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94 F. Muro and A. TonksRemark 1.2. There is a natural right action of C 0 on C 1 defined byc c 01 D c 1 Chc 0 ; @.c 1 /i:The axioms of a stable quadratic module imply that commutators in C 0 act trivially onC 1 , and that C 0 acts trivially on the image of h; i and on Ker @.The action gives @W C 1 ! C 0 the structure of a crossed module. Indeed a stablequadratic module is the same as a commutative monoid in the category of crossedmodules such that the monoid product of two elements in C 0 vanishes when one ofthem is a commutator, see [4, Lemma 4.18].A stable quadratic module can be defined by generators and relations in degree 0and 1. In the appendix we give details about the construction of a stable quadraticmodule defined by generators and relations. This is useful to understand Definition 1.3below from a purely group-theoretic perspective.We assume the reader has certain familiarity with Waldhausen categories and relatedconcepts. We refer to [12] for the basics, see also [11]. The following definition wasintroduced in [4].Definition 1.3. Let C be a Waldhausen category with distinguished zero object , andcofibrations and weak equivalences denoted by and !, respectively. A genericcofiber sequence is denoted byA B B=A:We define D C as the stable quadratic module generated in dimension zero by thesymbols• ŒA for any object in C,and in dimension one by• ŒA ! A 0 for any weak equivalence,• ŒA B B=A for any cofiber sequence,such that the following relations hold.(R1) @ŒA ! A 0 DŒA 0 C ŒA.(R2) @ŒA B B=A DŒB C ŒB=A C ŒA.(R3) Œ D 0.(R4) ŒA 1 A! A D 0.(R5) ŒA 1 A! A D 0, Œ A 1 A! A D 0.