Complete functors in homology I. Chain maps and endomorphisms

Extract

There is a sense in which the homology group HA of a free Abelian chain complex A may be said to be a ‘complete system of invariants’ of A, to within chain equivalence; certainly any graded Abelian group G is isomorphic to HA for a suitable A, and if HA and HB are isomorphic then A and B are chain equivalent. Such a result is useful in showing that it is fruitless to seek other homotopy invariants of A; whatever depends only on the homotopy class of A depends only on HA, so that we can, for instance, predict the existence of a formula giving H(A ⊗ G), to within isomorphism, in terms of HA and G. The theorem on the existence and uniqueness to within chain equivalence of projective resolutions of modules is a variant of the above theorem, more general in one direction and more special in another.