View source for Product (category theory)

{{short description|Generalized object in category theory}}

In [[Category theory|category theory]], the '''product''' of two (or more) objects in a [[Category (mathematics)|category]] is a notion designed to capture the essence behind constructions in other areas of [[Mathematics|mathematics]] such as the [[Cartesian product]] of [[Set (mathematics)|sets]], the [[Direct product|direct product]] of [[Group (mathematics)|groups]] or [[Ring (mathematics)|rings]], and the [[Product topology|product]] of [[Topological space|topological space]]s. Essentially, the product of a [[Indexed family|family]] of objects is the "most general" object which admits a [[Morphism|morphism]] to each of the given objects.

==Definition==

===Product of two objects===

Fix a category <math>C.</math>  Let <math>X_1</math> and <math>X_2</math> be objects of <math>C.</math> A product of <math>X_1</math> and <math>X_2</math> is an object <math>X,</math> typically denoted <math>X_1 \times X_2,</math> equipped with a pair of morphisms <math>\pi_1 : X \to X_1,</math> <math>\pi_2 : X \to X_2</math> satisfying the following [[Universal property|universal property]]:
* For every object <math>Y</math> and every pair of morphisms <math>f_1 : Y \to X_1,</math> <math>f_2 : Y \to X_2,</math> there exists a unique morphism <math>f : Y \to X_1 \times X_2</math> such that the following diagram [[Commutative diagram|commutes]]:
*:[[Image:CategoricalProduct-03.svg|280px|center|Universal property of the product]]

Whether a product exists may depend on <math>C</math> or on <math>X_1</math> and <math>X_2.</math>  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: if <math>X', \pi_1' , \pi_2' </math> is another product, there exists a unique isomorphism <math> h : X' \to X_1 \times X_2 </math> such that <math> \pi_1' = \pi_1 \circ h </math> and <math> \pi_2' = \pi_2 \circ h </math>.

The morphisms <math>\pi_1</math> and <math>\pi_2</math> are called the '''canonical projections''' or '''projection morphisms'''; the letter <math>\pi</math> (pronounced pi) alliterates with projection.  Given <math>Y</math> and <math>f_1,</math> <math>f_2,</math> 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>

===Product of an arbitrary family===

Instead of two objects, we can start with an arbitrary family of objects [[Index set|indexed]] by a set <math>I.</math>

Given a family <math>\left(X_i\right)_{i \in I}</math> of objects, a '''product''' of the family is an object <math>X</math> equipped with morphisms <math>\pi_i : X \to X_i,</math> satisfying the following universal property: 
*For every object <math>Y</math> and every <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>
*:[[Image:Cat product.svg|center|Universal product of the product]]

The product is denoted <math>\prod_{i \in I} X_i.</math> If <math>I = \{1, \ldots, n\},</math> then it is 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>

===Equational definition===

Alternatively, the product may be defined through equations. So, for example, for the binary product:
* Existence of <math>f</math> is guaranteed by existence of the operation <math>\langle \cdot,\cdot \rangle.</math>
* Commutativity of the diagrams above is guaranteed by the equality: for all <math>f_1, f_2</math> and all <math>i \in \{1, 2\},</math> <math>\pi_i \circ \left\langle f_1, f_2 \right\rangle = f_i</math>
* Uniqueness of <math>f</math> is guaranteed by the equality: for all <math>g : Y \to X_1 \times X_2,</math> <math>\left\langle \pi_1 \circ g, \pi_2 \circ g \right\rangle = g.</math><ref>{{cite book|author=Lambek J., Scott P. J.|title=Introduction to Higher-Order Categorical Logic|publisher=Cambridge University Press|year=1988|page=304}}</ref>

===As a limit===

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).

===Universal property===

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>\mathbf{J}</math> as the discrete category with two objects, so that <math>\mathbf{C}^{\mathbf{J}}</math> is simply the [[Product category|product category]] <math>\mathbf{C} \times \mathbf{C}.</math>  The [[Diagonal functor|diagonal functor]] <math>\Delta : \mathbf{C} \to \mathbf{C} \times \mathbf{C}</math> assigns to each object <math>X</math> the [[Ordered pair|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>C</math> is given by a universal morphism from the functor <math>\Delta</math> to the object <math>\left(X_1, X_2\right)</math> in <math>\mathbf{C} \times \mathbf{C}.</math> This universal morphism consists of an object <math>X</math> of <math>C</math> and a morphism <math>(X, X) \to \left(X_1, X_2\right)</math> which contains projections.

