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

Via topological group completion

For $C$ a permutative category its nerve/geometric realization $\vert C \vert$ (often denoted $B C$, 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 $\Omega B {\vert C\vert}$ is the algebraic K-theory spectrum of $C$ (see e.g. May, def 4.11)

In particular for $R$ a topological ring one considers $C$ a skeleton of the groupoid of (finitely generated) projective modules over $R$. Then the K-theory of $C$ is the algebraic K-theory of $R$ (e.g. May, p. 25)

By (Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7) the operation $\Omega B (-)$ 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 $FinSet^{*/}$ 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

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

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

References

Revised on September 23, 2014 08:39:15 by Urs Schreiber (185.26.182.25)