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| 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|cartesian product of sets]], the [[direct product of groups]], the [[direct product of rings]] and the [[product topology|product of topological spaces]]. Essentially, the product of a family of objects is the "most general" object which admits a [[morphism]] to each of the given objects.
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| ==Definition==
| | Here is my web blog: [http://prometeu.net clash of clans hacker] |
| Let <math>\mathcal C</math> be a category with some objects <math>X_1</math> and <math>X_2</math>. An object <math>X</math> is a product of <math>X_1</math> and <math>X_2</math>, denoted <math>X_1 \times X_2</math>, iff it satisfies this [[universal property]]:
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| : there exist morphisms <math>\pi_1 : X \to X_1, \pi_2 : X \to X_2</math> such that for every object <math>Y</math> and pair of morphisms <math>f_1 : Y \to X_1, f_2 : Y \to X_2</math> there exists a unique morphism <math>f : Y \to X</math> such that the following diagram [[commutative diagram|commutes]]:
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| [[Image:CategoricalProduct-03.svg|280px|center|Universal product of the product]]
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| The unique morphism <math>f</math> is called the '''product of morphisms''' <math>f_1</math> and <math>f_2</math> and is denoted <math>\langle f_1, f_2 \rangle</math>. The morphisms <math>\pi_1</math> and <math>\pi_2</math> are called the '''[[canonical projection]]s''' or '''projection morphisms'''.
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| Above we defined the '''binary product'''. Instead of two objects we can take an arbitrary [[indexed family|family]] of objects indexed by some set <math>I</math>. Then we obtain the definition of a '''product'''.
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| An object <math>X</math> is the product of a family <math>\{X\}_i</math> of objects iff there exist morphisms <math>\pi_i : X \to X_i</math>, such that for every object <math>Y</math> and a <math>I</math>-indexed family of morphisms <math>f_i : Y \to X_i</math> there exists a unique morphism <math>f : Y \to X</math> such that the following diagrams commute for all <math>i \in I</math>:
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| [[Image:CategoricalProduct-01.png|center|Universal product of the product]]
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| The product is denoted <math>\prod_{i\in I} X_i</math>; if <math>I = \{1,\ldots, n\}</math>, then denoted <math>X_1 \times \cdots \times X_n</math> and the product of morphisms is denoted <math>\langle f_1, \ldots, f_n \rangle</math>.
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| ===Equational definition===
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| Alternatively, the product may be defined through equations. So, for example, for the binary product:
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| * Existence of <math>f</math> is guaranteed by the operation <math>\lang -,- \rang</math>.
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| * Commutativity of the diagrams above is guaranteed by the equality <math>\forall f_1, \forall f_2, \forall i \in \{1,2\},\ \pi_i \circ \langle f_1, f_2\rangle = f_i</math>.
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| * Uniqueness of <math>f</math> is guaranteed by the equality <math>\forall f,\ \langle \pi_1 \circ f,\pi_2 \circ f \rangle = f</math>.<ref>{{cite book
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| | author = Lambek J., Scott P. J.
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| | title = Introduction to Higher-Order Categorical Logic
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| | publisher = Cambridge University Press
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| | year = 1988
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| | page = 304
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| }}</ref>
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| ===As a limit===
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| The product is a special case of a [[limit (category theory)|limit]]. This may be seen by using a [[discrete category]] (a family of objects without any morphisms, other than their identity morphisms) as the [[Diagram (category theory)|diagram]] 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 <math>I</math> considered as a discrete category. The definition of the product then coincides with the definition of the limit, <math>\{f\}_i</math> being a [[cone (category theory)|cone]] and projections being the limit (limiting cone).
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| ===Universal property===
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| Just as the limit is a special case of the [[universal construction]], so is the product. Starting with the definition given for the [[Limit_(category_theory)#Universal_property|universal property of limits]], take <math>J</math> as the discrete category with two objects, so that <math>{\mathcal C}^{\mathcal J}</math> is simply the [[product category]] <math>\mathcal C \times \mathcal C</math>. The [[diagonal functor]] <math>\Delta : \mathcal C \to \mathcal C \times \mathcal C</math> assigns to each object <math>X</math> the [[ordered pair]] <math>(X,X)</math> and to each morphism <math>f</math> the pair <math>(f,f)</math>. The product <math>X_1 \times X_2</math> in <math>\mathcal C</math> is given by a [[universal morphism]] from the functor <math>\Delta</math> to the object <math>(X_1,X_2)</math> in <math>\mathcal C \times \mathcal C</math>. This universal morphism consists of an object <math>X</math> of <math>
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| \mathcal C</math> and a morphism <math>(X,X) \to (X_1,X_2)</math> which contains projections.
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| == Examples ==
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| In the [[category of sets]], the product (in the category theoretic sense) is the [[cartesian product]]. Given a family of sets ''X<sub>i</sub>'' the product is defined as
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| :<math>\prod_{i \in I} X_i := \{(x_i)_{i \in I} | x_i \in X_i \, \forall i \in I\}</math>
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| with the canonical projections
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| :<math>\pi_j : \prod_{i \in I} X_i \to X_j \mathrm{ , } \quad \pi_j((x_i)_{i \in I}) := x_j</math>
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| {{anchor|Product function}}
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| Given any set ''Y'' with a family of functions
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| :<math>f_i : Y \to X_i</math>
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| the universal arrow ''f'' is defined as
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| :<math>f:Y \to \prod_{i \in I} X_i \mathrm{ , } \quad f(y) := (f_i(y))_{i \in I}</math> | |
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| Other examples:
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| * In the [[category of topological spaces]], the product is the space whose underlying set is the cartesian product and which carries the [[product topology]]. The product topology is the [[coarsest topology]] for which all the projections are [[continuous function (topology)|continuous]].
