On weak epimorphisms in homotopy theory (Q1822089)

A weak epimorphism e is a morphism such that if \(g\circ e=0\) then \(g=0\). This notion is the Eckmann-Hilton dual to weak monomorphism. Ganea exhibited examples, in the pointed homotopy category (PHC), of weak monomorphisms that are not monomorphisms. Here the author presents us with two examples, in the PHC, of weak epimorphisms that are not epimorphisms. This is done utilizing a known group H, introduced by Higman, and applying the Eilenberg-MacLane functor K(\,1) to a homomorphism into H. It is noted that in PHC, duals of (weak) monomorphisms are not necessarily (weak) epimorphisms. Thus Ganea's examples cannot be used to construct weak epimorphisms which are not epimorphisms. Moreover, in general, K(\,1) does not produce (weak) epimorphisms from (weak) epimorphisms. Indeed, the author had to prove that the morphism obtained by applying K(\,1) to the homomorphism into H is a weak epimorphism but not an epimorphism.