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

Idea

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

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

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

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 $\mathcal{K}$ 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 $R$ a commutative ring, and $R$Mod its category of modules (projective modules), then $\mathcal{K}(R Mod)$ is Quillen’s algebraic K-theory of $R$. 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.

Related concepts

• regulator in algebraic K-theory

• K-theory of a permutative category, K-theory of a bipermutative category

References

Traditional category theoretic

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

• Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.

See also

• Robert Thomason, First quadrant spectral sequences in algebraic K-theory via homotopy colimits. Comm. Algebra, 10(15):1589–1668,

  1982.

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

• Robert Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995), no. 5, 78–118.

and a new proof of that is in

• Michael Mandell, An Inverse K-Theory Functor (arXiv:1002.3622)

$\infty$-Category theoretic

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

• Ulrich Bunke, Georg Tamme, section 2.1 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)

• Thomas NikolausAlgebraic K-Theory of $\infty$-Operads (arXiv:1303.2198)

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, def. 6.1 in Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

• David Gepner, Moritz Groth, Thomas Nikolaus, Universality of multiplicative infinite loop space machines, arXiv:1305.4550.

• Ulrich Bunke, Georg Tamme, Multiplicative differential algebraic K-theory and applications (arXiv:1311.1421)