# Product category

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.

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