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K-theory of a permutative category

Contents

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 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\mathcal{C} a category, write |𝒞|{\vert \mathcal{C}\vert} \in sSet Qu\simeq_{Qu} Top for its nerve/geometric realization. (Beware that this is often denoted BCB C, instead, but that notation clashes with that for delooping, which we also need in the following.)

For 𝒞\mathcal{C} a permutative category its nerve/geometric realization |𝒞|\vert \mathcal{C} \vert is naturally a topological monoid (Quillen 70 see e.g. May 13, theorem 4.10), hence admits a bar construction/classifying space B|𝒞|B {\vert \mathcal{C}\vert}. The loop space of that

(𝕂𝒞) 0ΩB|𝒞|, (\mathbb{K}\mathcal{C})_0 \;\coloneqq\; \Omega B {\vert \mathcal{C}\vert} \,,

being an ∞-group, may be regarded as the homotopy theoretic group completion of the topological monoid |𝒞|{\vert \mathcal{C}\vert}.

This is the degree-0 space in the algebraic K-theory spectrum 𝕂𝒞\mathbb{K}\mathcal{C} of the permutative category 𝒞\mathcal{C} (see e.g. May 13, def 4.11).

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 (∞,1)-category.

In particular, under decategorification, its group of connected components is the actual Grothendieck group K()K(-) of the isomorphism classes of objects in 𝒞\mathcal{C}:

π 0(𝕂𝒞)K(𝒞/ ,) \pi_0 \left( \mathbb{K} \mathcal{C} \right) \;\simeq\; K\left( \mathcal{C}/_\sim, \oplus \right)

(recalled e.g. in Bohmann-Osorno 14, p. 14).

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 13, p. 25)

𝕂(RMod proj fin)KR. \mathbb{K}\left( R Mod^{fin}_{proj} \right) \;\simeq\; K R \,.

Via Gamma spaces

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

For CC a permutative category, there is naturally a functor

C¯ ():FinSet */Cat \widebar {C}_{(-)} \;\colon\; 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)

Properties

Monoidal functoriality

Proposition

The construction of K-theory spectra of permutative categories constitutes a multifunctor

𝕂:PermCatSpectra \mathbb{K} \;\colon\; PermCat \longrightarrow Spectra

between multicategories of permutative categories (under Deligne tensor product) and spectra (under smash product of spectra).

Hence for a multilinear functor of permutative categories

F:𝒜 1××𝒜 n F \;\colon\; \mathcal{A}_1 \times \cdots \times \mathcal{A}_n \longrightarrow \mathcal{B}

there is a compatibly induced morphism of K-spectra out of the smash product

𝕂(𝒜 1)𝕂(𝒜 n)𝕂(). \mathbb{K}(\mathcal{A}_1) \wedge \cdots \wedge \mathbb{K}(\mathcal{A}_n) \longrightarrow \mathbb{K}(\mathcal{B}) \,.

This implies that the construction further extends to a 2-functor from the 2-category PermCatCatPermCat Cat of enriched categories over the multicategory of permutative categories to that of enriched categories of spectra:

𝕂 :PermCatCatSpectraCat \mathbb{K}_\bullet \;\colon\; PermCat Cat \longrightarrow Spectra Cat

which applies 𝕂\mathbb{K} to each hom-object.

(May 80, theorem 1.6, Theorem 2.1, Elmendorf-Mandell 04, theorem 1.1, Guillou 10, Theorem 1.1

Examples

Example

(ordinary algebraic K-theory)

For RR a commutative ring, let 𝒞=RMod pr\mathcal{C} = R Mod_{pr} its category of finitely generated projective modules regarded as a permutative category. Then

K(RMod fin pr)KR K (R Mod^{pr}_{fin}) \;\simeq\; K R

is the classical algebraic K-theory spectrum of the ring RR.

(e.g. Elmendorf-Mandell 04, p. 10).

Example

(stable cohomotopy is K-theory of FinSet)

Let 𝒞=\mathcal{C} = FinSet be a skeleton of the category of finite sets, regarded as a permutative category. Then

K(FinSet)𝕊 K(FinSet) \;\simeq\; \mathbb{S}

is the sphere spectrum, hence represents the cohomology theory called stable cohomotopy.

(due to Segal 74, Prop. 3.5, see also Priddy 73)

Remark

(stable cohomotopy as algebraic K-theory over the field with one element)

Since (pointed) finite sets may be regarded as the modules over the “field with one element𝔽 1\mathbb{F}_1 (see there),

𝔽 1Mod=FinSet */ \mathbb{F}_1 Mod \;=\; FinSet^{\ast/}

one may read example in view of example as saying that stable cohomotopy is the algebraic K-theory of the field with one element:

K𝔽 1=𝕊. K \mathbb{F}_1 \;=\; \mathbb{S} \,.

This perspective is highlighted for instance in (Deitmar 06, p. 2, Guillot 06).

References

The interpretation of stable cohomotopy as the algebraic K-theory over the field with one element is adopted in

The generalization of K-theory of permutative categories to Mackey functors is discussed in

Generalization to equivariant stable homotopy theory and G-spectra is discussed in

Last revised on October 26, 2018 at 06:35:01. See the history of this page for a list of all contributions to it.