Kerodon

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Example 7.6.4.2. A (strictly) commutative diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_{1} \ar [r] & X }$

is a homotopy pullback square (in the sense of Definition 3.4.1.1) if and only if the induced diagram $\Delta ^1 \times \Delta ^1 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}$ is a pullback square in the $\infty$-category of spaces $\operatorname{\mathcal{S}}$. This follows by combining Propositions 7.5.4.13 and 7.5.4.5.