CANCELLATION OF LOCAL SPHERES WITH RESPECT TO WEDGE AND CARTESIAN PRODUCT

Let C be a category of (pointed) spaces. For $X, Y \in C$ we denote the wedge (or one point union) by $X \vee Y$ and the cartesian product by $X \times Y$. Let $Z \in C$; we say that Z cancels with respect to wedge (resp. cartesian product) and C, if for all $X, Y \in C$ the existence of a homotopy equivalence $X \vee Z \to Y \vee Z$ implies the existence of a homotopy equivalence $X \to Y$ (resp. for cartesian product). If this does not hold, we say that there is a non-cancellation phenomenon involving Z (and C).