Product category

"Product of categories" redirects here. For the operation on objects of a category, see Product (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 a straightforward extension of the concept of the Cartesian product of two sets.

Definition

The product category C × D has:

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


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