process

102 F. Muro and A. TonksWe now give an explicit definition of what we understand by a strict coproductfunctor. Let C be a Waldhausen category endowed with a symmetric monoidal structureC which is strictly associative.A C B/ C C D A C .B C C/;strictly unital with unit object , the distinguished zero object,CA D A D A C;but not necessarily strictly commutative, such thatA D A C !A C B CB D Bis always a coproduct diagram. Such a category will be called a Waldhausen categorywith a functorial coproduct. Then we define the sum-normalized stable quadraticmodule D CC as the quotient of D C by the extra relation(R10) ŒB i 2 A C B p 1 A D 0.Remark 4.1. In D C C, the relations (R2) and (R10) imply thatŒA C B D ŒA C ŒB:Furthermore, relation (R9) becomes equivalent to(R9 0 ) hŒA; ŒBi DŒ B;A W B C A Š ! A C B.Here B;A is the symmetry isomorphism of the symmetric monoidal structure. Thisequivalence follows from the commutative diagramA A i 2 p 1B C A B B;AŠi 1 A C B p2 Btogether with (R4), (R7) and (R10).Theorem 4.2. Let C be a Waldhausen category with a functorial coproduct such thatthere exists a set S which freely generates the monoid of objects of C under the operationC. Then the projectionp W D C D C Cis a weak equivalence. Indeed, it is part of a strong deformation retraction.We prove this theorem later in this section.Waldhausen categories satisfying the hypothesis of Theorem 4.2 are general enoughas the following proposition shows.