Proposition 4.4.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset which contains every vertex of $B$, and suppose we are given a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times A) \coprod _{ (\{ 1\} \times A) } (\{ 1 \} \times B) \ar [r]^-{h_0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \times B \ar [r]^-{ \overline{h} } \ar@ {-->}[ur]^{h} & \operatorname{\mathcal{D}}} \]
with the following property:
- $(\ast )$
For every simplex $\tau : \Delta ^{n} \rightarrow B$ which is not contained in $A$ having final vertex $b = \tau (n)$, the edge
\[ \Delta ^1 \simeq \Delta ^1 \times \{ b\} \xrightarrow {h_0} \operatorname{\mathcal{C}} \]
is an isomorphism in $\operatorname{\mathcal{C}}$.
Then $h_0$ can be extended to a diagram $h: \Delta ^1 \times B \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{h} = F \circ h$.
Proof.
We proceed as in the proof of Lemma 4.4.4.8, with some minor modifications. Let $P$ denote the collection of all pairs $(K, h_ K)$, where $K \subseteq B$ is a simplicial subset containing $A$ and $h_ K: \Delta ^1 \times K \rightarrow \operatorname{\mathcal{C}}$ is a morphism of simplicial sets satisfying
\[ h_ K |_{\Delta ^1 \times A } = h_0 |_{ \Delta ^1 \times A } \quad \quad h_{K} |_{ \{ 1\} \times K} = h_0 |_{ \{ 1\} \times K }. \]
We regard $P$ as partially ordered set, where $(K, h_ K) \leq (K', h_{K'} )$ if $K \subseteq K'$ and $h_ K = h_{K'} |_{\Delta ^1 \times K}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma, and therefore has a maximal element $(K_{\mathrm{max}}, h_{K_{\mathrm{max}}})$. We will complete the proof by showing that $K_{\mathrm{max}} = B$. Assume otherwise. Then there exists some nondegenerate $n$-simplex $\tau : \Delta ^{n} \rightarrow B$ whose image is not contained in $K_{ \mathrm{max} }$. Choosing $n$ as small as possible, we can assume that $\tau $ carries the boundary $\operatorname{\partial \Delta }^{n}$ into $K_{\mathrm{max}}$. Note that, since $A$ contains every vertex of $B$, we must have $n > 0$. 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
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [d] \ar [r] & K_{\mathrm{max}} \ar [d] \\ \Delta ^{n} \ar [r] & K'. } \]
We will complete the proof by showing that the lifting problem
\[ \xymatrix@R =50pt@C=75pt{ (\Delta ^1 \times K_{\mathrm{max}}) \coprod _{ (\{ 1\} \times K_{\mathrm{max}})} (\{ 1\} \times K') \ar [r]^-{(h_ K, h_0|_{\{ 1\} \times K'})} \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^1 \times K' \ar [r] \ar@ {-->}[ur] & \Delta ^0 } \]
admits a solution, where the dotted arrow carries each edge $\Delta ^1 \times \{ x\} $ to an isomorphism in $\operatorname{\mathcal{C}}$ (contradicting the maximality of the pair $(K_{\mathrm{max}}, h_{K_{\mathrm{max}}} )$). To prove this, we can replace the inclusion $K_{\mathrm{max}} \hookrightarrow K'$ by $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$. We are therefore reduced to proving Lemma 4.4.4.8 in the special case where $B = \Delta ^ n$ is a simplex and $A = \operatorname{\partial \Delta }^ n$ is its boundary.
Let
\[ (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \cup (\{ 1\} \times \Delta ^ n) = X(0) \subset X(1) \subset X(2) \subset \cdots \subset X(n+1) = \Delta ^{1} \times \Delta ^ n \]
be the sequence of simplicial subsets appearing in the proof of Lemma 3.1.2.12, so that $h_0$ can be identified with a morphism of simplicial sets from $X(0)$ to $\operatorname{\mathcal{C}}$. We will show that, for $0 \leq i \leq n+1$, there exists a morphism of simplicial sets $h_ i: X(i) \rightarrow \operatorname{\mathcal{C}}$ satisfying $h_ i|_{X(0)} = h_0$ and $F \circ h_ i = \overline{h}|_{ X(i)}$ (taking $i = n+1$, this will complete the proof of Proposition 4.4.5.8). We proceed by induction on $i$, the case $i=0$ being vacuous. Assume that $i > 0$ and that we have already constructed a morphism $h_{i-1}: X(i-1) \rightarrow \operatorname{\mathcal{C}}$ satisfying $h_{i-1}|_{X(0)} = h_0$ and $F \circ h_{i-1} = \overline{h}|_{ X(i-1)}$. By virtue of Lemma 3.1.2.12, we have a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i} \ar [r]^-{\sigma _0} \ar [d] & X(i-1) \ar [d] \\ \Delta ^{n+1} \ar [r]^-{\sigma } & X(i). } \]
Consequently, to prove the existence of $h_{i}$, it suffices to solve the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i} \ar [r]^-{ h_{i-1} \circ \sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{n+1} \ar [r]^-{ \overline{h} \circ \sigma } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. } \]
For $0 < i < n+1$, the existence of the desired solution follows from our assumption that $F$ is an inner fibration. In the case $i = n+1$, the existence follows from Proposition 4.4.2.13, since the map $\sigma : \Delta ^{n+1} \rightarrow \Delta ^1 \times \Delta ^ n$ carries the final edge $\operatorname{N}_{ \bullet }( \{ n < n+1 \} ) \subseteq \Delta ^{n+1}$ to the edge $\Delta ^1 \times \{ n\} \subseteq \Delta ^1 \times \Delta ^ n$, which $h_0$ carries to an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ by virtue of assumption $(\ast )$.
$\square$