Rahul Sharma (Editor)

Product category

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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.

Contents

Definition

The product category C × D has:

  • as objects:
  • 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:
  • as identities, pairs of identities from the contributing categories:
  • 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 : Cop × CSet.

    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

    Product category Wikipedia