Deloopings in Algebraic K-Theory

A crucial observation in Quillen’s definition of higher algebraic K-theory was that the right way to proceed is to define the higher K-groups as the homotopy groups of a space ([23]). Quillen gave two different space level models, one via the plus construction and the other via the Q-construction. The Q-construction version allowed Quillen to prove a number of important formal properties of the K-theory construction, namely localization, devissage, reduction by resolution, and the homotopy property. It was quickly realized that although the theory initially revolved around a functor K from the category of rings (or schemes) to the category Top of topological spaces, K in fact took its values in the category of infinite loop spaces and infinite loop maps ([1]). In fact, K is best thought of as a functor not to topological spaces, but to the category of spectra ([2, 11]). Recall that a spectrum is a family of based topological spaces {X _i } _i≥0, together with bonding maps σ _i : X _i → ΩX _i+1, which can be taken to be homeomorphisms. There is a great deal of value to this refinement of the functor K.