$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 5.4.3.11. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0: K_0 \rightarrow \operatorname{\mathcal{C}}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Let $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ denote the projection map, and suppose we are given a lifting problem
5.38
\begin{equation} \label{equation:slice-interior-fibration-cartesian} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \{ 1\} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d]^-{q} \\ \Delta ^{1} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}_{/f_0} } \end{gathered} \end{equation}
with the following property:
- $(\ast _0)$
For every vertex $x \in K_0$, the composition
\[ \Delta ^2 \simeq \Delta ^1 \star \{ x\} \hookrightarrow \Delta ^1 \star K_0 \xrightarrow { \overline{\sigma }} \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
Then there exists an edge $\sigma : \Delta ^1 \rightarrow \operatorname{\mathcal{C}}_{/f}$ which solves the lifting problem problem (5.38) and which satisfies the following stronger version of $(\ast _0)$:
- $(\ast )$
For every vertex $x \in K$, the composition
\[ \Delta ^2 \simeq \Delta ^1 \star \{ x\} \hookrightarrow \Delta ^1 \star K \xrightarrow { \sigma } \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
Proof.
Arguing as in the proof of Proposition 5.4.3.8, we can reduce to the case where $K = \Delta ^ n$ is a standard simplex and $K_0 = \operatorname{\partial \Delta }^ n$ is its boundary. In this case, the lifting problem (5.38) determines a morphism of simplicial sets
\[ \tau _0: ( \{ 1\} \star \Delta ^ n ) {\coprod }_{( \{ 1\} \star \operatorname{\partial \Delta }^ n) } (\Delta ^1 \star \operatorname{\partial \Delta }^ n) \rightarrow \operatorname{\mathcal{C}}, \]
whose source can be identified with the horn $\Lambda ^{n+2}_{1} \subseteq \Delta ^{n+2}$ (Lemma 4.3.6.15), and we wish to extend $\tau $ to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$. If $n > 0$, then the desired extension exists because $\tau _0$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (by virtue of assumption $(\ast _0)$). If $n = 0$, then our assumption that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category allows us to extend $\tau _0$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
$\square$