Special and general types

Special notions


Extra structure





Given a stable (∞,1)-category CC, its naive decategorification (its set of objects modulo the relation of equivalence) naturally inherits the structure of an abelian monoid from the biproducts in CC. That is, we define [a]+[b]=[ab][a]+[b] = [a\oplus b]. We can then further pass to the Grothendieck group, adding formal additive inverses to get an abelian group.

Note that such a biproduct sits in a split (co)fibration sequence aabba\to a\oplus b \to b. Often we would like to obtain an analogous additivity result for arbitrary (co)fibration sequences (a.k.a. distinguished triangles). This requires imposing the relation [c]=[a]+[b][c] = [a] + [b] for any fibration sequence acba\to c\to b, thereby passing to a quotient abelian group called K 0(C)K_0(C), the K-group or Grothendieck group of CC; see 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 K n(C)K_n(C), collectively denoted K(C)K(C). 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), for which K 0(C)K_0(C) is the set of connected components

CK(C)π 0K(C)=K 0(C). C \mapsto \mathbf{K}(C) \to \pi_0 \mathbf{K}(C) = K_0(C) \,.

and more generally K n(C)=π nK(C)K_n(C) = \pi_n \mathbf{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.


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.


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.


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. See also Blumberg-Gepner-Tabuada, section 7.

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


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.


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) \,.

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)


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

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 August 29, 2017 03:32:00 by David Corfield (