Reduced homotopy pushouts and cofibers

Remark 2.1.5. In practice, we will most be interested in the reduced variants of each of these constructions: if \(X\), \(Y\) and \(Z\) are pointed topological spaces and \(f\) and \(g\) preserve the basepoints, then the reduced homotopy pushout is the quotient of \(X \sqcup _Z^h Y\) in which we demand that \((z_0,t) \simeq (z_0,0)\) for all \(t \in [0,1]\), where \(z_0 \in Z\) is the basepoint of \(Z\). The reduced form of the homotopy cofiber and suspension are denoted by \(\widetilde {C}(f)\) and \(\Sigma X\), respectively.