Remark 5.19.
While the proof of the Blakers–Massey theorem is quite involved, the dual Blakers–Massey theorem is easy to prove. It states that for a pullback square
if the absolute pushout product \(f \square g\) lies in \(L\), then the cogap map \(X \sqcup_W Y \to Z\) lies in \(L\). Indeed, the cogap map is the base change of
\[f\square g\colon (X\times Z)\sqcup_{X\times Y}(Z\times Y)\longrightarrow Z\times Z\]
along the diagonal \(\Delta_Z\colon Z\to Z\times Z\). The claim therefore follows from stability of \(L\) under base change.