Category theory | K-theory

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 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 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 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 extended the notion of K-theory even further, to very different kinds of categories, including the category of topological spaces. (Wikipedia).

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Topological space | Abelian group | Semiorthogonal decomposition | Cotriple homology | Exact sequence | Forgetful functor | Waldhausen category | Sphere spectrum | Volodin space | Algebraic K-theory | Eilenberg–MacLane space | Spectrum (topology) | Simplicial set | Barratt–Priddy theorem | Glossary of category theory | Projective module | Stable ∞-category | Abelian category | Pushout (category theory) | Natural number | Q-construction | Exact category | Ring (mathematics) | Categorification | Hochschild homology | Category (mathematics) | Morita equivalence | Monomorphism | Grothendieck group | Module (mathematics)