K-theory of a category

In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups K_i(C) associated to it. If C is an abelian category, there is no need for extra data, but in general it only makes sense to speak of K-theory after specifying on C a structure of an exact category, or of a Waldhausen category, or of a dg-category, or possibly some other variants. Thus, there are several constructions of those groups, corresponding to various kinds of structures put on C. Traditionally, the K-theory of C is defined to be the result of a suitable construction, but in some contexts there are more conceptual definitions. For instance, the K-theory is a 'universal additive invariant' of dg-categories and small stable ∞-categories.

The motivation for this notion comes from algebraic K-theory of rings. For a ring R Daniel Quillen in Template:Harvtxt introduced two equivalent ways to find the higher K-theory. The plus construction expresses K_i(R) in terms of R directly, but it's hard to prove properties of the result, including basic ones like functoriality. The other way is to consider the exact category of projective modules over R and to set K_i(R) to be the K-theory of that category, defined using the Q-construction. This approach proved to be more useful, and could be applied to other exact categories as well. Later Friedhelm Waldhausen in Template:Harvtxt extended the notion of K-theory even further, to very different kinds of categories, including the category of topological spaces.

K-theory of Waldhausen categories

In algebra, the S-construction is a construction in algebraic K-theory that produces a model that can be used to define higher K-groups. It is due to Friedhelm Waldhausen and concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration can be thought of as analogous to a monomorphism, and a category with cofibrations is one in which, roughly speaking, monomorphisms are stable under pushouts. According to Waldhausen, the "S" was chosen to stand for Graeme B. Segal.

Unlike the Q-construction, which produces a topological space, the S-construction produces a simplicial set.

Details

The arrow category $Ar(C)$ of a category C is a category whose objects are morphisms in C and whose morphisms are squares in C. Let a finite ordered set $[n]=\{0<1<2<\cdots <n\}$ be viewed as a category in the usual way.

Let C be a category with cofibrations and let $S_{n}C$ be a category whose objects are functors $f:Ar[n]\to C$ such that, for $i\le j\le k$, $f(i=i)=*$, $f(i\le j)\to f(i\le k)$ is a cofibration, and $f(j\le k)$ is the pushout of $f(i\le j)\to f(i\le k)$ and $f(i\le j)\to f(j=j)=*$. The category $S_{n}C$ defined in this manner is itself a category with cofibrations. One can therefore iterate the construction, forming the sequence$S^{(m)}C=S\cdots SC$. This sequence is a spectrum called the K-theory spectrum of C.

The additivity theorem

Most basic properties of algebraic K-theory of categories are consequences of the following important theorem. There are versions of it in all available settings. Here's a statement for Waldhausen categories. Notably, it's used to show that the sequence of spaces obtained by the iterated S-construction is an Ω-spectrum.

Let C be a Waldhausen category. The category of extensions $ℰ(C)$ has as objects the sequences $A\rightarrowtail B\twoheadrightarrow A^{\prime }$ in C, where the first map is a cofibration, and $B\twoheadrightarrow A^{\prime }$ is a quotient map, i.e. a pushout of the first one along the zero map A → 0. This category has a natural Waldhausen structure, and the forgetful functor $[A\rightarrowtail B\twoheadrightarrow A^{\prime }]\mapsto (A,A^{\prime })$ from $ℰ(C)$ to C × C respects it. The additivity theorem says that the induced map on K-theory spaces $K(ℰ(C))\to K(C)\times K(C)$ is a homotopy equivalence.

For dg-categories the statement is similar. Let C be a small pretriangulated dg-category with a semiorthogonal decomposition $C\cong ⟨C_1,C_2⟩$. Then the map of K-theory spectra K(C) → K(C₁) ⊕ K(C₂) is a homotopy equivalence. In fact, K-theory is a universal functor satisfying this additivity property and Morita invariance.

Category of finite sets

Consider the category of pointed finite sets. This category has an object $k_{+}=\{0,1,\dots ,k\}$ for every natural number k, and the morphisms in this category are the functions $f:m_{+}\to n_{+}$ which preserve the zero element. A theorem of Barratt, Priddy and Quillen says that the algebraic K-theory of this category is a sphere spectrum.

Miscellaneous

More generally in abstract category theory, the K-theory of a category is a type of decategorification in which a set is created from an equivalence class of objects in a stable (∞,1)-category, where the elements of the set inherit an Abelian group structure from the exact sequences in the category.

Group completion method

The Grothendieck group construction is a functor from the category of rings to the category of abelian groups. The higher K-theory should then be a functor from the category of rings but to the category of higher objects such as simplicial abelian groups.

Topological Hochschild homology

Waldhausen introduced the idea of a trace map from the algebraic K-theory of a ring to its Hochschild homology; by way of this map, information can be obtained about the K-theory from the Hochschild homology. Bökstedt factorized this trace map, leading to the idea of a functor known as the topological Hochschild homology of the ring's Eilenberg–MacLane spectrum.

K-theory of a simplicial ring

If R is a constant simplicial ring, then this is the same thing as K-theory of a ring.

See also

• Volodin space
• Cotriple homology

References

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Further reading

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For the recent ∞-category approach, see

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