Phantom maps and homology theories

We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X, Y) of phantom maps X → Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X, HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolisʹs axiomatisation conjecture to an extension problem. We show that a certain functor J → A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.