Product (category theory)

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 $X\_1\; \backslash times\; X\_2,$ equipped with a pair of morphisms $\backslash pi\_1\; :\; X\; \backslash to\; X\_1,$ $\backslash pi\_2\; :\; X\; \backslash to\; X\_2$ satisfying the following universal property:

• For every object $Y$ and every pair of morphisms $f\_1\; :\; Y\; \backslash to\; X\_1,$ $f\_2\; :\; Y\; \backslash to\; X\_2,$ there exists a unique morphism $f\; :\; Y\; \backslash to\; X\_1\; \backslash times\; X\_2$ such that the following diagram commutes:
• :Image:CategoricalProduct-03.svg

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: if $X\text{'},\; \backslash pi\_1\text{'},\; \backslash pi\_2\text{'}$ is another product, there exists a unique isomorphism $h\; :\; X\text{'}\; \backslash to\; X\_1\; \backslash times\; X\_2$ such that $\backslash pi\_1\text{'}\; =\; \backslash pi\_1\; \backslash circ\; h$ and $\backslash pi\_2\text{'}\; =\; \backslash pi\_2\; \backslash circ\; h$.

The morphisms $\backslash pi\_1$ and $\backslash pi\_2$ are called the canonical projections or projection morphisms; the letter $\backslash pi$ alliterates with projection. Given $Y$ and $f\_1,$ $f\_2,$ the unique morphism $f$ is called the product of morphisms $f\_1$ and $f\_2$ and may be denoted $\backslash langle\; f\_1,\; f\_2\; \backslash rangle$, $f\_1\; \backslash times\; f\_2$, or $f\_1\; \backslash otimes\; f\_2$.

Instead of two objects, we can start with an arbitrary family of objects indexed by a set $I.$

Given a family $\backslash left(X\_i\backslash right)\_\{i\; \backslash in\; I\}$ of objects, a product of the family is an object $X$ equipped with morphisms $\backslash pi\_i\; :\; X\; \backslash to\; X\_i,$ satisfying the following universal property:

• For every object $Y$ and every $I$-indexed family of morphisms $f\_i\; :\; Y\; \backslash to\; X\_i,$ there exists a unique morphism $f\; :\; Y\; \backslash to\; X$ such that the following diagrams commute for all $i\; \backslash in\; I:$
• :Image:Cat product.svg

The product is denoted $\backslash prod\_\{i\; \backslash in\; I\}\; X\_i.$ If $I\; =\; \backslash \{1,\; \backslash ldots,\; n\backslash \},$ then it is denoted $X\_1\; \backslash times\; \backslash cdots\; \backslash times\; X\_n$ and the product of morphisms is denoted $\backslash langle\; f\_1,\; \backslash ldots,\; f\_n\; \backslash rangle.$

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 $\backslash langle\; \backslash cdot,\backslash cdot\; \backslash rangle.$
• Commutativity of the diagrams above is guaranteed by the equality: for all $f\_1,\; f\_2$ and all $i\; \backslash in\; \backslash \{1,\; 2\backslash \},$ $\backslash pi\_i\; \backslash circ\; \backslash left\backslash langle\; f\_1,\; f\_2\; \backslash right\backslash rangle\; =\; f\_i$
• Uniqueness of $f$ is guaranteed by the equality: for all $g\; :\; Y\; \backslash to\; X\_1\; \backslash times\; X\_2,$ $\backslash left\backslash langle\; \backslash pi\_1\; \backslash circ\; g,\; \backslash pi\_2\; \backslash circ\; g\; \backslash right\backslash rangle\; =\; g.${{cite book|author=Lambek J., Scott P. J.|title=Introduction to Higher-Order Categorical Logic|publisher=Cambridge University Press|year=1988|page=304}}

The product is a special case of a 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 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, $\backslash \{f\backslash \}\_i$ being a cone and projections being the limit (limiting cone).

Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take $\backslash mathbf\{J\}$ as the discrete category with two objects, so that $\backslash mathbf\{C\}^\{\backslash mathbf\{J\}\}$ is simply the product category $\backslash mathbf\{C\}\; \backslash times\; \backslash mathbf\{C\}.$ The diagonal functor $\backslash Delta\; :\; \backslash mathbf\{C\}\; \backslash to\; \backslash mathbf\{C\}\; \backslash times\; \backslash mathbf\{C\}$ assigns to each object $X$ the ordered pair $(X,\; X)$ and to each morphism $f$ the pair $(f,\; f).$ The product $X\_1\; \backslash times\; X\_2$ in $C$ is given by a universal morphism from the functor $\backslash Delta$ to the object $\backslash left(X\_1,\; X\_2\backslash right)$ in $\backslash mathbf\{C\}\; \backslash times\; \backslash mathbf\{C\}.$ This universal morphism consists of an object $X$ of $C$ and a morphism $(X,\; X)\; \backslash to\; \backslash left(X\_1,\; X\_2\backslash right)$ which contains projections.

