K-theory of a permutative category
Special and general types
To a permutative category is naturally associated a Gamma-space, hence a symmetric spectrum. The generalized (Eilenberg-Steenrod) cohomology theory represented by this is called the (algebraic) K-theory of (or represented by) .
If the category is is even a bipermutative category then the corresponding K-theory of a bipermutative category in addition has E-infinity ring structure, hence is a multiplicative cohomology theory.
Via topological group completion
For a permutative category its nerve/geometric realization (often denoted , but we avoid this here not to confuse with delooping) is naturally a topological monoid (Quillen 70 see e.g. May, theorem 4.10). Its group completion is the algebraic K-theory spectrum of (see e.g. May, def 4.11)
In particular for a topological ring one considers a skeleton of the groupoid of (finitely generated) projective modules over . Then the K-theory of is the algebraic K-theory of (e.g. May, p. 25)
By (Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7) the operation is the derived functor of group completion, so that this construction ought to be a model for the K-theory of a symmetric monoidal (infinity,1)-category.
Via Gamma spaces
Write for the category of pointed objects finite sets.
For a permutative category, there is naturally a functor
such that (…).
(Elmendorf-Mandell, theorem 4.2)
Accordingly, postcomposition with the nerve produces from a Gamma-space . To this corresponds a spectrum
This is the K-theory spectrum of .
(Elmendorf-Mandell, def. 4.3)
Daniel Quillen, Cohomology of groups, Proceedings of the international congress of mathematics 1970
Daniel Quillen, On the group completion of a simplicial monoid
Graeme Segal, Catgeories and cohomology theories, Topology vol 13 (1974) (pdf)
Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)
Peter May, -Spaces, group completions, and permutative categories (pdf)
William Dwyer, Daniel Kan, Simplicial localization of categories, Journal of pure and applied algebra 17 (1980) 267-284
Anthony Elmendorf, Michael Mandell, Permutative categories as a model of connective stable homotopy, in: Birgit Richter (ed.) Structured Ring spectra, Cambridge University Press (2004)
Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, K-Theory 0680 (web, pdf)
Anthony Elmendorf, Michael Mandell, Permutative categories, multicategories, and algebraic K-theory, Algebraic & Geometric Topology 9 (2009) 2391-2441 (arXiv:0710.0082v2)
Revised on September 23, 2014 08:39:15
by Urs Schreiber