dualcategory

Dualcategory refers to the opposite category of a given category, commonly denoted C^op. In the opposite, the objects are the same as in C, but all arrows are reversed: for objects A and B, morphisms from A to B in C^op correspond to morphisms from B to A in C. The identity morphisms and composition are defined so that (g ∘ f)^op = f^op ∘ g^op, and the morphism corresponding to f: A → B in C is f^op: B → A in C^op.

Construction and basic facts proceed by reversing arrows. If C is a category, its dual or opposite C^op has the same objects, but a morphism from A to B in C^op is precisely a morphism from B to A in C. This reversal preserves composition up to the op notation, and identities remain on the same objects, viewed appropriately in the opposite setting. Because of this reversal, many dual notions arise: limits in C correspond to colimits in C^op, and vice versa. A contravariant functor from C to D is the same as a covariant functor from C^op to D.

Applications and relevance are broad in category theory and its uses. The opposite category formalizes dual arguments and dual constructions, enabling systematic translation of statements about limits to statements about colimits. It also clarifies the relationship between covariant and contravariant functors, since contravariant functors C → D are equivalent to covariant functors C^op → D.

Examples illustrate the idea. If C is a poset viewed as a category, C^op corresponds to the same elements with the order reversed. In Set, Set^op has the same objects as Set but reversed arrows, which changes many categorical properties and typically makes Set^op unsuitable as a stand-alone universe for ordinary sets.

See also: opposite category, duality, contravariant functor.