An un-delooped version of algebraic K-theory

Abstract

Problems working with the Segal operations in algebraic K-theory of spaces—constructed by F. Waldhausen (1982)—arose from the absence of a nice groupcompletion on the category level. H. Grayson and D. Gillet (1987) introduced a combinatorial model G . for K-theory of exact categories. For dealing with K-theory of spaces we need an extension wG . of their result to the context of categories with cofibrations and weak equivalences. Our main result is that in the presence of a suspension functor—as in the case of retractive spaces—the wG . construction on the category of prespectra is an un-delooped version of the K-theory of the original category. In a sequel to this paper we show that Grayson's formula (1988) for Segal operations works as intended.