Kerodon

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

Exercise 9.1.5.6. Let $g: X \rightarrow S$ be Kan fibration between Kan complexes, where $S$ is connected and $X$ is contractible. Choose a vertex $s \in S$ and let $X_{s}$ denote the fiber $\{ s\} \times _{S} X$, so that we have a commutative diagram of Kan complexes

9.11
\begin{equation} \begin{gathered}\label{equation:impossible-lifting-problem} \xymatrix@R =50pt@C=50pt{ X_{s} \ar [r] \ar [d] & X \ar [d]^{f} \\ \{ s\} \ar [r] & S } \end{gathered} \end{equation}

Show that:

  • In the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, the lifting problem determined by (9.11) admits a solution only if $S$ is contractible.

  • In the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{S}}}$, the lifting problem determined by (9.11) always has a solution.