Infinite Loop G-Spaces Associated to Monoidal G-Graded Categories

Abstract

We construct a functor K G which takes each pair of monoidal G-graded categories (D,D f ) to an infinite loop G-space K G (D,D'). When D'=D, its homotopy groups n%K G (D,D) coincide with the equivariant K-groups K n RepD of D. Applications include the simple construction of equivariant infinite deloopings of the maps BO(G}-^BPL(G}-^> BTop(G} between equivariant classifying spaces. 0. Introduction Let G be a finite group. By a (simplicial) G-graded category we shall mean a (simplicial) category D equipped with a (simplicial) functor 7-from D to G which is regarded as a category with only one object. We often identify a simplicial G-graded category D with its realization rD; a topological G-graded category such that ob(rZ)) and mor (rD) are the geometric realizations of the simplicial sets [k~] i-> obZ) fe and [k~\ i- morD k respectively.