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K-theory

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Idea

Given a stable (∞,1)-category CC, its decategorification

K 0(C)={equivalenceclasses[c]ofobjectscC} K_0(C) = \{equivalence \,classes\; [c]\; of \,objects \,c \in C\}

naturally inherits the structure of an abelian group from the fibration sequences in CC:

for

axc a \to x \to c

a fibration sequence (i.e. a homotopy exact sequence) the abelian group operation

+:K 0(C)×K 0(C)K 0(C) + : K_0(C) \times K_0(C) \to K_0(C)

is such that

[x]=[a]+[c]. [x] = [a] + [c] \,.

The group K 0(C)K_0(C) is called the K-group of CC or the Grothendieck group of CC. See in particular the latter entry for more details.

The “K” is chosen by Grothendieck for the German word Klasse for “class”. The K-group of CC is the group of equivalence classes of CC: it is a group due to the existence of a notion of exact sequences in CC.

K-theory starts with the study of these K-groups and their higher analogues. Sometimes the K-groups themselves are called “K-theory”. One would say for instance: ”K(C)K(C) is the K-theory of CC.”

More generally, there is a symmetric groupal ∞-groupoid K(C)\mathbf{K}(C) – i.e. a connective spectrum – in between the decategorification from CC to K(C)K(C) of which K(C)K(C) is the set of connected components

CK(C)π 0K(C)=K(C). C \mapsto \mathbf{K}(C) \to \pi_0 \mathbf{K}(C) = K(C) \,.

In nice cases this is the degree 0 part of a non-connective spectrum which is then the K-theory spectrum of CC. This is also called the Waldhausen K-theory of CC.

Special cases and models

Much of the literature on K-theory discusses constructions that model the above abstract setup in terms of model categories, or just their homotopy categories, often of the derived catgeories type and then often expressed in terms of the abelian category or more generally Quillen exact category from which the derived category is derived.

Only a subset of the structure on a model category is necessary in order to conveniently extract the K-groups of the presented stable (∞,1)-category. For that reason the axioms of a Waldhausen category have been devised to provide just the necessary convenient prerequisites to compute the K-groups of the (∞,1)-category presented by the underlying homotopical category.

Definition

Recall that given a (∞,1)-category CC, we may regard it as a complete Segal space C ,C_{\bullet,\bullet}, a bisimplicial set. For instance if CC is originally given as a quasicategory then

C ,:[n],[m]Core(Func(Δ n,C)) m, C_{\bullet,\bullet} : [n],[m] \mapsto Core(Func(\Delta^n,C))_{m} \,,

where Core(Func(Δ n,C))Core(Func(\Delta^n,C)) denotes the maximal Kan complex inside the (∞,1)-category of (∞,1)-functors from Δ n\Delta^n to CC.

Definition

For 𝒞\mathcal{C} an (∞,1)-category and nn \in \mathbb{N}, write Gap(Δ n,𝒞)Gap(\Delta^n, \mathcal{C}) for the full sub-\infty-category on Func(Arr(Δ n),𝒞)Func(Arr(\Delta^n),\mathcal{C} ) on those objects FF for which

  • the diagonal F(n,n)F(n,n) is inhabited by zero objects, for all nn;

  • all diagrams of the form

    F(i,j) F(i,k) F(j,j) F(j,k) \array{ F(i,j) &\to& F(i,k) \\ \downarrow && \downarrow \\ F(j,j) &\to& F(j,k) }

    is an (∞,1)-pushout.

Definition

Let CC be a stable (∞,1)-category. Then its Waldhausen K-theory

K(C):=lim nCore(Gap(C Δ n)) \mathbf{K}(C) := \underset{\rightarrow}{\lim}_n Core(Gap(C^{\Delta^n}))

is the geometric realization of/homotopy colimit of the degreewise core of the GapGap, def. 1, of the corresponding complete Segal space (as a simplicial diagram of \infty-groupoids).

This is remark 11.4 in StCat.

This construction is also conjectured in the last section of Toen-Vezzosi’s A remark on K-theory .

Remark

In the case that CC is the simplicial localization of a Waldhausen category C¯\bar C the explicit way to obtain this is the Waldhausen S-construction.

Remark

It should be true that with this definition we have an isomorphism of groups

K(C)π 0K(C). K(C) \simeq \pi_0 \mathbf{K}(C) \,.
Remark

This Waldhausen/hocolim-construction gives the connective K-theory, taking values in connective spectra. The universal completion to functor that sends homotopy cofibers of stable (infinity,1)-categories to homotopy cofibers of spectra is the corresponding unconnective 𝕂\mathbb{K}-functor.

There is a universal characterization of the construction of the 𝕂\mathbb{K}-theory spectrum of a stable (,1)(\infty,1)-category AA:

there is an (,1)(\infty,1)-functor

U:(,1)StabCatN U : (\infty,1)StabCat \to N

to a stable (,1)(\infty,1)-category which is universal with the property that it respects colimits and exact sequences in a suitable way. Given any stable (,1)(\infty,1)-category AA, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

K(A)Hom(U(Sp),U(A)), K(A) \simeq Hom(U(Sp), U(A)) \,,

where SpSp denotes the stable (,1)(\infty,1)-category of compact spectra. (BGT)

References

It was in

that it was proven that the the Waldhausen S-construction of the K-theory spectrum depends precisely on the simplicial localization of the Waldhausen category, i.e. of the (∞,1)-category that it presents.

In view of this remark 11.4 in

interprets the construction of the K-theory spectrum as a natural operation of stable (∞,1)-categories, as described above.

The universal property of the (,1)(\infty,1)-categorical definition is studied in

The standard constructions of K-theory spectra from Quillen exact categories are discussed in detail in chapter 1 of

  • Eric M. Friedlander, Daniel R. Grayson, Handbook of K-theory, Springer Verlag .

A useful introduction to the definition and computation of K-groups (with a little on K-spectra) is

  • Charles Weibel, The K-book: An introduction to algebraic K-theory (web)

Revised on April 9, 2014 07:26:10 by Urs Schreiber (145.116.131.80)