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K-theory of a permutative category

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

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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

Via topological group completion

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

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

By (Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7) the operation ΩB()\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 */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}. K^{Seg} C \coloneqq \{N \widebar C_{S_\bullet^n}\} \,.

This is the K-theory spectrum of CC.

(Elmendorf-Mandell, def. 4.3)

References

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