In algebraic K-theory, the K-theory of a categoryC is a sequence of abelian groupsKi 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 onC. 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 smallstable ∞-categories. The motivation for this notion comes from algebraic K-theory of rings. For a ringRDaniel Quillen in introduced two equivalent ways to find the higher K-theory. The plus construction expresses Ki 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 Ki 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 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 be a category whose objects are functors such that, for,, is a cofibration, and is the pushout of and. The category defined in this manner is itself a category with cofibrations. One can therefore iterate the construction, forming the sequence. 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 has as objects the sequences in C, where the first map is a cofibration, and is a quotient map, i.e. a pushout of the first one along the zero mapA → 0. This category has a natural Waldhausen structure, and the forgetful functor from to C × C respects it. The additivity theorem says that the induced map on K-theory spaces is a homotopy equivalence. For dg-categories the statement is similar. Let C be a small pretriangulated dg-category with a semiorthogonal decomposition. Then the map of K-theory spectra K → K ⊕ K is a homotopy equivalence. In fact, K-theory is a universal functor satisfying this additivity property and Morita invariance.
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.
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.
