process

On K 1 of a Waldhausen category 101Proposition 3.2. A pair of weak cofiber sequences given by two times the same weakcofiber sequence is trivial.89C ˆ= B jˆ: C D 0:(S2)wr >;C 1We establish a further useful relation in the following proposition.Proposition 3.3. Relations (S1) and (S2) imply that the sum of two pairs of weakcofiber sequences coincides with their coproduct.89 89C 1 _ NC 1ˆ< CA _ AN >= ˆ= B Cˆ: >;ˆ: >;C 2 _ NC 2C 289ˆ< NC 1 C AN >= NB NC :ˆ: >; NC 2Proof. Apply relations (S1) and (S2) to the following pairs of weak cofiber sequencesA BC 1 C; C 2AN NB NC 1 NC 2NC;A _ ANC 1 _ NC 1 B _ NB C _ NC; C 2 _ NC 2p 2 AN 1 A i 1A _ ANA; Ni1p 2AN 1p 2 B i 1B _ NB i 1 p 2NB 1 NB 1NB;p 2 NC 1 C i 1C _ NCNC:i 1p 2 NC 14 Waldhausen categories with functorial coproductsWaldhausen categories admit finite coproducts A _ B. The operation _ need not bestrictly associative and unital. However, as we check below, any Waldhausen categorycan be replaced by an equivalent one with strict coproducts.