Product category

In the mathematical field of category theory, the product of two categories C and D, denoted C × D 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 C × D has:
 * as objects:
 * pairs of objects (A, B), where A is an object of C and B of D;
 * as arrows from (A1, B1) to (A2, B2):
 * pairs of arrows (f, g), where f : A1 → A2 is an arrow of C and g : B1 → B2 is an arrow of D;
 * as composition, component-wise composition from the contributing categories:
 * (f2, g2) o (f1, g1) = (f2 o f1, g2 o g1);
 * as identities, pairs of identities from the contributing categories:
 * 1(A, B) = (1A, 1B).

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:


 * Hom : Cop × C → Set.

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.