Incategory theory, theproduct of two (or more)objects in acategory is a notion designed to capture the essence behind constructions in other areas ofmathematics such as theCartesian product ofsets, thedirect product ofgroups orrings, and theproduct oftopological spaces. Essentially, the product of afamily of objects is the "most general" object which admits amorphism to each of the given objects.

Fix a category${\displaystyle C.}$ Let${\displaystyle X_1}$ and${\displaystyle X_2}$ be objects of${\displaystyle C.}$ A product of${\displaystyle X_1}$ and${\displaystyle X_2}$ is an object${\displaystyle X,}$ typically denoted${\displaystyle X_1\times X_2,}$ equipped with a pair of morphisms${\displaystyle {\pi }_1:X\to X_1,}$${\displaystyle {\pi }_2:X\to X_2}$ satisfying the followinguniversal property:

• For every object${\displaystyle Y}$ and every pair of morphisms${\displaystyle f_1:Y\to X_1,}$${\displaystyle f_2:Y\to X_2,}$ there exists a unique morphism${\displaystyle f:Y\to X_1\times X_2}$ such that the following diagramcommutes:

  

Whether a product exists may depend on${\displaystyle C}$ or on${\displaystyle X_1}$ and${\displaystyle X_2.}$ If it does exist, it is uniqueup tocanonical isomorphism, because of the universal property, so one may speak ofthe product. This has the following meaning: if${\displaystyle X^{\prime },{\pi }_1^{\prime },{\pi }_2^{\prime }}$ is another product, there exists a unique isomorphism${\displaystyle h:X^{\prime }\to X_1\times X_2}$ such that${\displaystyle {\pi }_1^{\prime }={\pi }_1\circ h}$ and${\displaystyle {\pi }_2^{\prime }={\pi }_2\circ h}$.

The morphisms${\displaystyle {\pi }_1}$ and${\displaystyle {\pi }_2}$ are called thecanonical projections orprojection morphisms; the letter${\displaystyle \pi }$ alliterates with projection. Given${\displaystyle Y}$ and${\displaystyle f_1,}$${\displaystyle f_2,}$ the unique morphism${\displaystyle f}$ is called theproduct of morphisms${\displaystyle f_1}$ and${\displaystyle f_2}$ and may be denoted${\displaystyle ⟨f_1,f_2⟩}$,${\displaystyle f_1\times f_2}$, or${\displaystyle f_1\otimes f_2}$.

Instead of two objects, we can start with an arbitrary family of objectsindexed by a set${\displaystyle I.}$

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

• For every object${\displaystyle Y}$ and every${\displaystyle I}$-indexed family of morphisms${\displaystyle f_i:Y\to X_i,}$ there exists a unique morphism${\displaystyle f:Y\to X}$ such that the following diagrams commute for all${\displaystyle i\in I:}$

  

The product is denoted${\displaystyle \prod _{i\in I}{X}_i.}$ If${\displaystyle I=\{1,\dots ,n\},}$ then it is denoted${\displaystyle X_1\times \cdots \times X_n}$ and the product of morphisms is denoted${\displaystyle ⟨f_1,\dots ,f_n⟩.}$

Alternatively, the product may be defined through equations. So, for example, for the binary product:

• Existence of${\displaystyle f}$ is guaranteed by existence of the operation${\displaystyle ⟨\cdot ,\cdot ⟩.}$
• Commutativity of the diagrams above is guaranteed by the equality: for all${\displaystyle f_1,f_2}$ and all${\displaystyle i\in \{1,2\},}$${\displaystyle {\pi }_i\circ ⟨f_1,f_2⟩=f_i}$
• Uniqueness of${\displaystyle f}$ is guaranteed by the equality: for all${\displaystyle g:Y\to X_1\times X_2,}$${\displaystyle ⟨{\pi }_1\circ g,{\pi }_2\circ g⟩=g.}$^[1]

The product is a special case of alimit. This may be seen by using adiscrete category (a family of objects without any morphisms, other than their identity morphisms) as thediagram 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${\displaystyle I}$ considered as a discrete category. The definition of the product then coincides with the definition of the limit,${\displaystyle \{f{\}}_i}$ being acone and projections being the limit (limiting cone).

Just as the limit is a special case of theuniversal construction, so is the product. Starting with the definition given for theuniversal property of limits, take${\displaystyle \mathbf{J}}$ as the discrete category with two objects, so that${\displaystyle {\mathbf{C}}^{\mathbf{J}}}$ is simply theproduct category${\displaystyle \mathbf{C}\times \mathbf{C}.}$ Thediagonal functor${\displaystyle \mathrm{\Delta }:\mathbf{C}\to \mathbf{C}\times \mathbf{C}}$ assigns to each object${\displaystyle X}$ theordered pair${\displaystyle (X,X)}$ and to each morphism${\displaystyle f}$ the pair${\displaystyle (f,f).}$ The product${\displaystyle X_1\times X_2}$ in${\displaystyle C}$ is given by auniversal morphism from the functor${\displaystyle \mathrm{\Delta }}$ to the object${\displaystyle \left(X_1,X_2\right)}$ in${\displaystyle \mathbf{C}\times \mathbf{C}.}$ This universal morphism consists of an object${\displaystyle X}$ of${\displaystyle C}$ and a morphism${\displaystyle (X,X)\to \left(X_1,X_2\right)}$ which contains projections.

