Kerodon

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

Example 3.4.1.5. Suppose we are given a commutative diagram of simplicial sets

3.42
\begin{equation} \label{diagram:homotopy-pullback-square3} \begin{gathered} \xymatrix { Y \ar [r] \ar [d] & X \ar [d]^{f} \\ T \ar [r] & S, } \end{gathered}\end{equation}

where $f$ is a Kan fibration. Then (3.42) is homotopy Cartesian if and only if the induced map $Y \rightarrow T \times _{S} X$ is a weak homotopy equivalence. In particular, if (3.42) is a pullback diagram, then it is also a homotopy pullback diagram. Beware that this conclusion is generally false if $f$ is not a Kan fibration.