Science Fair Project Encyclopedia
Dual (category theory)
In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in Cop being the morphisms from Y to X in C. Hence, the dual of a dual of a category is itself. The dual category is also called the opposite category.
- x ≤new y if and only if y ≤ x.
For example, there are opposite pairs child/parent, or descendant/ancestor.
This is a special case, since partial orders correspond to a certain kind of category in which Mor(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws.
Generalising that observation, limits and colimits are interchanged when one passes to the opposite category. This is immediately useful, when one can identify the opposite category in concrete terms. For example the category of affine schemes is equivalent to the opposite of the category of commutative rings. The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff abelian topological groups and the opposite of the category of (discrete) abelian groups. The category of Stone spaces and continuous functions is equivalent to the opposite of the category of Boolean algebras and homomorphisms.
See also: Duality (mathematics)
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details