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K-theory of a permutative category

Idea

To a permutative category $C$ 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) $C$.

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.

Definition

Write ${\mathrm{FinSet}}^{*/}$ for the category of pointed objects finite sets.

For $C$ a permutative category, there is naturally a functor

$${\overline{C}}_{(-)}:{\mathrm{FinSet}}^{*/}\to \mathrm{Cat}$$\widebar {C}_{(-)} : FinSet^{*/} \to Cat

$$A\mapsto {\overline{C}}_A$$A \mapsto \widebar C_A

such that (…).

(Elmendorf-Mandell, theorem 4.2)

Accordingly, postcomposition with the nerve $N:\mathrm{Cat}\to \mathrm{sSet}$ produces from $C$ a Gamma-space $N\overline{C}$. To this corresponds a spectrum

$$K^{\mathrm{Seg}}C≔\{N{\overline{C}}_{S_{•}^n}{\}}_n.$$K^{Seg} C \coloneqq \{N \widebar C_{S_\bullet^n}\}_n \,.

This is the K-theory spectrum of $C$.

(Elmendorf-Mandell, def. 4.3)

Related concepts

• K-theory of a bipermutative category

References

• Graeme Segal, Catgeories and cohomology theories, Topology vol 13 (1974) (pdf)

• Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)

• 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)