TheInfoList

In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, the product of two (or more)
objects Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
in a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
is a notion designed to capture the essence behind constructions in other areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
such as the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of sets, the
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
or
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
, and the product of
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s. Essentially, the product of a
family In human society A society is a Social group, group of individuals involved in persistent Social relation, social interaction, or a large social group sharing the same spatial or social territory, typically subject to the same Politic ...
of objects is the "most general" object which admits a
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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 category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
: * 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 : *: 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. The morphisms $\pi_1$ and $\pi_2$ are called the
canonical projection Canonical may refer to: Science and technology * Canonical form, a natural unique representation of an object, or a preferred notation for some object Mathematics * Canonical coordinates, sets of coordinates that can be used to describe a physica ...
s 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

indexed Index may refer to: Arts, entertainment, and media Fictional entities * Index (A Certain Magical Index), Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo (megastr ...
by a set $I.$ Given a family $\left\left(X_i\right\right)_$ 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 Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
. This may be seen by using a
discrete categoryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

(a family of objects without any morphisms, other than their identity morphisms) as the
diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...
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 A cone is a three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:Ï€Î ...
and projections being the limit (limiting cone).

## Universal property

Just as the limit is a special case of the
universal construction In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
, 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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objec ...

$\mathbf \times \mathbf.$ The
diagonal functor In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...

$\Delta : \mathbf \to \mathbf \times \mathbf$ assigns to each object $X$ the
ordered pair In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

$\left(X, X\right)$ and to each morphism $f$ the pair $\left(f, f\right).$ The product $X_1 \times X_2$ in $C$ is given by a
universal morphism In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
from the functor $\Delta$ to the object $\left\left(X_1, X_2\right\right)$ in $\mathbf \times \mathbf.$ This universal morphism consists of an object $X$ of $C$ and a morphism $\left(X, X\right) \to \left\left(X_1, X_2\right\right)$ which contains projections.

# Examples

In the
category of sets In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, 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 $f\left(y\right) := \left\left(f_i\left(y\right)\right\right)_.$ Other examples: * In the
category of topological spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, the product is the space whose underlying set is the Cartesian product and which carries the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
. The product topology is the
coarsest topologyIn topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the c ...
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 ga ...
. * In the
category of modulesIn algebra Algebra (from ar, Ø§Ù„Ø¬Ø¨Ø±, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
over some ring $R,$ the product is the Cartesian product with addition defined componentwise and distributive multiplication. * In the
category of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, the product is the
direct product of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 ''G'' Ã— ''H'' of graphs ''G'' and ''H'' is a graph such that * the vertex set of ''G'' Ã— ''H'' is the Cartesian product ''V''(''G'') Ã— ''V''(''H''); and * vertices (''g,h'') and (''g',h) are adjacent in ''G'' ...
. * In the
category of relations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, the product is given by the
disjoint union In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. (This may come as a bit of a surprise given that the category of sets is a
subcategory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the category of relations.) * In the category of
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
, the product is given by the
Segre embeddingIn 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 :\s ...
. * In the category of semi-abelian monoids, the product is given by the
history monoidIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. * A
partially ordered set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
can be treated as a category, using the order relation as the morphisms. In this case the products and
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
s correspond to greatest lower bounds ( meets) and least upper bounds (
joins Join may refer to: * Join (law) In law, a joinder is the joining of two or more legal issues together. Procedurally, a joinder allows multiple issues to be heard in one hearing or trial and is done when the issues or parties involved overlap su ...
).

# 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(that is, $I$ is the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

) is the same as a
terminal object In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

$\mathbf^I \to \mathbf.$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. For $f_1 : X \to Y_1, f_2 : X_2 \to Y_2$ 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. 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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...
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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
; a Cartesian category with its finite products is an example of a
symmetric monoidal categoryIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
.

# Distributivity

For any objects $X, Y, \text Z$ of a category with finite products and coproducts, there is a
canonical Canonical may refer to: Science and technology * Canonical form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geo ...

morphism $X \times Y + X \times Z \to X \times \left(Y + Z\right),$ where the plus sign here denotes the
coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
. 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 $X \times \left(Y + Z\right)$ then guarantees a unique morphism $X \times Y + X \times Z \to X \times \left(Y + Z\right)$ 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 $X\times (Y + Z)\simeq (X\times Y) + (X \times Z).$

*
Coproduct In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
â€“ the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of the product *
Diagonal functor In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...

â€“ the
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs o ...
of the product functor. * * * * *

# References

* * Chapter 5. * * Definition 2.1.1 in