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]. Vol. 1. Cambridge University Press. p. 22. ISBN0-521-44178-1.
Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 36–40. ISBN1441931236. OCLC 851741862.
This category theory-related article is a stub. You can help Wikipedia by expanding it.
product, category, product, categories, redirects, here, operation, objects, category, product, category, theory, mathematical, field, category, theory, product, categories, denoted, called, product, category, extension, concept, cartesian, product, sets, prod. 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 an extension of the concept of the Cartesian product of two sets Product categories are used to define bifunctors and multifunctors 1 Contents 1 Definition 2 Relation to other categorical concepts 3 Generalization to several arguments 4 ReferencesDefinition EditThe 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 EditFor 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 EditJust 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 Edit 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 at the 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 This category theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Product category amp oldid 1084454707, wikipedia, wiki, book, books, library,
