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 a straightforward extension of the concept of the Cartesian product of two sets.
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) = (f2o f1, g2o 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: