Well-pointed models for pointed homotopical constructions

Remark 2.4.13. A pointed topological space \((X,x_0)\) is called well-pointed if \((X,\{x_0\})\) is an NDR-pair. Under this hypothesis, the familiar reduced models compute the constructions in the preceding proposition. More precisely, for every pointed map \(f\colon X\to Y\), the collapse maps \(C(f)\to \widetilde C(f)\) and \(SX\to \Sigma X\) are homotopy equivalences. Likewise, if every \(X_i\) is well-pointed, then the strict wedge \(\bigvee _iX_i\) computes the homotopy wedge \(\bigvee _i^hX_i\). Thus the conclusions of the proposition apply in particular to reduced cofibers, reduced suspensions and strict wedges of pointed cell complexes whose basepoints are \(0\)-cells.