Kerodon

Proof. Suppose we are given integers $0 < i < n$; we wish to show that every lifting problem

admits a solution. Let $\alpha $ denote the composite map

If $\alpha $ is a constant morphism, then the existence of $\sigma $ is immediate. We may therefore assume without loss of generality that $\alpha $ is not constant. Write $\sigma _0 = (\alpha _0, \tau _0)$, where $\alpha _0 = \alpha |_{ \Lambda ^{n}_{i} }$ and $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{E}}$ is a morphism of simplicial sets, and let $\overline{\tau }$ denote the composite map $\Delta ^ n \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$. Since $U$ is an inner fibration, the lifting problem

admits a solution. We now observe that the pair $\sigma = (\alpha , \tau )$ can be regarded as an $n$-simplex of $\Delta ^1 \times \operatorname{\mathcal{E}}$ which solves the lifting problem (5.18). $\square$

Proof. Unwinding the definitions, we see that $F$ factors as a composition

where $G$ is a pullback of the inner fibration $\operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}$ of Lemma 5.2.3.12 and $H$ is a pullback of the inner fibration $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star \operatorname{\mathcal{D}}'$ of Proposition 4.3.3.24. $\square$

Proof of Proposition 5.2.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Applying Lemma 5.2.3.13, we see that the natural map

is an inner fibration of simplicial sets. Since $\Delta ^1$ is an $\infty $-category, it follows that $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category (Remark 4.1.1.9). $\square$

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

where $\sigma _0$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{n}_{0}$ to the edge $e$. If $\sigma '$ is contained in the simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$, then the lifting problem (5.23) admits a solution by virtue of assumption $(1)$. Let us therefore assume that $\sigma '$ is not contained in $\operatorname{\mathcal{C}}'$. Let $\rho : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be as in $(2)$, and define $\rho ': \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}'$ similarly. Unwinding the definitions, we can rewrite (5.23) as a lifting problem

which admits a solution by virtue of assumption $(2)$. $\square$

Proof. It follows from Lemma 5.2.3.12 that $W$ is an inner fibration of simplicial sets. Let us say that an edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special if it satisfies conditions $(1)$ and $(2)$ of Lemma 5.2.3.16, so that every special edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $W$-cocartesian. We consider three cases:

• Suppose that $X$ belongs to $\operatorname{\mathcal{C}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{C}}'$. In this case, our assumption that $U$ is a cocartesian fibration guarantees that we can lift $\overline{e}$ to $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Since $F(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{D}}$, the edge $e$ is special.

• Suppose that $X$ belongs to $\operatorname{\mathcal{C}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{D}}'$. In this case, we can identify $\overline{e}$ with an edge $\overline{e}_0: V( F(X) ) \rightarrow \overline{Y}$ of the simplicial set $\operatorname{\mathcal{D}}'$. Since $V$ is a cocartesian fibration, we can lift $\overline{e}_0$ to a $V$-cocartesian morphism $e_0: F(X) \rightarrow Y$ of $\operatorname{\mathcal{D}}$, which we can identify with a special edge $e: X \rightarrow Y$ of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $W(e) = \overline{e}$.

• Suppose that $X$ belongs to $\operatorname{\mathcal{D}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{D}}'$. In this case, our assumption that $V$ is a cocartesian fibration guarantees that we can lift $\overline{e}$ to $V$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which is then special when regarded as an edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$.

To complete the proof, it will suffice to show that every $W$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special. Applying the preceding argument, we can choose a special edge $e': X \rightarrow Y'$ satisfying $W(e') = W(e)$. Set $\overline{Y} = W(Y) = W(Y')$. Since $e$ and $e'$ are both $W$-cocartesian, Remark 5.1.3.8 supplies a $2$-simplex $\sigma $ of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ with boundary given by

where $u$ is an isomorphism in the $\infty $-category $\{ \overline{Y} \} \times _{ ( \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}' ) } (\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}})$. Applying Remark 5.1.3.8 to the cocartesian fibrations $U$ and $V$, we deduce that the edge $e$ is also special. $\square$

Proof of Proposition 5.2.3.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, so that the projection map $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^{1}$ of Example 5.2.3.18 is a cocartesian fibration. Note that the morphism

satisfies $H|_{ \{ 0\} \times \operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and $H|_{ \{ 1\} \times \operatorname{\mathcal{C}}} = F$. To complete the proof, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the restriction $H|_{ \Delta ^1 \times \{ C\} }$ is a $\pi $-cocartesian morphism $e: X \rightarrow F(X)$ in the $\infty $-category $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. This follows from the criterion of Example 5.2.3.18, since $e$ corresponds to the identity morphism $\operatorname{id}_{ F(X) }: F(X) \rightarrow F(X)$ under the identification of Remark 5.2.3.14. $\square$