Kerodon

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Example 7.6.3.4. Let $q: X \rightarrow S$ be a Kan fibration between Kan complexes, let $s \in S$ be a vertex, and let $X_{s}$ denote the fiber $\{ s\} \times _{S} X$. Then the commutative diagram of simplicial sets

7.57
\begin{equation} \begin{gathered}\label{equation:disagreement-of-fiber-products} \xymatrix@R =50pt@C=50pt{ X_{s} \ar [r] \ar [d] & X \ar [d]^-{q} \\ \{ s\} \ar [r] & S } \end{gathered} \end{equation}

is a homotopy pullback square (Example 3.4.1.3), and therefore induces a pullback square in the $\infty $-category $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})$ (see Example 7.6.4.2). However, if $X$ is contractible and $X_{s}$ is not, then (7.57) is not a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.