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, ca ...
, 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 ''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\t ...
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 ai ...
:
*: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 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 ...
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 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 of ...
and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
,
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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