nLab K-theory of a permutative category

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 (…).

(Elmendorf-Mandell, theorem 4.2)

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 $C$.

(Elmendorf-Mandell, def. 4.3)

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.

(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 $R$ 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.

(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 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).

Related concepts

• K-theory of a bipermutative category

• algebraic K-theory of a symmetric monoidal (∞,1)-category

References

• Daniel Quillen, Cohomology of groups, Proceedings of the international congress of mathematics 1970

• Daniel Quillen, On the group completion of a simplicial monoid

• Stewart Priddy, Transfer, symmetric groups, and stable homotopy theory, in Higher K-Theories, Springer, Berlin, Heidelberg, 1973. 244-255 (pdf)

• Graeme Segal, Catgeories and cohomology theories, Topology vol 13 (1974) (pdf)

• Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)

• Peter May, $E_\infty$-Spaces, group completions, and permutative categories, in Graeme Segal (ed.) New Developments in Topology, Cambirdge University Press 2013 (pdf, doi:10.1017/CBO9780511662607.008)

• Peter May, Pairings of categories and spectra, J. Pure Appl. Algebra, 19:299–346, 1980

• William Dwyer, Daniel Kan, Simplicial localization of categories, Journal of pure and applied algebra 17 (1980) 267-284

• Anthony Elmendorf, Michael Mandell, Permutative categories as a model of connective stable homotopy, in: Birgit Richter (ed.) Structured Ring spectra, Cambridge University Press (2004)

• Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, Adv. in Math. 205 (2006), no. 1, 163-228 (arXiv:math/0403403)

• Anthony Elmendorf, Michael Mandell, Permutative categories, multicategories, and algebraic K-theory, Algebraic & Geometric Topology 9 (2009) 2391-2441 (arXiv:0710.0082, euclid:1513797088)

• Bertrand Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ., 24:No. 20, 564–579, 2010.

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

• Anton Deitmar, Remarks on zeta functions and K-theory over $\mathbb{F}_1$ (arXiv:math/0605429)

• Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)

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

• Anna Marie Bohmann, Angélica Osorno, Constructing equivariant spectra via categorical Mackey functors, Algebraic & Geometric Topology 15.1 (2015): 537-563 (arXiv:1405.6126)

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

• Bert Guillou, Peter May, around section 4.4 of Permutative $G$-categories in equivariant infinite loop space theory (arXiv:1207.3459)