Lemma 4.4.4.8. Let $r: Y \rightarrow S$ be an inner fibration of simplicial sets, let $\overline{F}: B \rightarrow S$ be any morphism of simplicial sets, let $A$ be a simplicial subset of $B$, let $n \geq 2$ be an integer. Let $\pi : B \times \Delta ^ n \rightarrow B$ be the projection map and suppose we are given a lifting problem
4.18
\begin{equation} \begin{gathered}\label{equation:isomorphism-extension} \xymatrix@R =50pt@C=50pt{ (A \times \Delta ^ n) \coprod _{ (A \times \Lambda ^{n}_{0} ) } (B \times \Lambda ^{n}_0) \ar [r]^-{F_0} \ar [d] & Y \ar [d]^{r} \\ B \times \Delta ^ n \ar [r]^-{ \overline{F} \circ \pi } \ar@ {-->}[ur]^{F} & S. } \end{gathered} \end{equation}
Assume that, for every vertex $b \in B$, the edge
\[ \Delta ^1 \simeq \{ b\} \times \operatorname{N}_{\bullet }( \{ 0,1\} ) \hookrightarrow B \times \Lambda ^{n}_{0} \xrightarrow {F_0} \{ \overline{F}(b) \} \times _{S} Y \]
is an isomorphism in the $\infty $-category $Y_{b} = \{ \overline{F}(b) \} \times _{S} X$. Then the lifting problem (4.18) admits a solution $F: B \times \Delta ^ n \rightarrow Y$.
Proof.
Let $P$ denote the collection of all pairs $(K, F_ K)$, where $K \subseteq B$ is a simplicial subset containing $A$ and $F_ K: K \times \Delta ^ n \rightarrow Y$ is a morphism of simplicial sets satisfying $F_{K}|_{ A \times \Delta ^ n} = F_0 |_{ A \times \Delta ^ n }$, $F_ K|_{ K \times \Lambda ^ n_0 } = F_0 |_{ K \times \Lambda ^{n}_0}$, and $r \circ F_{K} = (\overline{F} \circ \pi )|_{ K \times \Delta ^ n}$. We regard $P$ as partially ordered set, where $(K, F_ K) \leq (K', F_{K'} )$ if $K \subseteq K'$ and $F_ K = F_{K'} |_{K \times \Delta ^ n}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma, and therefore has a maximal element $(K_{\mathrm{max}}, F_{K_{\mathrm{max}}})$. We will complete the proof by showing that $K_{\mathrm{max}} = B$. Assume otherwise. Then there exists some nondegenerate $m$-simplex $\tau : \Delta ^{m} \rightarrow B$ whose image is not contained in $K_{ \mathrm{max} }$. Choosing $m$ as small as possible, we can assume that $\tau $ carries the boundary $\operatorname{\partial \Delta }^{m}$ into $K_{\mathrm{max}}$. Let $K' \subseteq B$ be the union of $K_{\mathrm{max}}$ with the image of $\tau $, so that we have a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [d] \ar [r] & K_{\mathrm{max}} \ar [d] \\ \Delta ^{m} \ar [r] & K'. } \]
We will complete the proof by showing that the lifting problem
\[ \xymatrix@R =50pt@C=75pt{ (K_{\mathrm{max}} \times \Delta ^ n) \coprod _{ (K_{\mathrm{max}} \times \Lambda ^ n_0 )} (K' \times \Lambda ^ n_0) \ar [r]^-{(F_{K_{\mathrm{max}}}, F_0|_{K' \times \Lambda ^{n}_{0}})} \ar [d] & Y \ar [d]^{r} \\ K' \times \Delta ^ n \ar [r] \ar@ {-->}[ur] & S } \]
admits a solution (contradicting the maximality of the pair $(K_{\mathrm{max}}, F_{K_{\mathrm{max}}} )$). To prove this, we can replace the inclusion $K_{\mathrm{max}} \hookrightarrow K'$ by $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^ m$. We are therefore reduced to proving Lemma 4.4.4.8 in the special case where $B = \Delta ^ m$ is a simplex and $A = \operatorname{\partial \Delta }^ m$ is its boundary. Replacing $r$ by the projection map $\Delta ^ m \times _{S} Y \rightarrow \Delta ^ m$, we may further assume that $S$ is an $\infty $-category.
Choose a sequence of simplicial subsets
\[ X(0) \subset X(1) \subset X(2) \subset \cdots \subset X(t) = \Delta ^{m} \times \Delta ^{n} \]
satisfying the requirements of Lemma 4.4.4.7, so that $F_0$ can be identified with a morphism $X(0) \rightarrow Y$. We will show that, for $0 \leq s \leq t$, there exists a morphism of simplicial sets $F_{s}: X(s) \rightarrow Y$ satisfying $F_ s|_{ X(0)} = F_0$ and $r \circ F_{s} = (\overline{F} \circ \pi )|_{X(s)}$ (taking $s = t$, this will complete the proof of Lemma 4.4.4.8). We proceed by induction on $s$, the case $s=0$ being vacuous. Assume that $s > 0$ and that we have already constructed a morphism $F_{s-1}: X(s-1) \rightarrow Y$ satisfying $F_{s-1}|_{X(0)} = F_0$ and $r \circ F_{s-1} = (\overline{F} \circ \pi )|_{X(s-1)}$ By construction, there exists integers $q \geq 2$, $0 \leq p < q$, and a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{q}_{p} \ar [r]^-{\sigma _0} \ar [d] & X(s-1) \ar [d] \\ \Delta ^{q} \ar [r]^-{\sigma } & X(s). } \]
Moreover, in the special case $p=0$, we can assume that $\sigma (0) = (0,0)$ and $\sigma (1) = (0,1)$, so that the composite map
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^{q}_{p} \xrightarrow {\sigma _0} X(s-1) \xrightarrow { F_{s-1} } Y \]
corresponds to an isomorphism in $Y$. To construct the desired extension $F_{s}: X(s) \rightarrow Y$, it will suffice to solve a lifting problem of the form
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{q}_{p} \ar [r] \ar [d] & Y \ar [d]^{r} \\ \Delta ^{q} \ar [r] \ar@ {-->}[ur] & S. } \]
In the case $0 < p < q$, this lifting problem admits a solution by virtue of our assumption that $r$ is an inner fibration of simplicial sets. In the special case $p=0$, it follows from Proposition 4.4.2.13.
$\square$