process

100 F. Muro and A. Tonksthat we denote for the sake of simplicity as A , B , C , 0 , , and 00 , respectively.Moreover, assume that for i D 1; 2 there exists a commutative diagramA0 j A iA r A iA 00 iwiAA 00j 0 iB 0 j B ij iBr B iL| iBi00w B ij 00ir 0 iCi 0 r i Lr i O| i Or iC iCi00 Ow iB 00 r 00iOCi00w 0 iw iLw iw 00iC 0 j C iCr C iLC 00iw C iC 00such that the induced map j i [ A 0 jiB W A[ A 0 B 0 B is a cofibration. Notice that six ofthe eight weak cofiber sequences in this diagram are already in the six diagrams above.The other two weak cofiber sequences are just assumed to exist with the property ofmaking the diagram commutative.Theorem 3.1. In the situation above the following equation holds˚A ˚ B C ˚ C D ˚ 0 fg C ˚ 00 : (S1)Proof. For i D 1; 2 let A i , B i , C i , 0 i , i, and 00ibe the upper and the lower weakcofiber sequences of the pairs above, respectively. By Proposition 1.6Œ A 1 ŒC 1 ŒA Œ 1 C Œ B 1 C Œ00 1 ŒB0 C Œ 0 1 DhŒC 0 ; ŒA 00 iD Œ A 2 ŒC 2 ŒA Œ 2 C Œ B 2 C Œ00 2 ŒB0 C Œ 0 2 :Since @. C 1 / D @.C 2/ and @.001 / D @.00 2/ the actions cancel,Œ A 1 ŒC 1 Œ 1CŒ B 1 CŒ00 1 CŒ0 1 D ŒA 2 ŒC 2 Œ 2CŒ B 2 CŒ00 2 CŒ0 2 :Now one can rearrange the terms in this equation, by using that pairs of weak cofibersequences are central in D 1 C, obtaining the equation in the statement.Theorem 3.1 encodes the most relevant relation satisfied by pairs of weak cofibersequences in K 1 C. They satisfy a further relation which follows from the very definitionin (2.1), but which we would like to record as a proposition by analogy with [6].