==Examples==
In the [[Category of sets|category of sets]], the product (in the category theoretic sense) is the Cartesian product. Given a family of sets <math>X_i</math> the product is defined as
<math display=block>\prod_{i \in I} X_i := \left\{\left(x_i\right)_{i \in I} : x_i \in X_i \text{ for all } i \in I\right\}</math>
with the canonical projections
<math display=block>\pi_j : \prod_{i \in I} X_i \to X_j, \quad \pi_j\left(\left(x_i\right)_{i \in I}\right) := x_j.</math>
{{anchor|Product function}}
Given any set <math>Y</math> with a family of functions <math>f_i : Y \to X_i,</math> 
the universal arrow <math>f : Y \to \prod_{i \in I} X_i</math> is defined by <math>f(y) := \left(f_i(y)\right)_{i \in I}.</math>

Other examples:
* In the [[Category of topological spaces|category of topological spaces]], the product is the space whose underlying set is the Cartesian product and which carries the [[Product topology|product topology]]. The product topology is the coarsest topology for which all the projections are continuous.
* In the [[Category of modules|category of modules]] over some ring <math>R,</math> the product is the Cartesian product with addition defined componentwise and distributive multiplication.
* In the [[Category of groups|category of groups]], the product is the [[Direct product of groups|direct product of groups]] given by the Cartesian product with multiplication defined componentwise.
* In the [[Graph homomorphism#Structure of homomorphisms|category of graphs]], the product is the tensor product of graphs.
* In the [[Category of relations|category of relations]], the product is given by the [[Disjoint union|disjoint union]]. (This may come as a bit of a surprise given that the category of sets is a [[Subcategory|subcategory]] of the category of relations.)
* In the category of [[Algebraic variety|algebraic varieties]], the product is given by the [[Segre embedding]].
* In the category of [[Trace monoid|semi-abelian monoid]]s, the product is given by the [[History monoid|history monoid]].
* In the category of Banach spaces and short maps, the product carries the [[L-infinity|{{math|''l''<sup>&infin;</sup>}}]] norm.<ref name=Ban1Cat>{{cite web|website=Annoying Precision|title=Banach spaces (and Lawvere metrics, and closed categories)|date=June 23, 2012|author=Qiaochu Yuan|url=https://qchu.wordpress.com/2012/06/23/banach-spaces-and-lawvere-metrics-and-closed-categories/}}</ref>  
* A [[Partially ordered set|partially ordered set]] can be treated as a category, using the order relation as the morphisms.  In this case the products and [[Coproduct|coproduct]]s correspond to greatest lower bounds (meets) and least upper bounds (joins).

==Discussion==

An example in which the product does not exist: In the category of fields, the product <math>\Q \times F_p</math> does not exist, since there is no field with homomorphisms to both <math>\Q</math> and <math>F_p.</math>

Another example: An [[Empty product|empty product]] (that is, <math>I</math> is the [[Empty set|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>\Z \to G,</math> so <math>G</math> cannot be terminal.

If <math>I</math> is a set such that all products for families indexed with <math>I</math> exist, then one can treat each product as a [[Functor|functor]] <math>\mathbf{C}^I \to \mathbf{C}.</math><ref>{{cite book|last1=Lane|first1=S. Mac|title=Categories for the working mathematician|date=1988|publisher=Springer-Verlag|location=New York|isbn=0-387-90035-7|page=37|edition=1st}}</ref> 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. 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>\left\langle f_1 \circ \pi_1, f_2 \circ \pi_2 \right\rangle.</math> This operation on morphisms is called '''Cartesian product of morphisms'''.<ref name="esslli">{{cite book|author=Michael Barr, Charles Wells|title=Category Theory – Lecture Notes for ESSLLI|year=1999|url=http://www.let.uu.nl/esslli/Courses/barr/barrwells.ps|page=62|url-status=dead|archive-url=https://web.archive.org/web/20110413051026/http://www.let.uu.nl/esslli/Courses/barr/barrwells.ps|archive-date=2011-04-13}}</ref> Second, consider the general product functor. For families <math>\left\{X\right\}_i, \left\{Y\right\}_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>\left\{f_i \circ \pi_i\right\}_i.</math>

A category where every finite set of objects has a product is sometimes called a '''Cartesian category'''<ref name="esslli"/>
(although some authors use this phrase to mean "a category with all finite limits").

