Kerodon

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

Remark 4.5.5.17. Suppose we are given a lifting problem in the category of simplicial sets

4.39
\begin{equation} \begin{gathered}\label{equation:typical-lifting-problem} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f_0} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{ \overline{f} } \ar@ {-->}[ur] & S, } \end{gathered} \end{equation}

where $q$ is an isofibration and $i$ is a monomorphism. It follows from Corollary 4.5.5.16 that, if we regard the morphisms $q$, $i$, and $\overline{f}$ as fixed, then the existence of a solution to the lifting problem (4.39) depends only on the isomorphism class of $f$ as an object of the $\infty $-category $\operatorname{Fun}_{/S}(A,X)$.