abstract duality: opposite category,
Instances of “dualities” relating two different, maybe opposing, but to some extent equivalent concepts or phenomena are ubiquitous in mathematics (and in mathematical physics, see at dualities in physics).
In terms of general abstract concepts in category theory instances of dualities might be (and have been) organized as follows:
abstract duality – the operation of sending a category to its opposite category is such an involution on Cat itself (and in fact this is the only non-trivial automorphism of Cat, see here). This has been called abstract duality. While the construction is a priori tautologous, any given opposite category often is equivalent to a category known by other means, which makes abstract duality interesting.
concrete duality – given a closed category and any object of it, then the operation obtained by forming the internal hom into sends each object to something like a -dual object. This is particularly so if is indeed a dualizing object in a closed category in that applying this operation twice yields an equivalence of categories (so that is a (contravariant) involution on ). If is in addition a closed monoidal category then under some conditions on (but not in general) this kind of concrete dualization coincides with the concept of forming dual objects in monoidal categories.
From (Lawvere-Rosebrugh, chapter 7):
Not every statement will be taken into its formal dual by the process of dualizing with respect to , and indeed a large part of the study of mathematics
space vs. quantity
and of logic
theory vs. example
may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite correspond or fail to correspond. (p. 122)
adjunction – another categorical concept of duality is that of adjunction, as in pairs of adjoint functors. Via the many incarnations of universal constructions in category theory this accounts for all dualities that arise as instances as the dual pairs
left and right Kan extension
(Given that the saying has it that “Everything in mathematics is a Kan extension”, this goes some way in explaining the ubiquity of duality in mathematics.)
Adjunctions and specifically dual adjunctions (“Galois connections”) may be thought of as a generalized version of the above abstract duality: every dual adjunction induces a maximal dual equivalence between subcategories.
the duality between space and quantity
Poincaré duality for finite dimensional (oriented) closed manifolds
Verdier duality for abelian categories of sheaves; e.g. for a category of sheaves of abelian groups.
Of particular interest are concrete dualities between concrete categories , i.e. categories equipped with faithful functors
H.-E. Porst, W. Tholen, Concrete Dualities in Category Theory at Work, Herrlich, Porst (eds.) pdf
wikipedia duality (mathematics)