process

On K 1 of a Waldhausen category 105Here we have used (R8) for the composable cofibrationsB A C B S C A C B:It is clear from the definition of ˛ that the relation p˛ D 0 holds. It remains to see thatjp factors through D C C, that is,jp.c 1 / D 0if c 1 D ŒB A C B A:Let c 0 D ŒA C B so that c 0 C @.c 1 / D ŒA C ŒB. Then by Lemma 4.5 (b) we havejp.c 1 / D ˛.c 0 / C c 1 C ˛.c 0 C @.c 1 //D ˛.ŒA C B/ C ŒB A C B A C ˛.ŒA C ŒB/:This is zero by (4.1).We finish this section with four lemmas which show useful relations in D CC.Lemma 4.6. The following equality holds in D CC.ŒA C A 0 B C B 0 B=A C B 0 =A 0 C C C 0 D ŒA B B=A C ŒB0 C ŒA 0 B 0 B 0 =A 0C 0 ChŒA; ŒC 0 i:Proof. Use (R4), (R10) and Proposition 1.6 applied to the commutative diagramA0 B 0 B 0 =A0 C 0A C A 0 B C B 0B=A C B 0 =A 0 C C C 0A B B=A CAs special cases we haveA B B=A C .Lemma 4.7. The following equality holds in D C C.ŒA C A 0 B C B 0 B=A C B 0 =A 0 D ŒA B B=A ŒB0 C ŒA 0 B 0 B 0 =A 0 ChŒA; ŒB 0 =A 0 i:Lemma 4.8. The following equality holds in D C C.ŒA C B ! A 0 C B 0 D ŒA ! A 0 ŒB0 C ŒB ! B 0 :