Product category

Short description: Product of two categories, in category theory

In the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.^[1]

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

↑ Mac Lane 1978, p. 37.

Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1. https://archive.org/details/handbookofcatego0000borc/page/22.
Product category in nLab
Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 36-40. ISBN 1441931236. OCLC 851741862.