process

110 F. Muro and A. TonksThey satisfy two kind of relations. The first, analogous to (S2), says that a generatorvanishes provided j 1 D j 2 and r 1 D r 2 . The second is a simplification of (S1): givensix pairs of short exact sequencesA 0A 0j A 1 rA 1 A A 00 ; B 0 j A 2j 0 2r A 2 j 0 1 r0 1B 0 C 0 ;r 0 2j B 1j B 2A j 1 Bj 2rB 1B B 00 ; C 0 r B 2j C 1j C 2rC 1C C 00 ;r C 2r 1j C; A 00 001 r001 B 00 C 00 ;r 2denoted for simplicity as A , B , C , 0 , , and 00 , such that the diagramj 002r 002A 0 j iAj 0 iB 0 j B ir 0i C 0 j C iA j ir A i A00j 00iB rB iB 0 ir i C r C C00ir 00icommutes for i D 1; 2 then˚A ˚ B C ˚ C D ˚ 0 fg C ˚ 00 :Proposition 6.1. For any exact category E there is a natural isomorphismD.E/ Š K1 wcs E:Proof. There is a clear homomorphism D.E/ ! K wcs´A j 1 Bj 2μr 1Cr 21E defined by89r ˆ= B Cj ˆ: 2r 2 C 1 : >;Exact categories regarded as Waldhausen categories have the particular feature thatall weak equivalences are isomorphisms, hence one can also define a homomorphismK wcs1E ! D.E/ as8ˆ=w 1 w C 2r 2 Š>;C 27!´A j 1 Bj 2w1 1r1w 12 r 2μC :We leave the reader to check the compatibility with the relations and the fact that thesetwo homomorphisms are inverse of each other.