Dual (category theory)
Duality in category theory is the idea that many statements about a category C have a twin statement about its opposite category Cop, created by reversing every arrow and the order of composition.
The opposite category Cop is formed by reversing every morphism of C. If a statement S about C is true, its dual Sop about Cop is also true; if S is false in C, Sop is false in Cop. In other words, S is true in C if and only if Sop is true in Cop.
Sometimes Cop is abstract, so another category D is said to be in duality with C if D is equivalent to Cop. If C is equivalent to Cop, the category is self-dual.
The basic language used in category theory has two sorts: objects and morphisms. It includes which objects are sources or targets of morphisms, and how to compose morphisms. The dual of a statement is formed by reversing arrows and reversing the order of composition.
As an example, a morphism f in C is a monomorphism if and only if its reverse fop in Cop is an epimorphism.
Partial orders are a simple special case: they correspond to categories where there is at most one morphism between any two objects. In this setting, duality simply reverses the order.
In logic and lattice theory, duality often behaves like negation: it swaps operations such as meets and joins in lattices, which is related to De Morgan-type laws.
This page was last edited on 3 February 2026, at 09:27 (CET).