Product category
In the mathematical field of category theory, the product of two categories C and D, denoted and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.Definition
The product category has:
- as objects:
- :pairs of objects, where A is an object of C and B of D;
- as arrows from to :
- :pairs of arrows, where is an arrow of C and is an arrow of D;
- as composition, component-wise composition from the contributing categories:
- :;
- as identities, pairs of identities from the contributing categories:
- :1 =.
Relation to other categorical concepts
For small categories, this is the same as the action on objects of the categorical product in the category Cat. A functor whose domain is a product category is known as a bifunctor. An important example is the Hom functor, which has the product of the opposite of some category with the original category as domain:Generalization to several arguments
Just as the binary Cartesian product is readily generalized to an n-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of n categories. The product operation on categories is commutative and associative, up to isomorphism, and so this generalization brings nothing new from a theoretical point of view.
