Codirectedness

Codirectedness is the dual notion of directedness in order theory and related areas. In a partially ordered set (P, ≤), a nonempty subset D is codirected if for any a, b in D there exists c in D such that c ≤ a and c ≤ b. Equivalently, every finite subset of D has a lower bound in D.

In category theory, the dual notion of a filtered category is a cofiltered (codirected) category; diagrams indexed by such categories are used to define inverse limits. Roughly, a cofiltered index category provides a means to refine systems in the opposite direction, enabling the construction of limits that “go backwards” along a diagram.

Codirectedness contrasts with directedness. A directed subset has a common upper bound for any two of its elements, while a codirected subset has a common lower bound. This duality makes codirectedness a natural tool for arguments about refinement, convergence from above, or inverse-type constructions.

Examples help illustrate the idea. In the lattice of ideals of a ring ordered by inclusion, a subset D is codirected if for any I, J in D there exists K in D with K ⊆ I ∩ J; in particular, if D is closed under finite intersections, it is codirected. The natural numbers with the usual order form a codirected set, since min(i, j) is a lower bound in N for any i, j.

Applications of codirectedness appear in the study of inverse limits, pro-objects, and completion procedures in algebra and topology. As the dual counterpart to directedness, codirectedness provides a framework for handling refinements and limits taken in the reverse or backward direction.