On the functors \(CW_ A\) and \(P_ A\)
Given a pointed space \(A\) and a map \(f:X\to Y\) consider its factorization \(X\to Y'@>p>>Y\), where \((Y',X)\) is a relative \(A\)-CW-complex and \(p\) a map inducing a weak equivalence \(p_*:\text{TOP}(A,Y')\to \text{TOP}(A,Y)\) of mapping spaces. In particular, for the maps \(*\to X\) and \(X\to*\) we get maps \(CW_AX\to X\) and \(X\to P_AX\), respectively. The author applies the results by \textit{E. Dror Farjoun} [Cellular inequalities, Contemp. Math. 181, 159-181 (1995)] to study the relationship between the functors \(CW_A\) and \(P_A\). It is shown that the kernel of \(P_A\) is the closure, under taking extensions by fibrations, of the image of \(CW_A\) and various weak cellular inequalities are presented.
scientific article; zbMATH DE number 943720