From a fundamental category theoretic perspective, there are two fundamental kinds of dualities:
abstract duality: formally nothing but the reversal of arrows in a category , i.e. the passage from to the opposite category ;
concrete duality: the oppositely-oriented image of some diagram under a contravariant internal hom functor .
(See Chapter 7 of Lawvere and Rosebrugh’s Sets for Mathematics.)
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)
A duality or dual equivalence is an equivalence between a category and the abstract dual (i.e. opposite) of a category .
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Of particular interest are concrete dualities between concrete categories , i.e. categories equipped with a faithful functor , to Set, which are represented by objects , with the same underlying set . Such objects are known as dualizing objects.