Duality (mathematics)

Duality (mathematics)

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. As involutions sometimes have fixed points, the dual of A is sometimes A itself. For example, Desargues' theorem in projective geometry is self-dual in this sense.

In mathematical contexts, duality has numerous meanings, and although it is “a very pervasive and important concept in (modern) mathematics” and “an important general theme that has manifestations in almost every area of mathematics”, there is no single universally agreed definition that unifies all concepts of duality.

Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.

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: VW, its dual f*: V*W*.

Read more about Duality (mathematics):  Order-reversing Dualities, Dimension-reversing Dualities, Duality in Logic and Set Theory, Dual Objects, Analytic Dualities, Poincaré-style Dualities