process

Categorical aspects of bivariant K-theory 19Alternatively, we may also require additivity for coproducts (that is, free products).Of course, this only makes sense if S is closed under coproducts in G-C alg. Thecoproduct A t B in G-C alg comes with canonical maps A A t B B as well;the maps A W A ! A t B and B W B ! A t B are the coordinate embeddings, themaps A W A t B ! A and B W A t B ! B restrict to .id A ;0/ and .0; id B / on Aand B, respectively.Definition 43. We call F additive on coproducts if it maps coproduct diagrams A A t B B to direct sum diagrams in C.The coproduct and product are related by a canonical G-equivariant -homomorphism' W A t B A B that is compatible with the maps to and from A and B, that is,' ı A D A , A ı ' D A , and similarly for B. There is no map backwards, but thereis a correspondence W A B Ü A t B, which is induced by the G-equivariant -homomorphism A .a/ 0A B ! M 2 .A t B/; .a;b/ 7!:0 B .b/It is easy to see that the composite correspondence ' ı is equal to the identitycorrespondence on AB. The other composite ı' is not the identity correspondence,but it is homotopic to it (see [9], [10]). This yields:Proposition 44. If F is C -stable and homotopy invariant, then the canonical mapF.'/W F.At B/ ! F.A B/ is invertible. Therefore, additivity and additivity forcoproducts are equivalent for such functors.The correspondence exists because the stabilisation creates enough room to replace A and B by homotopic homomorphisms with orthogonal ranges. We can achievethe same effect by a suspension (shift A and B to the open intervals .0; 1=2/ and .1=2;1/,respectively). Therefore, any homotopy invariant functor satisfies F Sus.A t B/ ŠF Sus.A B/ .3.4.2 Split-exact functorsDefinition 45. We call F split-exact if, for any split extension K i E p Q withsection s W Q ! E, the map F .i/; F .s/ W F.K/˚ F.Q/ ! F.E/is invertible.It is clear that split-exact functors are additive.Split-exactness is useful because of the following construction of Joachim Cuntz [9].Let B G E be a G-invariant ideal and let f C ;f W A E be G-equivariant -homomorphisms with f C .a/ f .a/ 2 B for all a 2 A. Equivalently, f C and fboth lift the same morphism f N W A ! E=B. The data .A; f C ;f ;E;B/ is called aquasi-homomorphism from A to B.Pulling back the extension B E E=B along f N , we get an extension B E 0 A with two sections fC 0 ;f0 W A E 0 . The split-exactness of F shows that