Introduction

Controlled algebra is a powerful tool to prove statements about the algebraic K-theory of a ring R. While early on it was used in [PW85] to construct a nonconnective delooping of K(R)—a space such that π i(K(R)) = Ki−1(R)—it is a crucial ingredient in recent progress of the so-called Farrell-Jones Conjecture. Our aim here is to construct for a simplicial ring R, and a so-called “control space” X, a category of “controlled simplicial R-modules over a X”. It should be regarded as a generalization of controlled algebra from rings to simplicial rings.

The category of “controlled simplicial modules” supports a homotopy theory which is formally very similar to the homotopy theory of CW-complexes. In particular we have a “cylinder object” which yields a notion of homotopy and therefore the category has homotopy equivalences. Waldhausen nicely summarized a minimal set of axioms to do homotopy theory in his notion of a Waldhausen category, called ’category with cofibrations and weak equivalences in [Wal85]. He did this to define algebraic K-theory of such a category. Our category satisfies Waldhausen's axioms, which is our main result:

Theorem.Let X be a control space and R a simplicial ring. The category of controlled simplicial modules over X, C(X;R), together with the homotopy equivalences and a suitable class of cofibrations is a “category with cofibrations and weak equivalences” in the sense of Waldhausen ([Wal85]). Therefore Waldhausen's algebraic K-theory of C(X;R) is defined.