Remark 3.4.0.11. The notions of homotopy pullback and homotopy pushout diagrams can be regarded as homotopy-invariant replacements for the usual notion of pullback and pushout diagrams, respectively. We will later make this heuristic precise by showing that a commutative diagram in the ordinary category of Kan complexes
\[ \xymatrix@C =50pt@R=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_1 \ar [r] & X } \]
is a homotopy pullback square (homotopy pushout square) if and only if it is a pullback square (pushout square) when regarded as a diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ of Kan complexes (Construction 5.5.1.1); see Examples 7.6.4.2 and 7.6.4.3.