Kerodon

Proof. Replacing $\operatorname{\mathcal{C}}$ by a full simplicial subset if necessary, we may assume that $R$ contains every degenerate edge of $\operatorname{\mathcal{C}}$. Choose integers $0 < i < n$; we wish to show that every lifting problem

admits a solution. Using Proposition 8.1.3.7, we can rewrite (8.23) as a lifting problem

where the morphism $F$ is required to satisfy the following additional condition:

$(\ast )$

For every pair of integers $0 \leq i \leq j \leq n$, the morphism $F$ carries $(j,j) \rightarrow (i,j)$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.

Replacing $\operatorname{\mathcal{E}}$ by the fiber product $\operatorname{Tw}(\Delta ^ n) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, we can reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{Tw}( \Delta ^ n )$, and $\overline{F}$ is the identity morphism. Since $U$ is an inner fibration, it follows that $\operatorname{\mathcal{E}}$ is an $\infty $-category.

Suppose first that $n \geq 3$. In this case, $F_0$ determines a commutative diagram

in the $\infty $-category $\operatorname{\mathcal{E}}$. Using our assumption that $f_0$ factors through $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R}}(\operatorname{\mathcal{E}}) \subseteq \operatorname{Cospan}(\operatorname{\mathcal{E}})$ (together with Corollary 5.1.2.4), we deduce that the horizontal maps the diagram (8.24) are $U$-cocartesian. In particular, (8.24) is a $U$-colimit diagram (Proposition 7.6.2.26). Since the image of (8.24) in $\operatorname{Tw}(\Delta ^ n)$ is a pushout square, it is a pushout diagram in $\operatorname{\mathcal{E}}$ (Corollary 7.1.6.17). Applying Proposition 8.1.4.2, we deduce that $f_0|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{E}})$, and therefore also of $\operatorname{Cospan}^{ \mathrm{all}, \widetilde{R} }(\operatorname{\mathcal{E}})$ (Remark 8.1.6.4). It follows that $f_0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R} }( \operatorname{\mathcal{E}})$, which we can identify with a functor $F: \operatorname{Tw}( \Delta ^ n ) \rightarrow \operatorname{\mathcal{E}}$ satisfying condition $(\ast )$ and the identity $F|_{ \operatorname{Tw}( \Lambda ^{n}_{i} ) } = F_0$. The equality $U \circ F = \overline{F}$ is automatic, since $\operatorname{Tw}(\Delta ^ n)$ is the nerve of a partially ordered set and $\operatorname{Tw}( \Lambda ^{n}_{i} )$ contains every vertex of $\operatorname{Tw}( \Delta ^ n )$.

We now treat the case $n = 2$ (so that $i = 1$). In this case, we can identify $F_0$ with a diagram

in the $\infty $-category $\operatorname{\mathcal{E}}$, where the morphisms $u$ and $v$ are $U$-cocartesian. Our assumption that $\overline{f}$ factors through $\operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ guarantees that the morphism $(2,2) \rightarrow (0,2)$ belongs to $R$. Since morphisms of $R$ admit $U$-cocartesian lifts, we can choose a $U$-cocartesian morphism $w': X_{2,2} \rightarrow X_{0,2}$ in $\operatorname{\mathcal{E}}$, where $X_{0,2}$ belongs to the fiber over the object $(0,2) \in \operatorname{\mathcal{C}}$. Since $v$ is also $U$-cocartesian, we can choose a $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram

Since $\operatorname{\mathcal{E}}$ is an $\infty $-category, we can choose another $2$-simplex $\sigma _1$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram

Invoking our assumption that $u$ is $U$-cocartesian, we can choose another $2$-simplex $\sigma _2$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram

Using the fact that $\operatorname{\mathcal{E}}$ is an $\infty $-category, we obtain another $2$-simplex $\sigma _3$ of $\operatorname{\mathcal{E}}$ with boundary indicated in the diagram

The $2$-simplices $\sigma _0$, $\sigma _1$, $\sigma _2$, and $\sigma _3$ determine a functor $F: \operatorname{Tw}( \Delta ^2) \rightarrow \operatorname{\mathcal{C}}$ extending $F_0$, which we display informally as a diagram

Since the morphism $w'$ is $U$-cocartesian, the functor $F$ satisfies condition $(\ast )$ and can therefore be viewed as a solution to the lifting problem (8.23). $\square$

Proof of Proposition 8.1.9.1. Combine Proposition 8.1.9.9 with Example 8.1.9.7. $\square$

Proof. Let $f: X \rightarrow X_1$ be a morphism of $\operatorname{\mathcal{E}}$ which belongs to $\widetilde{L}$, and let $g': X \rightarrow X_0$ be a morphism of $\operatorname{\mathcal{E}}$ which belongs to $\widetilde{R}$. We wish to show that there exists a pushout diagram

in the $\infty $-category $\operatorname{\mathcal{E}}$, where $f'$ belongs to $\widetilde{L}$ and $g$ belongs to $\widetilde{R}$. Since $L$ and $R$ are pushout compatible, there exists a pushout diagram

in the $\infty $-category $\operatorname{\mathcal{C}}$, where $\overline{f}'$ belongs to $L$ and $\overline{g}$ belongs to $R$. Our assumption on $U$ guarantees that $\overline{g}$ can be lifted to a $U$-cocartesian morphism $g: X_0 \rightarrow X_{01}$ of $\operatorname{\mathcal{E}}$. Since $U$ is an inner fibration, the lower left half of (8.26) can be lifted to a $2$-simplex $\sigma $ of $\operatorname{\mathcal{E}}$ which we display as a diagram

Since $g'$ is $U$-cocartesian, we can then lift the upper right half of (8.26) to a $2$-simplex $\tau $ of $\operatorname{\mathcal{E}}$ which we display as a diagram

Amalgamating $\sigma $ and $\tau $, we obtain a diagram of the form (8.25), where $f' \in \widetilde{L}$ and $g \in \widetilde{R}$. We will complete the proof by showing that this diagram is a pushout square in the $\infty $-category $\operatorname{\mathcal{E}}$. Since (8.26) is a pushout square in $\operatorname{\mathcal{C}}$, it will suffice to show that 8.25 is a $U$-pushout square (Corollary 7.1.6.18). This is a special case of Proposition 7.6.2.26, since the horizontal morphisms appearing in the diagram are $U$-cocartesian. $\square$