In the mathematical field of category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, the product of two categories C and D, denoted and called a product category, is a straightforward extension of the concept of the Cartesian product

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

 of two sets.

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_{(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

Category of small categories

In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories...

. A functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

 whose domain is a product category is known as a bifunctor. An important example is the Hom functor

Hom functor

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...

, which has the product of the opposite

Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...

 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.