In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, the product of two
categories ''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 and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of two
sets. Product categories are used to define
bifunctors and multifunctors.
Definition
The product category has:
*as
objects:
*: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 the category
Cat. 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. An important example is the
Hom functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
, 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. Perhaps most familiar as a pr ...
and
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
,
up to isomorphism, and so this generalization brings nothing new from a theoretical point of view.
References
* Definition 1.6.5 in
*
*
Category theory
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