An un-delooped version of algebraic \(K\)-theory (Q1192620)

This paper extends the construction, by Gillet and Grayson, of a combinatorial model \(G_ \bullet\) for the \(K\)-theory of exact categories in the sense of Quillen, to the context of categories with cofibrations and weak equivalences. The main result is that in the presence of a suspension functor, the extended \(wG_ \bullet\) construction on the category of prespectra is an undelooped version of the \(K\)-theory of the original category. These constructions allow the definition and proofs of formulas for \(\lambda-\) and Segal operations in the algebraic \(K\)-theory of spaces.