Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 3.4.1.3. Suppose we are given a commutative diagram of simplicial sets

3.39
\begin{equation} \label{diagram:homotopy-pullback-square3} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X, } \end{gathered}\end{equation}

where $q$ is a Kan fibration. Applying Proposition 3.4.1.2 to the factorization $q = q \circ \operatorname{id}_{X_0}$, we see that (3.39) is a homotopy pullback square if and only if the induced map $X_{01} \rightarrow X_0 \times _{X} X_{1}$ is a weak homotopy equivalence. In particular, if (3.39) is a pullback diagram, then it is also a homotopy pullback diagram. Beware that this conclusion is generally false when $q$ is not a Kan fibration.