Call two maps, f,g from C to C', of chain complexes absolutely homologous if for any additive functor F, the induced Ff and Fg are homologous (induce the same map on homology). It is known that the identity is absolutely homologous to 0 iff it is homotopic to 0 and tempting to conjecture that f and g are absolutely homologous iff they are homotopic. This conjecture is false, but there is an equational characterization of absolute homology. I also characterize left absolute and right absolute (in which F is quantified over left or right exact functors).
Keywords: absolutely homologous chain maps
2000 MSC: 18G35
Theory and Applications of Categories,
Vol. 14, 2005,
No. 3, pp 53-59.