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

Contents

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

  • 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

\infty-Category theoretic

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

Last revised on January 28, 2015 at 12:04:38. See the history of this page for a list of all contributions to it.