The product is associative. Suppose <math>C</math> is a Cartesian category, product functors have been chosen as above, and <math>1</math> denotes a terminal object of <math>C.</math> We then have natural isomorphisms
<math display=block>X \times (Y \times Z) \simeq (X\times Y) \times Z \simeq X \times Y \times Z,</math>
<math display=block>X \times 1 \simeq 1 \times X \simeq X,</math>
<math display=block>X \times Y \simeq Y \times X.</math>
These properties are formally similar to those of a commutative [[Monoid|monoid]]; a Cartesian category with its finite products is an example of a symmetric monoidal category.

==Distributivity==
{{Main|Distributive category}}
For any objects <math>X, Y, \text{ and } Z</math> of a category with finite products and coproducts, there is a [[List of mathematical jargon#canonical|canonical]] morphism <math>X \times Y + X \times Z \to X \times (Y + Z),</math> where the plus sign here denotes the [[Coproduct|coproduct]].  To see this, note that the universal property of the coproduct <math>X \times Y + X \times Z</math> guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):

[[File:Product-Coproduct Distributivity SVG.svg|400px|center]]

The universal property of the product <math>X \times (Y + Z)</math> then guarantees a unique morphism <math>X \times Y + X \times Z \to X \times (Y + Z)</math> induced by the dashed arrows in the above diagram.  A [[Distributive category|distributive category]] is one in which this morphism is actually an isomorphism.  Thus in a distributive category, there is the canonical isomorphism
<math display=block>X\times (Y + Z)\simeq (X\times Y) + (X \times Z).</math>

==See also==

* [[Coproduct]]&nbsp;– the [[Dual (category theory)|dual]] of the product
* [[Diagonal functor]]&nbsp;– the left adjoint of the product functor.
* {{annotated link|Limit (category theory)|Limit and colimits}}
* {{annotated link|Equaliser (mathematics)|Equalizer}}
* {{annotated link|Inverse limit}}
* {{annotated link|Cartesian closed category}}

==References==

{{reflist}}

* {{cite book|last=Adámek|first=Jiří|author2=Horst Herrlich|author3=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}}
* {{cite book|last=Barr|first=Michael|author2=Charles Wells|title=Category Theory for Computing Science|year=1999|url=http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf|publisher=Les Publications CRM Montreal (publication PM023)|access-date=2016-03-21|archive-url=https://web.archive.org/web/20160304031956/http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf|archive-date=2016-03-04|url-status=dead}} Chapter 5.
* {{cite book|first=Saunders|last=Mac Lane|year=1998|title=[[Categories for the Working Mathematician]]|series=[[Graduate Texts in Mathematics]] '''5'''|edition=2nd|publisher=Springer|isbn=0-387-98403-8}}
* Definition 2.1.1 in {{Cite book|publisher=Cambridge University Press|isbn=0-521-44178-1|volume=1|last=Borceux|first=Francis|title=Handbook of categorical algebra|series=Encyclopedia of mathematics and its applications 50–51, 53 [i.e. 52]|date=1994|page=[https://archive.org/details/handbookofcatego0000borc/page/39 39]| url-access=registration|url=https://archive.org/details/handbookofcatego0000borc/page/39}}

==External links==
* [https://web.archive.org/web/20080916162345/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 [https://web.archive.org/web/20081223001815/http://www.j-paine.org/ Jocelyn Paine].
* {{nlab|id=product|title=Product}}

{{Category theory}}

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[[Category:Limits (category theory)]]

{{Sourceattribution|Product (category theory)}}