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In
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 (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 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\tim ...
of sets, the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of groups or
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film an ...
, and the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s. Essentially, the product of a
family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Ideall ...
of objects is the "most general" object which admits a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
to each of the given objects.


Definition


Product of two objects

Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the following diagram commutes: *: Whether a product exists may depend on C or on X_1 and X_2. If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of ''the'' product. This has the following meaning: let X', \pi_1' , \pi_2' be another cartesian product, there exists a unique isomorphism h : X' \to X_1 \times X_2 such that \pi_1' = \pi_1 \circ h and \pi_2' = \pi_2 \circ h . The morphisms \pi_1 and \pi_2 are called the canonical projections or projection morphisms. Given Y and f_1, f_2, the unique morphism f is called the product of morphisms f_1 and f_2 and is denoted \langle f_1, f_2 \rangle.


Product of an arbitrary family

Instead of two objects, we can start with an arbitrary family of objects indexed by a set I. Given a family \left(X_i\right)_ of objects, a product of the family is an object X equipped with morphisms \pi_i : X \to X_i, satisfying the following universal property: *For every object Y and every I-indexed family of morphisms f_i : Y \to X_i, there exists a unique morphism f : Y \to X such that the following diagrams commute for all i \in I: *: The product is denoted \prod_ X_i. If I = \, then it is denoted X_1 \times \cdots \times X_n and the product of morphisms is denoted \langle f_1, \ldots, f_n \rangle.


Equational definition

Alternatively, the product may be defined through equations. So, for example, for the binary product: * Existence of f is guaranteed by existence of the operation \langle \cdot,\cdot \rangle. * Commutativity of the diagrams above is guaranteed by the equality: for all f_1, f_2 and all i \in \, \pi_i \circ \left\langle f_1, f_2 \right\rangle = f_i * Uniqueness of f is guaranteed by the equality: for all g : Y \to X_1 \times X_2, \left\langle \pi_1 \circ g, \pi_2 \circ g \right\rangle = g.


As a limit

The product is a special case of a limit. This may be seen by using a
discrete category In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: :hom''C''(''X'', ''X'') = {id''X''} for all objects ''X'' :hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ ''Y ...
(a family of objects without any morphisms, other than their identity morphisms) as the
diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a thr ...
required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set I considered as a discrete category. The definition of the product then coincides with the definition of the limit, \_i being a cone and projections being the limit (limiting cone).


Universal property

Just as the limit is a special case of the
universal construction Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...
, so is the product. Starting with the definition given for the universal property of limits, take \mathbf as the discrete category with two objects, so that \mathbf^ is simply the
product category In the mathematical field of 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 of two sets. Product categories are used to define bifun ...
\mathbf \times \mathbf. The
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct a ...
\Delta : \mathbf \to \mathbf \times \mathbf assigns to each object X the
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
(X, X) and to each morphism f the pair (f, f). The product X_1 \times X_2 in C is given by a
universal morphism In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
from the functor \Delta to the object \left(X_1, X_2\right) in \mathbf \times \mathbf. This universal morphism consists of an object X of C and a morphism (X, X) \to \left(X_1, X_2\right) which contains projections.


Examples

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets X_i the product is defined as \prod_ X_i := \left\ with the canonical projections \pi_j : \prod_ X_i \to X_j, \quad \pi_j\left(\left(x_i\right)_\right) := x_j. Given any set Y with a family of functions f_i : Y \to X_i, the universal arrow f : Y \to \prod_ X_i is defined by f(y) := \left(f_i(y)\right)_. Other examples: * In the category of topological spaces, the product is the space whose underlying set is the Cartesian product and which carries the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
. The product topology is the coarsest topology for which all the projections are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
. * In the category of modules over some ring R, the product is the Cartesian product with addition defined componentwise and distributive multiplication. * In the category of groups, the product is the
direct product of groups In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is o ...
given by the Cartesian product with multiplication defined componentwise. * In the category of graphs, the product is the
tensor product of graphs In graph theory, the tensor product of graphs and is a graph such that * the vertex set of is the Cartesian product ; and * vertices and are adjacent in if and only if ** is adjacent to in , and ** is adjacent to in . The tensor pro ...
. * In the
category of relations In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so . The composition of two re ...
, the product is given by the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
. (This may come as a bit of a surprise given that the category of sets is a
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of the category of relations.) * In the category of algebraic varieties, the product is given by the
Segre embedding In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. Definition The Segre map may be defined as the map :\ ...
. * In the category of semi-abelian monoids, the product is given by the
history monoid In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid p ...
. * In the category of
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
and
short map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 19 ...
s, the product carries the norm. * A
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
can be treated as a category, using the order relation as the morphisms. In this case the products and
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
s correspond to greatest lower bounds ( meets) and least upper bounds (
joins Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two to ...
).


Discussion

An example in which the product does not exist: In the category of fields, the product \Q \times F_p does not exist, since there is no field with homomorphisms to both \Q and F_p. Another example: An
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
(that is, I is the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
) is the same as a
terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group G there are infinitely many morphisms \Z \to G, so G cannot be terminal. If I is a set such that all products for families indexed with I exist, then one can treat each product as 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 ...
\mathbf^I \to \mathbf. How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is 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 ...
. For f_1 : X_1 \to Y_1, f_2 : X_2 \to Y_2 we should find a morphism X_1 \times X_2 \to Y_1 \times Y_2. We choose \left\langle f_1 \circ \pi_1, f_2 \circ \pi_2 \right\rangle. This operation on morphisms is called Cartesian product of morphisms. Second, consider the general product functor. For families \left\_i, \left\_i, f_i : X_i \to Y_i we should find a morphism \prod_ X_i \to \prod_ Y_i. We choose the product of morphisms \left\_i. A category where every finite set of objects has a product is sometimes called a Cartesian category (although some authors use this phrase to mean "a category with all finite limits"). The product is
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 ...
. Suppose C is a Cartesian category, product functors have been chosen as above, and 1 denotes a terminal object of C. We then have
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
s X \times (Y \times Z) \simeq (X\times Y) \times Z \simeq X \times Y \times Z, X \times 1 \simeq 1 \times X \simeq X, X \times Y \simeq Y \times X. These properties are formally similar to those of a commutative
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
; a Cartesian category with its finite products is an example of a
symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...
.


Distributivity

For any objects X, Y, \text Z of a category with finite products and coproducts, there is a
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical exampl ...
morphism X \times Y + X \times Z \to X \times (Y + Z), where the plus sign here denotes the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
. To see this, note that the universal property of the coproduct X \times Y + X \times Z guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed): The universal property of the product X \times (Y + Z) then guarantees a unique morphism X \times Y + X \times Z \to X \times (Y + Z) induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, there is the canonical isomorphism X\times (Y + Z)\simeq (X\times Y) + (X \times Z).


See also

*
Coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
 – the dual of the product *
Diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct a ...
 – the
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
of the product functor. * * * * *


References

* * Chapter 5. * * Definition 2.1.1 in


External links


Interactive Web page
which generates examples of products in the category of finite sets. Written b
Jocelyn Paine
* {{DEFAULTSORT:Product (Category Theory) Limits (category theory)