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Duality (mathematics)
URLﾘﾝｸ(en.wikipedia.org)
In mathematics, a duality, generally speaking, translates concepts,
theorems or mathematical structures into other concepts,
theorems or structures, in a one-to-one fashion,
often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A.
Such involutions sometimes have fixed points, so that the dual of A is A itself.
For example, Desargues' theorem in projective geometry is self-dual in this sense.
Duality can also be seen as a functor, at least in the realm of vector spaces. There it is allowed to assign to each space its dual space and the pullback construction allows to assign for each arrow f: V → W, its dual f?: W? → V?.
Contents
1 Order-reversing dualities
2 Dimension-reversing dualities
3 Duality in logic and set theory
4 Dual objects
5 Dual categories
5.1 Opposite category and adjoint functors
5.2 Examples
6 Analytic dualities
7 Poincare-style dualities
8 See also
9 Notes
10 References
10.1 Duality in general
10.2 Duality in algebraic topology
10.3 Specific dualities