cohomology

# Contents

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

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

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

such that (…).

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

$K^{Seg} C \coloneqq \{N \widebar C_{S_\bullet^n}\}_n \,.$

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

## References

• Graeme Segal, Catgeories and cohomology theories, Topology vol 13 (1974) (pdf)
Revised on April 28, 2014 06:02:21 by Urs Schreiber (88.128.80.161)