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

$$\backslash prod\_\{i\; \backslash in\; I\}\; X\_i\; :=\; \backslash left\backslash \{\backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}\; :\; x\_i\; \backslash in\; X\_i\; \backslash text\{\; for\; all\; \}\; i\; \backslash in\; I\backslash right\backslash \}$$

with the canonical projections

$$\backslash pi\_j\; :\; \backslash prod\_\{i\; \backslash in\; I\}\; X\_i\; \backslash to\; X\_j,\; \backslash quad\; \backslash pi\_j\backslash left(\backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}\backslash right)\; :=\; x\_j.$$

{{anchor|Product function}}

Given any set $Y$ with a family of functions $f\_i\; :\; Y\; \backslash to\; X\_i,$

the universal arrow $f\; :\; Y\; \backslash to\; \backslash prod\_\{i\; \backslash in\; I\}\; X\_i$ is defined by $f(y)\; :=\; \backslash left(f\_i(y)\backslash right)\_\{i\; \backslash in\; I\}.$

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. The product topology is the coarsest topology for which all the projections are continuous.
• 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 given by the Cartesian product with multiplication defined componentwise.
• In the category of graphs, the product is the tensor product of graphs.
• In the category of relations, the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets is a subcategory of the category of relations.)
• In the category of algebraic varieties, the product is given by the Segre embedding.
• In the category of semi-abelian monoids, the product is given by the history monoid.
• In the category of Banach spaces and short maps, the product carries the L-infinity norm.{{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/}}
• A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).

An example in which the product does not exist: In the category of fields, the product $\backslash Q\; \backslash times\; F\_p$ does not exist, since there is no field with homomorphisms to both $\backslash Q$ and $F\_p.$

Another example: An empty product (that is, $I$ 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 $G$ there are infinitely many morphisms $\backslash Z\; \backslash to\; G,$ 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 $\backslash mathbf\{C\}^I\; \backslash to\; \backslash mathbf\{C\}.${{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}} 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 $f\_1\; :\; X\_1\; \backslash to\; Y\_1,\; f\_2\; :\; X\_2\; \backslash to\; Y\_2$ we should find a morphism $X\_1\; \backslash times\; X\_2\; \backslash to\; Y\_1\; \backslash times\; Y\_2.$ We choose $\backslash left\backslash langle\; f\_1\; \backslash circ\; \backslash pi\_1,\; f\_2\; \backslash circ\; \backslash pi\_2\; \backslash right\backslash rangle.$ This operation on morphisms is called Cartesian product of morphisms.{{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}} Second, consider the general product functor. For families $\backslash left\backslash \{X\backslash right\backslash \}\_i,\; \backslash left\backslash \{Y\backslash right\backslash \}\_i,\; f\_i\; :\; X\_i\; \backslash to\; Y\_i$ we should find a morphism $\backslash prod\_\{i\; \backslash in\; I\}\; X\_i\; \backslash to\; \backslash prod\_\{i\; \backslash in\; I\}\; Y\_i.$ We choose the product of morphisms $\backslash left\backslash \{f\_i\; \backslash circ\; \backslash pi\_i\backslash right\backslash \}\_i.$

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. 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 isomorphisms

$$X\; \backslash times\; (Y\; \backslash times\; Z)\; \backslash simeq\; (X\backslash times\; Y)\; \backslash times\; Z\; \backslash simeq\; X\; \backslash times\; Y\; \backslash times\; Z,$$

$$X\; \backslash times\; 1\; \backslash simeq\; 1\; \backslash times\; X\; \backslash simeq\; X,$$

$$X\; \backslash times\; Y\; \backslash simeq\; Y\; \backslash times\; X.$$

These properties are formally similar to those of a commutative monoid; a Cartesian category with its finite products is an example of a symmetric monoidal category.

{{Main|Distributive category}}

For any objects $X,\; Y,\; \backslash text\{\; and\; \}\; Z$ of a category with finite products and coproducts, there is a canonical morphism $X\; \backslash times\; Y\; +\; X\; \backslash times\; Z\; \backslash to\; X\; \backslash times\; (Y\; +\; Z),$ where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct $X\; \backslash times\; Y\; +\; X\; \backslash times\; Z$ guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):

File:Product-Coproduct Distributivity SVG.svg

The universal property of the product $X\; \backslash times\; (Y\; +\; Z)$ then guarantees a unique morphism $X\; \backslash times\; Y\; +\; X\; \backslash times\; Z\; \backslash to\; X\; \backslash times\; (Y\; +\; Z)$ 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

$$X\backslash times\; (Y\; +\; Z)\backslash simeq\; (X\backslash times\; Y)\; +\; (X\; \backslash times\; Z).$$

• Coproduct – the dual of the product
• Diagonal functor – 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}}
• {{annotated link|Categorical pullback}}

{{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|author-link=Saunders 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}}

• [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)