Simplicial groups that are models for algebraic K-theory

In this paper, we present and compare some simplicial groups, functorially associated to a ring R, whose homotopy groups are Quillen’s K-groups of R. The first such simplicial group is the group Ω(NQPR), where Ω is the loop space construction of Clemens Berger, applied to the simplicial set NQPR (the nerve of Quillen’s category QPR). The second is a subgroup GR of the simplicial group Ω(NQPR). This second group is compared to Kan’s construction [12] of a loop group for a connected simplicial set, and shown to be isomorphic to it as a simplicial group. Other simplicial groups that are models for algebraic K-theory are also presented; in particular, the subgroup G(s.PR) of Ω(s.PR); here, s.PR is Waldhausen’s simplicial set [25], [26]. We initially give an exposition of Berger’s construction in general; then, we present the construction of GR and a summary of Kan’s construction. Next, we point out that GR is an infinite loop object in the category of simplicial groups, and draw some corollaries. We then compare directly the homotopy groups thus constructed with the classical K-theory in degrees 0 and 1. The final section compares various models.