The Linearisation Map in Algebraic K-Theory

Abstract

Let X be a pointed connected simplicial set with loop group G. The linearisation map in K-theory as defined by Waldhausen uses G-equivariant spaces. This paper gives an alternative description using presheaves of sets and abelian groups on the simplex category of X. In other words, the linearisation map is defined in terms of X only, avoiding the use of the less geometric loop group. The paper also includes a comparison of categorical finiteness with the more geometric notion of finite CW objects in cofibrantly generated model categories. The application to the linearisation map employs a model structure on the category of abelian group objects of retractive spaces over X.