Product category

In the mathematical field of category theory, the product of two categories "C" and "D", denoted nowrap| "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 nowrap| "C" × "D" has:
*as objects:
*:pairs of objects nowrap| ("A", "B"), where "A" is an object of "C" and "B" of "D";
*as arrows from nowrap| ("A"₁, "B"₁) to nowrap| ("A"₂, "B"₂):
*:pairs of arrows nowrap| ("f", "g"), where nowrap| "f" : "A"₁ → "A"₂ is an arrow of "C" and nowrap| "g" : "B"₁ → "B"₂ is an arrow of "D";
*as composition, component-wise composition from the contributing categories:
*:nowrap|1= ("f"₂, "g"₂) o ("f"₁, "g"₁) = ("f"₂ o "f"₁, "g"₂ o "g"₁);
*as identities, pairs of identities from the contributing categories:
*:1_{("A", "B")} = (1_"A", 1_"B").

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 : "C"^op × "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.