Mathematical category formed by reversing morphisms
In category theory, a branch of mathematics, the opposite category or dual categoryCop of a given categoryC is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, .
An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by
x ≤opy if and only if y ≤ x.
The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/up-set, ideal/filter etc. This order theoretic duality is in turn a special case of the construction of opposite categories as every ordered set can be understood as a category.
Given a semigroup (S, ·), one usually defines the opposite semigroup as (S, ·)op = (S, *) where x*y ≔ y·x for all x,y in S. So also for semigroups there is a strong duality principle. Clearly, the same construction works for groups, as well, and is known in ring theory, too, where it is applied to the multiplicative semigroup of the ring to give the opposite ring. Again this process can be described by completing a semigroup to a monoid, taking the corresponding opposite category, and then possibly removing the unit from that monoid.
Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 33. ISBN1441931236. OCLC851741862.