process

On K 1 of a Waldhausen category 93the Gillet–Grayson construction for an arbitrary Waldhausen category W. The largestknown class over which the Gillet–Grayson construction works is the class of pseudoadditivecategories, which extends the class of exact categories but still does not coverall Waldhausen categories, see [3]. We use instead the algebraic model D W definedin [4] for the 1-type of the Waldhausen K-theory spectrum KW. The algebraic objectD W is a chain complex of non-abelian groups concentrated in dimensions i D 0; 1whose homology is K i W,K 1 W .D 0 W/ ab ˝ .D 0 W/ abh;i D 1 W@ D 0 W K 0 W;i.e. the lower row is exact. This non-abelian chain complex is equipped with a bilinearmap h; i which determines the commutators. This makes D W a stable quadraticmodule in the sense of [1]. This stable quadratic module is defined in [4] in terms ofgenerators and relations. Generators correspond simply to objects, weak equivalences,and cofiber sequences in W.Acknowledgement. The authors are very grateful to Grigory Garkusha for askingabout the relation between the algebraic model for Waldhausen K-theory defined in [4]and Nenashev’s presentation of K 1 of an exact category in [7].(A)1 The algebraic model for K 0 and K 1Definition 1.1. A stable quadratic module C is a diagram of group homomorphismsC0 ab ˝ C 0absuch that given c i ;d i 2 C i , i D 0; 1,1. @hc 0 ;d 0 iDŒd 0 ;c 0 ,2. h@.c 1 /; @.d 1 /iDŒd 1 ;c 1 ,h;i! C1@! C03. hc 0 ;d 0 iChd 0 ;c 0 iD0.Here Œx; y D x y C x C y is the commutator of two elements x;y 2 K in anygroup K, and K ab is the abelianization of K. It follows from the axioms that the imageof h; i and Ker @ are central in C 1 , the groups C 0 and C 1 have nilpotency class 2, and@.C 1 / is a normal subgroup of C 0 .A morphism f W C ! D of stable quadratic modules is given by group homomorphismsf i W C i ! D i , i D 0; 1, compatible with the structure homomorphisms ofC and D , i.e. f 0 @ D @f 1 and f 1 h; iDhf 0 ;f 0 i.Stable quadratic modules were introduced in [1, Definition IV.C.1]. Notice, however,that we adopt the opposite convention for the homomorphism h; i, i.e. we composewith the symmetry isomorphism on the tensor square.