Kerodon

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

Warning 9.1.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Every lifting problem

9.9
\begin{equation} \begin{gathered}\label{equation:lifting-problem-in-homotopy} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u_0} & X \ar [d]^{g} \\ B \ar [r]^-{ \overline{u} } \ar@ {-->}[ur] & Y } \end{gathered} \end{equation}

in $\operatorname{\mathcal{C}}$ determines a lifting problem in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, given by the diagram

9.10
\begin{equation} \begin{gathered}\label{equation:lifting-problem-in-homotopy2} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{[f]} \ar [r]^-{[u_0]} & X \ar [d]^{[g]} \\ B \ar [r]^-{ [\overline{u}] } \ar@ {-->}[ur] & Y. } \end{gathered} \end{equation}

Moreover, every solution to the lifting problem (9.9) determines a solution to the lifting problem (9.10). Beware that the converse is false: it is possible for the lifting problem (9.10) to admit a solution when the lifting problem (9.9) does not (Exercise 9.1.5.6).