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.
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) = (1A, 1B).
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 × 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.
- 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
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