process

On K 1 of a Waldhausen category 109Moreover, C0 ab is a free abelian group, hence the middle row is a short exact sequence,see [1, I.4]. Therefore any element y 2 C 1 which is both in the image of c 1and the kernel of @ is in the image of c 1 N as required.The group D C 0C is free of nilpotency class 2, therefore by the previous lemmay Dha; ai for some a 2 D C 0C. By the laws of a stable quadratic module the elementha; ai only depends on a mod 2, compare [4, Definition 1.8], and so we can supposethat a is a sum of basis elements with coefficient C1. Hence by Remark 4.1 we cantake a D ŒM for some object M in C,y DhŒM ; ŒM i:The element hŒM ; ŒM i is itself a pair of weak cofiber sequences89p ˆ=M C M1 M ;i ˆ: 1p 2 M >;therefore x is also a pair of weak cofiber sequences, by Proposition 3.3, and the proofof Theorem 2.1 is complete.6 Comparison with Nenashev’s approach for exact categoriesFor any Waldhausen category C we denote by K wcs1C the abelian group generated bypairs of weak cofiber sequences8ˆ=B C >;C 2modulo the relations (S1) and (S2) in Section 3. Theorem 3.1 and Proposition 3.2 showthe existence of a natural homomorphismK1 wcs C K 1 C (6.1)which is surjective by Theorem 2.1.Given an exact category E Nenashev defines in [6] an abelian group D.E/ bygenerators and relations which surjects naturally to K 1 E. Moreover, he shows in [7]that this natural surjection is indeed a natural isomorphismGenerators of D.E/ are pairs of short exact sequences´μA j 1 r 1 B C :j 2 r 2D.E/ Š K 1 E: (6.2)