Filtered categoryIn , filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category, which will be recalled below. A is filtered when it is not empty, for every two objects and in there exists an object and two arrows and in , for every two parallel arrows in , there exists an object and an arrow such that . A filtered colimit is a colimit of a functor where is a filtered category.
Algebraic K-theoryAlgebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
Disjoint unionIn mathematics, a disjoint union (or discriminated union) of a family of sets is a set often denoted by with an injection of each into such that the of these injections form a partition of (that is, each element of belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In , the disjoint union is the coproduct of the , and thus defined up to a bijection. In this context, the notation is often used. The disjoint union of two sets and is written with infix notation as .
Limit (category theory)In , a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as , and inverse limits. The of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, s and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
