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 an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.[1]


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 : A1A2 is an arrow of C and g : B1B2 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) = (1A, 1B).

Relation to other categorical conceptsEdit

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 : Cop × CSet.

Generalization to several argumentsEdit

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.


  1. ^ 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]. Vol. 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1.
  • 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.