Kerodon

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

Example 3.4.1.4. Suppose we are given a commutative diagram of simplicial sets

3.40
\begin{equation} \label{diagram:homotopy-pullback-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r] & S, } \end{gathered}\end{equation}

where $q$ and $q'$ are Kan fibrations. Then (3.40) is a homotopy pullback square if and only if, for each vertex $s' \in S'$ having image $s \in S$, the induced map of fibers $X'_{s'} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes. This is essentially a restatement of Proposition 3.3.7.1 (by virtue of Proposition 3.4.1.2).