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 Ki(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[1] and small stable ∞-categories.
[2] The motivation for this notion comes from algebraic K-theory of rings.
For a ring R Daniel Quillen in Quillen (1973) introduced two equivalent ways to find the higher K-theory.
The plus construction expresses Ki(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 Ki(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 Waldhausen (1985) extended the notion of K-theory even further, to very different kinds of categories, including the category of topological spaces.
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.
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.
[3] According to Waldhausen, the "S" was chosen to stand for Graeme B.
One can therefore iterate the construction, forming the sequence
This sequence is a spectrum called the K-theory spectrum of C. Most basic properties of algebraic K-theory of categories are consequences of the following important theorem.
Notably, it's used to show that the sequence of spaces obtained by the iterated S-construction is an Ω-spectrum.
This category has a natural Waldhausen structure, and the forgetful functor
The additivity theorem says that the induced map on K-theory spaces
Let C be a small pretriangulated dg-category with a semiorthogonal decomposition
Then the map of K-theory spectra K(C) → K(C1) ⊕ K(C2) is a homotopy equivalence.
[7] In fact, K-theory is a universal functor satisfying this additivity property and Morita invariance.
for every natural number k, and the morphisms in this category are the functions
A theorem of Barratt, Priddy and Quillen says that the algebraic K-theory of this category is a sphere spectrum.
[4] 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.
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.