nLab
K-theory

Idea

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

$$K_0(C)=\{\mathrm{equivalence}\mathrm{classes}[c]\mathrm{of}\mathrm{objects}c\in C\}$$K_0(C) = \{equivalence \,classes\; [c]\; of \,objects \,c \in C\}

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

for

$$a\to x\to c$$a \to x \to c

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

$$+:K_0(C)\times K_0(C)\to K_0(C)$$+ : K_0(C) \times K_0(C) \to K_0(C)

is such that

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

The group $K_0(C)$ is called the K-group of $C$ or the Grothendieck group of $C$. 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 $C$ is the group of equivalence classes of $C$: it is a group due to the existence of a notion of exact sequences in $C$.

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

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

$$C\mapsto K(C)\to {\pi }_{0}K(C)=K(C).$$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 $C$. This is also called the Waldhausen K-theory of $C$.

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 prerequesites to compute the K-groups of the (∞,1)-category presented by the underlying homotopical category.

• In particular, the K-group associated to the stable (∞,1)-category ${\mathrm{Ch}}^b(A)$ of bounded chain complexes in an abelian category or exact category $A$ is often called the K-group of $A$ itself and just denoted

  $$K(A):=K({\mathrm{Ch}}^b(A)).$$K(A) := K(Ch^b(A)) \,.

  Most explicit constructions of K-theory spectra start with the data of an exact category, such as notably Quillen’s Q-construction? and the Waldhausen S-construction.

• In particular if the exact category $A$ is that of vector bundles on a topological space $X$

  $$A=\mathrm{VectBund}(X)$$A = VectBund(X)

  the corresponding K-group is degree 0 topological K-theory. This was the original of the notion and the term K-theory.

Definition

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

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

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

Definition

Let $C$ be a stable (∞,1)-category. Then its Waldhausen K-theory $K(C)$ is given by the the diagonal simplicial set

$$K(C)=\mathrm{diag}{C}_{•,•}$$\mathbf{K}(C) = diag C_{\bullet, \bullet}

where $C_{•,•}$ is the corresponding complete Segal space.

This is remark 11.4 in StCat in view of the lemma above that and using the discussion on p. 27 of CobHyp .

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

In the case that $C$ is the simplicial localization of a Waldhausen category $\overline{C}$ the explicit way to obtain this is the Waldhausen S-construction.

Proposition

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

$$K(C)\simeq {\pi }_{0}K(C).$$K(C) \simeq \pi_0 \mathbf{K}(C) \,.

Related entries

• Grothendieck group

• K-theory spectrum

• topological K-theory

• algebraic K-theory

• Karoubi K-theory

• twisted K-theory

• differential K-theory

• K-theory and physics

References

It was in

• Bertrand Toen, Gabriele Vezzosi, A remark on K-theory and $S$-categories (arXiv).

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

• Jacob Lurie, Stable ∞-Categories .

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

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)