Contents

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

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

$(\mathbb{K}\mathcal{C})_0 \;\coloneqq\; \Omega B {\vert \mathcal{C}\vert} \,,$

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

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

By (Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7) the operation $\Omega B (-)$ 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 $K(-)$ of the isomorphism classes of objects in $\mathcal{C}$:

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

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

In particular for $R$ 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 13, p. 25)

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

Via Gamma spaces

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

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

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

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 $C$.

Properties

Monoidal functoriality

Proposition

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

$\mathbb{K} \;\colon\; PermCat \longrightarrow Spectra$

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

Hence for a multilinear functor of permutative categories

$F \;\colon\; \mathcal{A}_1 \times \cdots \times \mathcal{A}_n \longrightarrow \mathcal{B}$

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

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

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

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

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

Examples

Example

(ordinary algebraic K-theory)

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

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

is the classical algebraic K-theory spectrum of the ring $R$.

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

Example

(stable cohomotopy is K-theory of FinSet)

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

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

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

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 element$\mathbb{F}_1$ (see there),

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

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

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

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 10:35:01. See the history of this page for a list of all contributions to it.