Monomorphisms and epimorphisms in a homotopy category

Question

What do the epimorphisms and mononomorphisms look like in the homotopy category of $CW-$complexes?

Answer 1

It's a difficult question to handle in full generality, but suffice it to say there aren't all that many. Hilton and Roitberg in their Note on Monomorphisms and Epimorphisms in Homotopy Theory give a sort of Cantor-Bernstein-Schroder theorem: under finiteness conditions, if $X$ and $Y$ admit epis in both directions, then they are equivalent, while if $X$ and $Y$ admit monos in both directions then they are equivalent up to coverings. You can find a few papers on the topic by Googling around, but most efforts seem to have gone to prove that the homotopy category is balanced, which is a relatively novel result; it seems beyond mortal ken to really understand the monos and epis fully. It is worth noting that monos and epis in stable homotopy are totally trivial-in any triangulated category, such as that of spectra, all monos and epis are split.