In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, the product of two

categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...

''C'' and ''D'', denoted and called a product category, is an extension of the concept of the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

of two sets. Product categories are used to define bifunctors and multifunctors.

Definition

The product category has: *as

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

: *:pairs of objects , where ''A'' is an object of ''C'' and ''B'' of ''D''; *as arrows from to : *:pairs of arrows , where is an arrow of ''C'' and is an arrow of ''D''; *as composition, component-wise composition from the contributing categories: *:; *as identities, pairs of identities from the contributing categories: *:1_{(''A'', ''B'')} = (1_''A'', 1_''B'').

Relation to other categorical concepts

For small categories, this is the same as the action on objects of the

categorical product In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or ring ...

in the category

Cat The cat (''Felis catus'') is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of ...

. A

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

whose domain is a product category is known as a

bifunctor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

. An important example is the

Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...

, which has the product of the opposite of some category with the original category as domain: :Hom : ''C''^op × ''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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

and associative,

up to isomorphism Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...

, and so this generalization brings nothing new from a theoretical point of view.

References

* Definition 1.6.5 in * * Category theory {{Categorytheory-stub