Deloopings in Algebraic K-Theory

Gunnar Carlsson ()
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Gunnar Carlsson: Stanford University, Department of Mathematics

Chapter I.1 in Handbook of K-Theory, 2005, pp 3-37 from Springer

Abstract: Abstract 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.

Keywords: Abelian Group; Homotopy Group; Loop Space; Weak Equivalence; Exact Functor (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/978-3-540-27855-9_1

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