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K-theory of a permutative category

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

To a permutative category CC 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) CC.

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 FinSet */FinSet^{*/} for the category of pointed objects finite sets.

For CC a permutative category, there is naturally a functor

C¯ ():FinSet */Cat \widebar {C}_{(-)} : FinSet^{*/} \to Cat
AC¯ A A \mapsto \widebar C_A

such that (…).

(Elmendorf-Mandell, theorem 4.2)

Accordingly, postcomposition with the nerve N:CatsSetN : Cat \to sSet produces from CC a Gamma-space NC¯N \widebar C. To this corresponds a spectrum

K SegC{NC¯ S n} n. K^{Seg} C \coloneqq \{N \widebar C_{S_\bullet^n}\}_n \,.

This is the K-theory spectrum of CC.

(Elmendorf-Mandell, def. 4.3)

References

  • Graeme Segal, Catgeories and cohomology theories, Topology vol 13 (1974) (pdf)
Revised on July 13, 2012 13:19:43 by Urs Schreiber (89.204.130.60)