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| * In the [[category of modules]] over some ring R, the product is the cartesian product with addition defined componentwise and distributive multiplication.
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| * In the [[category of groups]], the product is the [[direct product of groups]] given by the cartesian product with multiplication defined componentwise.
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| * In the [[category of relations]] ('''Rel'''), the product is given by the [[disjoint union]]. (This may come as a bit of a surprise given that the category of sets ('''Set''') is a [[subcategory]] of '''Rel'''.)
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| * In the category of [[algebraic variety|algebraic varieties]], the categorical product is given by the [[Segre embedding]].
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| * In the category of [[Trace monoid|semi-abelian monoid]]s, the categorical product is given by the [[history monoid]].
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| * A [[partially ordered set]] can be treated as a category, using the order relation as the morphisms. In this case the products and [[coproduct]]s correspond to greatest lower bounds ([[Meet (mathematics)|meets]]) and least upper bounds ([[Join (mathematics)|joins]]).
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| == Discussion ==
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| The product does not necessarily exist. For example, an [[empty product]] (i.e. <math>I</math> is the [[empty set]]) is the same as a [[terminal object]], and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group <math>G</math> there are infinitely many morphisms <math>\mathbb{Z} \to G</math>, so <math>G</math> cannot be terminal.
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| If <math>I</math> is a set such that all products for families indexed with <math>I</math> exist, then it is possible to choose the products in a compatible fashion so that the product turns into a [[functor]] <math>{\mathcal C}^I \to \mathcal C</math>. How this functor maps objects is obvious. Mapping of morphisms is subtle, because product of morphisms defined above does not fit. First, consider binary product functor, which is a [[bifunctor]]. For <math>f_1:X_1\to Y_1, f_2:X_2\to Y_2</math> we should find a morphism <math>X_1\times X_2 \to Y_1\times Y_2</math>. We choose <math>\langle f_1 \circ \pi_1, f_2 \circ \pi_2 \rangle</math>. This operation on morphisms is called '''cartesian product of morphisms'''.<ref name="esslli">{{cite book
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| | author = Michael Barr, Charles Wells
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| | title = Category Theory - Lecture Notes for ESSLLI
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| | year = 1999
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| | url = http://www.let.uu.nl/esslli/Courses/barr/barrwells.ps
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| | page = 62
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| }}</ref> Second, consider product functor. For families <math>\{X\}_i, \{Y\}_i, f_i : X_i \to Y_i</math> we should find a morphism <math>\prod_{i\in I}X_i \to \prod_{i\in I}Y_i</math>. We choose the product of morphisms <math>\{f_i \circ \pi_i\}_i</math>.
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| A category where every finite set of objects has a product is sometimes called a '''cartesian category'''<ref name="esslli"/>
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| (although some authors use this phrase to mean "a category with all finite limits").
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| The product is associative. Suppose <math>\mathcal C</math> is a cartesian category, product functors have been chosen as above, and <math>1</math> denotes the terminal object of <math>\mathcal C</math>. We then have [[natural isomorphism]]s
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| :<math>X\times (Y \times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z</math>
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| :<math>X\times 1 \simeq 1\times X \simeq X</math>
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| :<math>X\times Y \simeq Y\times X</math>
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| These properties are formally similar to those of a commutative [[monoid]]; a category with its finite products constitutes a [[symmetric monoidal category|symmetric]] [[monoidal category]].
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| == Distributivity ==
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| In a category with finite products and coproducts, there is a canonical morphism ''X''×''Y''+''X''×''Z'' → ''X''×(''Y''+''Z''), where the plus sign here denotes the [[coproduct]]. To see this, note that we have various canonical projections and injections which fill out the diagram
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| [[File:Product-Coproduct Distributivity.png|center]]
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| The universal property for ''X''×(''Y''+''Z'') then guarantees a unique morphism ''X''×''Y''+''X''×''Z'' → ''X''×(''Y''+''Z''). A [[distributive category]] is one in which this morphism is actually an isomorphism. Thus in a distributive category, one has the canonical isomorphism
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| :<math>X\times (Y + Z)\simeq (X\times Y)+ (X \times Z).</math>
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| ==See also==
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| * [[Coproduct]] – the [[Dual (category theory)|dual]] of the product
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| * [[Diagonal functor]] – the [[left adjoint]] of the product functor.
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| * [[Limit (category theory)|Limit and colimits]]
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| * [[Equaliser (mathematics)|Equalizer]]
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| * [[Inverse limit]]
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| * [[Cartesian closed category]]
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| * [[Categorical pullback]]
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| == References ==
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| <references/>
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| *{{cite book | last = Adámek | first = Jiří | coauthors = Horst Herrlich, and George E. Strecker | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories|publisher = John Wiley & Sons | isbn = 0-471-60922-6}}
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| *{{cite book | last = Barr | first = Michael | coauthors = Charles Wells | title = Category Theory for Computing Science | year = 1999 | publisher = [http://www.crm.umontreal.ca/pub/Ventes/CatalogueEng.html Les Publications CRM Montreal] (publication PM023)}} Chapter 5.
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| *{{cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series = Graduate Texts in Mathematics '''5''' | edition = 2nd ed. | publisher = Springer | isbn = 0-387-98403-8}}
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| == External links ==
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| *[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of products in the category of finite sets. Written by [http://www.j-paine.org/ Jocelyn Paine].
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| {{DEFAULTSORT:Product (Category Theory)}}
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| [[Category:Limits (category theory)]]
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