process

On K 1 of a Waldhausen category 107if the theorem holds for a certain Waldhausen category then it holds for all equivalentones. In particular by Proposition 4.3 it is enough to prove the theorem for any Waldhausencategory C with a functorial coproduct such that the monoid of objects is freelygenerated by a set S. By Theorem 4.2 we can work in the sum-normalized constructionD C C in this case.Any element x 2 D C 1C is a sum of weak equivalences and cofiber sequences withcoefficients C1 or 1. By Lemma 4.8 modulo the image of h; i we can collect onthe one hand all weak equivalences with coefficient C1 and on the other all weakequivalences with coefficient 1. Moreover, by Lemma 4.7 we can do the same forcofiber sequences. Therefore the following equation holds modulo the image of h; i.x D ŒV 1! V2 ŒX 1 X 2 X 3 C ŒY 1 Y 2 Y 3 C ŒW 1! W2 (R4), (R10), 4.8, 4.7 DŒV 1 C W 1 ! V2 C W 1 ŒX 1 C Y 1 X 2 C Y 3 C Y 1 X 3 C Y 3 C ŒX 1 C Y 1 X 3 C X 1 C Y 2 X 3 C Y 3 C ŒV 1 C W 1 ! V1 C W 2 renaming D ŒL ! L 1 ŒA E 1 DC ŒA E 2 D C ŒL ! L 2 mod h; i:Suppose that @.x/ D 0 modulo commutators. Then modulo Œ; 0 D ŒL C ŒL 1 ŒA ŒD C ŒE 1 ŒE 2 C ŒD C ŒA ŒL 2 C ŒLD ŒL 1 C ŒE 1 ŒE 2 ŒL 2 mod Œ; ;i.e. ŒL 1 C E 1 D ŒL 2 C E 2 mod Œ; :(5.1)The quotient of D C 0C by the commutator subgroup is the free abelian group on S,hence by (5.1) there are objects S 1 ;:::;S n 2 S and a permutation 2 Sym.n/ suchthatL 1 C E 1 D S 1 CCS n ;L 2 C E 2 D S 1 CCS n ;so there is an isomorphism S1 ;:::;S nW L 2 C E 2 ! L1 C E 1 :Again by Lemmas 4.8 and 4.7 modulo the image of h; ix D ŒL C D ! L 1 C D ŒA L 1 C E 1 L 1 C DC ŒA L 2 C E 2 L 2 C D C ŒL C D ! L 2 C D4.9 D ŒL C D ! L 1 C D ŒA L 1 C E 1 L 1 C D C