process

On K 1 of a Waldhausen category 95(R6) For any pair of composable weak equivalences A ! B ! C ,ŒA ! C D ŒB ! CC ŒA ! B:(R7) For any commutative diagram in C as followsA BB=A A 0 B0 B 0 =A 0we haveŒA ! A 0 C ŒB=A ! B 0 =A 0 ŒA D ŒA 0 B 0 B 0 =A 0 C ŒB ! B 0 C ŒA B B=A:(R8) For any commutative diagram consisting of four cofiber sequences in C as followsC=BB=A C=AA B Cwe haveŒB C C=BC ŒA B B=AD ŒA C C=AC ŒB=A C=A C=B ŒA :(R9) For any pair of objects A; B in CHerehŒA; ŒBi D ŒB i 2 A _ B p 1 A C ŒA i 1 A _ B p 2 B:A i 1 A _ Bp 1are the inclusions and projections of a coproduct in C.Remark 1.4. These relations are quite natural, and some illustration of their meaningis given in [4, Figure 2]. They are not however minimal: relation (R3) follows from(R2) and (R5), and (R4) follows from (R6). Also (R5) is equivalent toi 2p 2B