In thecategory of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets${\displaystyle X_i}$ the product is defined as$${\displaystyle \prod _{i\in I}{X}_i:=\left\{{\left(x_i\right)}_{i\in I}:x_i\in X_i\text{ for all }i\in I\right\}}$$with the canonical projections$${\displaystyle {\pi }_j:\prod _{i\in I}{X}_i\to X_j,{\pi }_j\left({\left(x_i\right)}_{i\in I}\right):=x_j.}$$Given any set${\displaystyle Y}$ with a family of functions${\displaystyle f_i:Y\to X_i,}$ the universal arrow${\displaystyle f:Y\to \prod _{i\in I}{X}_i}$ is defined by${\displaystyle f(y):={\left(f_i(y)\right)}_{i\in I}.}$

Other examples:

• In thecategory of topological spaces, the product is the space whose underlying set is the Cartesian product and which carries theproduct topology. The product topology is thecoarsest topology for which all the projections arecontinuous.
• In thecategory of modules over some ring${\displaystyle R,}$ the product is the Cartesian product with addition defined componentwise and distributive multiplication.
• In thecategory of groups, the product is thedirect product of groups given by the Cartesian product with multiplication defined componentwise.
• In thecategory of graphs, the product is thetensor product of graphs.
• In thecategory of relations, the product is given by thedisjoint union. (This may come as a bit of a surprise given that the category of sets is asubcategory of the category of relations.)
• In the category ofalgebraic varieties, the product is given by theSegre embedding.
• In the category ofsemi-abelian monoids, the product is given by thehistory monoid.
• In the category ofBanach spaces andshort maps, the product carries thel^∞ norm.^[2]
• Apartially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products andcoproducts 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${\displaystyle \mathbb{Q}\times F_p}$ does not exist, since there is no field with homomorphisms to both${\displaystyle \mathbb{Q}}$ and${\displaystyle F_p.}$

Another example: Anempty product (that is,${\displaystyle I}$ is theempty set) is the same as aterminal object, and some categories, such as the category ofinfinite groups, do not have a terminal object: given any infinite group${\displaystyle G}$ there are infinitely many morphisms${\displaystyle \mathbb{Z}\to G,}$ so${\displaystyle G}$ cannot be terminal.

If${\displaystyle I}$ is a set such that all products for families indexed with${\displaystyle I}$ exist, then one can treat each product as afunctor${\displaystyle {\mathbf{C}}^I\to \mathbf{C}.}$^[3] 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 abifunctor. For${\displaystyle f_1:X_1\to Y_1,f_2:X_2\to Y_2}$ we should find a morphism${\displaystyle X_1\times X_2\to Y_1\times Y_2.}$ We choose${\displaystyle ⟨f_1\circ {\pi }_1,f_2\circ {\pi }_2⟩.}$ This operation on morphisms is calledCartesian product of morphisms.^[4] Second, consider the general product functor. For families${\displaystyle {\left\{X\right\}}_i,{\left\{Y\right\}}_i,f_i:X_i\to Y_i}$ we should find a morphism${\displaystyle \prod _{i\in I}{X}_i\to \prod _{i\in I}{Y}_i.}$ We choose the product of morphisms${\displaystyle {\left\{f_i\circ {\pi }_i\right\}}_i.}$

A category where every finite set of objects has a product is sometimes called aCartesian category^[4](although some authors use this phrase to mean "a category with all finite limits").

The product isassociative. Suppose${\displaystyle C}$ is a Cartesian category, product functors have been chosen as above, and${\displaystyle 1}$ denotes a terminal object of${\displaystyle C.}$ We then havenatural isomorphisms$${\displaystyle X\times (Y\times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z,}$$$${\displaystyle X\times 1\simeq 1\times X\simeq X,}$$$${\displaystyle X\times Y\simeq Y\times X.}$$These properties are formally similar to those of a commutativemonoid; a Cartesian category with its finite products is an example of asymmetric monoidal category.

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

The universal property of the product${\displaystyle X\times (Y+Z)}$ then guarantees a unique morphism${\displaystyle X\times Y+X\times Z\to X\times (Y+Z)}$ induced by the dashed arrows in the above diagram. Adistributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, there is the canonical isomorphism$${\displaystyle X\times (Y+Z)\simeq (X\times Y)+(X\times Z).}$$

• Coproduct – thedual of the product
• Diagonal functor – theleft adjoint of the product functor.
• Limit and colimits – Mathematical concept
• Equalizer – Set of arguments where two or more functions have the same value
• Inverse limit – Construction in category theory
• Cartesian closed category – Type of category in category theory
• Categorical pullback – Most general completion of a commutative square given two morphisms with same codomainPages displaying short descriptions of redirect targets

• Adámek, Jiří; Horst Herrlich; George E. Strecker (1990).Abstract and Concrete Categories(PDF). John Wiley & Sons.ISBN 0-471-60922-6.
• Barr, Michael; Charles Wells (1999).Category Theory for Computing Science(PDF). