process

On K 1 of a Waldhausen category 103Proposition 4.3. For anyWaldhausen category C there is anotherWaldhausen categorySum.C/ with a functorial coproduct whose monoid of objects is freely generated bythe objects of C except from . Moreover, there are natural mutually inverse exactequivalences of categories'Sum.C/ C .Proof. The statement already says which are the objects of Sum.C/. The functor 'sends an object Y in Sum.C/, which can be uniquely written as a formal sum of nonzeroobjects in C, Y D X 1 CCX n , to an arbitrarily chosen coproduct of theseobjects in C'.Y / D X 1 __X n :For n D 0; 1 we make special choices, namely for n D 0, '.Y / D, and for n D 1we set '.Y / D X 1 . Morphisms in Sum.C/ are defined in the unique possible way sothat ' is fully faithful. Then the formal sum defines a functorial coproduct on Sum.C/.The functor sends to the zero object of the free monoid and any other object in Cto the corresponding object with a single summand in Sum.C/, so that ' D 1. Theinverse natural isomorphism 1 Š ',X 1 CCX n Š .X 1 __X n /;is the unique isomorphism in Sum.C/ which ' maps to the identity. Cofibrationsand weak equivalences in Sum.C/ are the morphisms which ' maps to cofibrationsand weak equivalences, respectively. This Waldhausen category structure in Sum.C/makes the functors ' and exact.Let us recall the notion of homotopy in the category of stable quadratic modules.Definition 4.4. Given morphisms f; g W C ! C 0 of stable quadratic modules, ahomotopy f ' ˛g from f to g is a function ˛ W C 0 ! C1 0 satisfying1. ˛.c 0 C d 0 / D ˛.c 0 / f 0.d 0 / C ˛.d 0 /,2. g 0 .c 0 / D f 0 .c 0 / C @˛.c 0 /,3. g 1 .c 1 / D f 1 .c 1 / C ˛@.c 1 /.The following lemma then follows from the laws of stable quadratic modules.Lemma 4.5. A homotopy ˛ as in Definition 4.4 satisfies˛.Œc 0 ;d 0 / D hf 0 .d 0 /; f 0 .c 0 /iChg 0 .d 0 /; g 0 .c 0 /i; (a)˛.c 0 / C g 1 .c 1 / D f 1 .c 1 / C ˛.c 0 C @.c 1 //:(b)Now we are ready to prove Theorem 4.2.