I am not sure this is exactly what Michael Barr is doing but I had some results in this direction in Batanin M.A., Mappings of spectral sequences and the generalized homotopy axiom, Siberian Math. Journal, 5, (1987), 22-31. I hope they can be useful so I give a brief formulation. Let M and K are chain complexes in an abelian category A with sufficiently many injective objects. Let T be an additive functor and let M,K and T satisfy the conditions of convergence of spectral sequences (*) E_2^p^q(M) = T^pH^q(M) ===> H^{p+q}TM and similar for homology of K. Now if, f:M --> K is such that Hf = 0 then for every q we have a short exact sequence (**) 0--> H^{q-1}K ---> H^q Cf ---> H^q M ---> 0 where Cf is a cone of f. Suppose also that second and higher differentials of spectral sequences (*) vanish. Then one of the results of my paper tells that H^nTf = 0 if and only if all boundary homomorphisms T^pH^q (M) ---) T^{p+1}H^{q-1} (K), p+q = n generated by (**) are trivial. (if some differential are nontrivial we can continue the process of deriving and can get some morphisms E^p^q(M) ---> E^{p+r}^{q-r}(K) but they are not so nicely calculated). If the condition above are applicable to Hom functor in A this immediately gives that f is absolutely homologous to 0 iff (**) split in every dimension. Hopefully this will be helpful. Michael Batanin. on 1/4/02 11:29 PM, Michael Barr at barr@barrs.org wrote:
Say that two maps f and g between two chain complexes in some abelian category are absolutely homologous if for any additive functor F to another abelian category, Ff is homologous to Fg. I have found an equational characterization of this (no it is not that they are homotopic, although it is not that different) and I wonder if anyone knows if this has been done.