Codirected

Codirected is an adjective used mainly in order theory and category theory to describe the dual notion of directedness. In a poset, a relation is codirected when every finite subset has a common lower bound; equivalently, the dual order is directed. This means that for any finite collection of elements, there exists another element that is less than or equal to each of them.

In practice, codirectedness is often discussed in the context of index categories and diagrams. A category is called cofiltered if it is nonempty, and for any pair of objects there exists a third object with morphisms to both; and for any pair of parallel arrows f, g: j1 → j2, there exists a morphism from j2 to some j3 that equalizes them (the dual of the directed/filtered condition). A diagram indexed by a cofiltered category is called codirected. Such diagrams are the dual notion of filtered diagrams and are used to form inverse limits, pro-objects, and other limit constructions.

Examples include posets in which every pair of elements has a common lower bound; for instance, any total order (where finite subsets have minimum elements) is codirected. In category theory, the opposite category of a filtered category is cofiltered.

See also: directed set, cofiltered category, inverse limit, pro-object, filtered category.