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) = (f₂ o f₁, g₂ o g₁);

• 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.

References

• Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Volume 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1. 
• Product category in nLab