Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Lemma 5.1.2.6. Let $q: X \rightarrow S$ be a left fibration of simplicial sets, let $s \in S$ be a vertex having the property that the Kan complex $X_{s} = \{ s\} \times _{S} X$ is contractible, and let $\overline{\sigma }: \Delta ^ n \rightarrow S$ be an $n$-simplex of $S$ satisfying $\overline{\sigma }(n) = s$. Then every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r]^-{ \sigma _0 } \ar [d] & X \ar [d]^{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^-{\sigma } & S } \]

admits a solution.

Proof. When $n=0$, the desired result follows from the fact that the fiber $X_{s}$ is nonempty. We may therefore assume without loss of generality that $n > 0$. Replacing $q$ by the projection map $\Delta ^{n} \times _{S} X \rightarrow \Delta ^ n$, we may further reduce to the special case where $S = \Delta ^ n$ and $\overline{\sigma }$ is the identity map. In this case, our assumption that $q$ is a left fibration guarantees that $X$ is an $\infty $-category (Remark 4.1.1.9).

Let $\overline{h}: \Delta ^{1} \times \Delta ^{n} \rightarrow \Delta ^ n$ be the morphism given on vertices by $h(i,j) = \begin{cases} j & \textnormal{ if } i=0 \\ n & \textnormal{ if } i=1. \end{cases}$ Since the inclusion $\{ 0\} \times \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^1 \times \operatorname{\partial \Delta }^{n}$ is left anodyne (Proposition 4.2.3.3), our assumption that $q$ is a left fibration guarantees the existence of a morphism $h': \Delta ^1 \times \operatorname{\partial \Delta }^ n \rightarrow X$ satisfying $h'|_{ \{ 0\} \times \operatorname{\partial \Delta }^ n} = \sigma _0$ and $q \circ h' = \overline{h}|_{ \Delta ^1 \times \operatorname{\partial \Delta }^ n}$. We will complete the proof by showing that $h'$ can be extended to a map $h: \Delta ^1 \times \Delta ^ n \rightarrow X$ satisfying $q \circ h = \overline{h}$ (in this case, our original lifting problem admits the solution $\sigma = h|_{ \{ 0\} \times \Delta ^ n}$).

Let $Y(0) \subset Y(1) \subset Y(2) \subset \cdots \subset Y(n+1) = \Delta ^1 \times \Delta ^ n$ denote the filtration constructed in the proof of Lemma 3.1.2.9. Then $Y(0)$ can be described as the pushout

\[ (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \coprod _{( \{ 1\} \times \operatorname{\partial \Delta }^ n) } ( \{ 1\} \times \Delta ^ n ). \]

Using our assumption that the fiber $X_{s}$ is a contractible Kan complex, we see that $h'$ can be extended to a morphism of simplicial sets $h_0: Y(0) \rightarrow X$ satisfying $q \circ h_0 = \overline{h}|_{Y(0)}$. We claim that $h_0$ can be extended to a compatible sequence of maps $h_{i}: Y(i) \rightarrow X$ satisfying $q \circ h_ i = \overline{h}|_{Y(i)}$. To prove this, we recall that each $Y(i+1)$ can be realized as a pushout of the horn inclusion $\Lambda ^{n+1}_{i+1} \hookrightarrow \Delta ^{n+1}$, so that the construction of $h_{i+1}$ from $h_{i}$ can be rephrased as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i+1} \ar [r]^-{f_ i} \ar [d] & X \ar [d]^{q} \\ \Delta ^{n+1} \ar [r] \ar@ {-->}[ur] & S. } \]

For $0 \leq i < n$, this lifting problem is automatically solvable by virtue of our assumption that $q$ is a left fibration. In the case $i = n$, the edge

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n, n+1\} ) \hookrightarrow \Lambda ^{n+1}_{n+1} \xrightarrow {f_ i} X \]

is an edge of the Kan complex $X_{s}$, and is therefore an isomorphism in the $\infty $-category $X$ (Proposition 1.3.6.11). In this case, the existence of the desired extension follows from Theorem 4.4.2.5. We complete the proof by taking $h = h_{n+1}$. $\square$