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

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

(,1)(\infty,1)-Category theory

Contents

Idea

The algebraic K-theory 𝒦(𝒞)\mathcal{K}(\mathcal{C}) of a symmetric monoidal (∞,1)-category 𝒞\mathcal{C} is the generalized (Eilenberg-Steenrod) cohomology theory represented by the ∞-group completion of the commutative ∞-monoid which is the core of 𝒞\mathcal{C}.

This construction subsumes various other construction in algebraic K-theory. Specifically when restricted to 1-categories it reproduces the classical construction by (Segal 74) described at K-theory of a permutative category, see below.

Definition

Write

𝒦:CMon (Cat)CMon (core)CMon (Gpd)FAbGrp (Grpd)Spectra \mathcal{K} \;\colon\; CMon_\infty(\infty Cat) \stackrel{CMon_\infty(core)}{\longrightarrow} CMon_\infty(\infty Gpd) \stackrel{F}{\longrightarrow} AbGrp_\infty(\infty Grpd) \hookrightarrow Spectra

for the composite of

  1. the core functor from symmetric monoidal (∞,1)-categories to E-∞ spaces;

  2. their ∞-group completion to abelian ∞-groups;

  3. the inclusion of “abelian ∞-groups”, hence connective spectra into all spectra.

This 𝒦\mathcal{K} is the algebraic K-theory of symmetric monoidal (∞,1)-categories. (Nikolaus 13, below remark 5.3, Bunke-Nikolaus-Völkl 13, def.6.1)

More generally, one may start with construction with other objects that map to Picard ∞-groups, such as (∞,1)-operads (Nikolaus 13). Also, instead of just group-completing one may “ring complete” to produce K-theory spectra equipped with the structure of E-∞ rings (Bunke-Tamme 13, section 2.4).

Properties

Relation to classical algebraic K-theory

For RR a commutative ring, and RRMod its category of modules (projective modules), then 𝒦(RMod)\mathcal{K}(R Mod) is Quillen’s algebraic K-theory of RR. More generally this reproduces the K-theory of a permutative category etc. (Nikolaus 13, section 6).

In particular, applied to the stack of algebraic vector bundles this produces the sheaf of spectra of algebraic K-theory of schemes (Bunke-Tamme 12, section 3.3), see at differential algebraic K-theory – Algebraic K-theory sheaf of spectra.

Examples

On monoidal stacks

Algebraic K-theory is traditionally applied to single symmetric monoidal/stable (∞,1)-categories, but to the extent that it is functorial it may just as well be applied to (∞,1)-sheaves with values in these.

Notably, applied to the monoidal stack of vector bundles (with connection) on the site of smooth manifolds, the K-theory of a monoidal category-functor produces a sheaf of spectra which is a form of differential K-theory and whose geometric realization is the topological K-theory spectrum. For more on this see at differential cohomology hexagon – Differential K-theory.

References

Traditional category theoretic

The functor from symmetric monoidal categories to connective spectra was originally given in

See also

The following article showed that this construction produces all connective spectra, up to equivalence

and a new proof of that is in

\infty-Category theoretic

The natural generalization of the construction to symmetric monoidal (∞,1)-categories appears in