\operatorname{\mathcal{C}}$. $\square$

Proof. Assertion $(1)$ follows from Proposition 5.3.6.6 (after passing to opposite simplicial sets). To prove $(2)$, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$ and $\pi (e)$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$. In this case, the simplicial sets $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are $\infty $-categories, and we can identify the edge $e$ with a morphism of simplicial sets $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ satisfying $V \circ E = \operatorname{id}_{\operatorname{\mathcal{D}}}$. Let $u: D \rightarrow D'$ be a morphism in the $\infty $-category $\operatorname{\mathcal{D}}_1 = \{ 1\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$; we wish to show that $E(u)$ is a $V$-cocartesian morphism of $\operatorname{\mathcal{E}}$. To prove this, let $G: \operatorname{\mathcal{D}}_{1} \rightarrow \operatorname{\mathcal{D}}_0 = \{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ be given by contravariant transport along the nondegenerate edge of $\operatorname{\mathcal{C}}$, so that we have a commutative diagram

in the $\infty $-category where the horizontal maps are $U$-cartesian. Our assumption that $e$ is $\pi $-cocartesian guarantees that the functor $E$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ (Proposition 5.3.6.6). We therefore obtain a commutative diagram

where the horizontal maps are $V$-cocartesian. By virtue of Corollary 5.1.2.4, it will suffice to show that the morphism $(E \circ G)(u)$ is $V$-cocartesian, which follows from our assumption that $X$ belongs to $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}})$. This completes the proof of $(2)$; assertions $(3)$ and $(4)$ then follow by applying Proposition 5.1.4.16. $\square$

Proof. Let $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ denote the projection map and let $f: \operatorname{\mathcal{C}}' \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ be a morphism satisfying $\pi \circ f = W$, corresponding to a morphism of simplicial sets $F: \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ for which $V \circ F$ is given by projection to the second factor. Note that we can regard $F$ as a vertex of the simplicial subset $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ if and only if it satisfies the following condition:

$(a)$

For every edge $(e',e)$ of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ for which $e'$ is a $W$-cocartesian edge of $\operatorname{\mathcal{C}}'$, the image $F(e',e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.

We wish to show that $(a)$ is equivalent to the following pair of conditions:

$(b)$

The morphism $f$ factors through the full simplicial subset $\operatorname{Res}^{\operatorname{CCart}}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. In other words, for every edge $(e',e)$ of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ for which $e'$ is a degenerate edge of $\operatorname{\mathcal{C}}'$, the image $F(e',e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.

$(c)$

For every $W$-cocartesian edge $e'$ of $\operatorname{\mathcal{C}}'$, the image $f(e')$ is a $\pi |_{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{E}}) }$-cocartesian edge of $\operatorname{Res}^{\operatorname{CCart}}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. By virtue of Propositions 5.3.7.9 and 5.3.6.6, this is equivalent to the assertion that for every edge $(e',e)$ of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ where $e'$ is $W$-cocartesian and $e$ is $U$-cartesian, the image $F(e',e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$.

The implications $(a) \Rightarrow (b)$ and $(a) \Rightarrow (c)$ are clear. For the converse, suppose that $(b)$ and $(c)$ are satisfied; we wish to prove $(a)$. Let $(e',e): (X',X) \rightarrow (Z',Z)$ be an edge of the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, where $e': X' \rightarrow Z'$ is $W$-cocartesian. Let $\overline{e} = U(e) = W(e')$ denote the corresponding edge of $\operatorname{\mathcal{C}}$. Since $U$ is a cartesian fibration, there exists a $U$-cartesian morphism $f: Y \rightarrow Z$ satisfying $U(f) = \overline{e}$. Let $\overline{\sigma }$ denote the left-degenerate $2$-simplex $s^{1}_0(\overline{e})$. Since $f$ is $U$-cartesian, we can lift $\overline{\sigma }$ to a $2$-simplex of $\operatorname{\mathcal{D}}$ as indicated in the diagram

Writing $\sigma '$ for the left-degenerate $2$-simplex $s^{1}_0(e')$ of $\operatorname{\mathcal{C}}'$, we obtain a $2$-simplex $\tau = F( \sigma ', \sigma )$ of $\operatorname{\mathcal{E}}$. It follows from assumption $(b)$ that the restriction $\tau |_{ \operatorname{N}_{\bullet }( \{ 0 < 1\} ) }$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$, and from assumption $(c)$ that the restriction $\tau |_{ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) }$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$. Applying Proposition 5.1.4.12, we conclude that $F(e',e) = \tau |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) }$ is also a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$. $\square$

Proof of Theorem 5.3.7.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, and let $\delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ be the diagonal embedding. Since $U$ is an isofibration (Proposition 5.1.4.8), the restriction map $\overline{\theta }: \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is also an isofibration (Corollary 4.5.5.16). Because $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ is a replete full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$, it follows that $\overline{\theta }$ restricts to an isofibration $\theta : \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. To prove Theorem 5.3.7.6, we will show that $\theta $ is an equivalence of $\infty $-categories (it is then automatically a trivial Kan fibration of simplicial sets: see Proposition 4.5.5.20).

Note that the functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ induces cocartesian fibrations $U': \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ and $U'': \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Let $\pi ': \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be given by projection onto the first factor, so that $\pi '$ is a cartesian fibration (Corollary 5.3.7.3). Let $\operatorname{\mathcal{M}}$ denote the cocartesian direct image $\operatorname{Res}^{\operatorname{CCart}}_{ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}})$ and let $T: \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map. Precomposition with the diagonal embedding $\delta : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ induces a restriction functor

which fits into a commutative diagram

It follows from Proposition 5.3.7.9 that $T$ is a cocartesian fibration and that $\delta ^{\ast }$ carries $T$-cocartesian morphisms of $\operatorname{\mathcal{M}}$ to $U''$-cocartesian morphisms of $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{E}}$. Using Proposition 5.3.7.10, we can identify $\theta $ with the map

given by postcomposition with $\delta ^{\ast }$. Consequently, to show that $\theta $ is an equivalence of $\infty $-categories, it will suffice to show that $\delta ^{\ast }$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. By virtue of Proposition 5.1.7.14), this can be checked fiberwise: that is, it suffices to show that for each object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers

is an equivalence of $\infty $-categories. This is a special case of Corollary 5.3.1.22, since $\delta (C)$ is an initial object of the $\infty $-category $\{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ (Proposition 4.6.7.22). $\square$