On the Functors Cwa and Pa

suspensions of A and attaching cones on suspensions of A (see [5, corollary 3.7]). If A = S, then a relative S–CW–complex is essentially an ordinary relative CW– complex. Any pointed map f : X → Y can be factored as a composition (X → Y ′ p → Y ), where (Y ′, X) is a relative A–CW–complex and p induces a weak equivalence of mapping spaces p? : map?(A, Y ′)→ map?(A, Y ). Let X be a pointed space. By factoring ?→ X, we get a map CWAX → X, where (CWAX, ?) is a relative A–CW–complex and the induced map map?(A,CWAX) → map?(A,X) is a weak equivalence. The assignment X 7→ CWAX can be made functorial, in such a way that the map CWAX → X is natural. By factoring X → ?, we get a map X → PAX, where (PAX,X) is a relative A– CW–complex and the space map?(A,PAX) is weakly contractible. The assignment X 7→ PAX can be made functorial, in such a way that the map X → PAX is natural. The functors CWA and PA are crucial in studying spaces through “the eyes” of A. The functor CWA assigns to a space X the largest sub–object CWAX → X, which is totally “visible” by A. While the functor PA associates with X the largest quotient X → PAX, which is totally “invisible” by A. The space CWAX contains all the information about X that can be detected by A, while PAX contains all the information about X, that can not be detected by A at all.