ABSTRACT
The aim of this chapter is to extend the constructions in chapter 10 from exact categories to Waldhausen categories. In a way analogous to what was done in chapter 10, we shall construct, for any Waldhausen category W and any finite group G, functors KGn (−W ),KGn (−,W, Y ) and PGn (−,W, Y ) : GSets → ZMod as Mackey functors for all n ≥ 0, and under suitable hypothesis on W show that KGn (−,W ),KGn (−W,Y ) are Green functors. We then highlight some consequences of these facts and also obtain equivariant versions of WaldhausenK-theory additivity and fibration theorems. Our main applications are to Thomason’s “complicial” bi-Waldhausen categories of the form Chb(C) (see example 5.4.1(ii) and example 5.4.2) where C is an exact category and hence to chain complexes of modules over grouprings. In particular, we prove (see theorem 13.4.1) that the Waldhausen’s K-groups of the category (Chb(M(RG), w) of bounded chain complexes of finitely generated RG-modules with suitable quasi-isomorphism as weak equivalences are finite Abelian groups (see theorem 13.4.1(2)). We also present an equivariant approximation theorem for complicial bi-Waldhausen categories.
