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 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 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 = \ 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 \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$S^C = S \cdots SC$. 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 $\mathcal(C)$ has as objects the sequences $A \rightarrowtail B \twoheadrightarrow A'$ in ''C'', where the first map is a cofibration, and $B \twoheadrightarrow A'$ 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' \mapsto (A, A') from $\mathcal(C)$ to ''C'' × ''C'' respects it. The additivity theorem says that the induced map on K-theory spaces $K(\mathcal(C)) \to K(C) \times K(C)$ is a homotopy equivalence.

For dg-categories the statement is similar. Let ''C'' be a small pretriangulated dg-category with a semiorthogonal decomposition $C \cong \langle C_1, C_2 \rangle$. Then the map of K-theory spectra K(''C'') → K(''C''₁) ⊕ K(''C''₂) 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_+ = \$ 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

Notes

References

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Further reading

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For the recent ∞-category approach, see
*{{cite book, last1=Dyckerhoff, first1=Tobias, last2=Kapranov, first2=Mikhail, title=Higher Segal spaces I, chapter=, series=Lecture Notes in Mathematics, year=2019, volume=2244, publisher=Springer , location=Cham , doi=10.1007/978-3-030-27124-4, arxiv=1212.3563, isbn=978-3-030-27122-0, s2cid=117874919

Category theory
K-theory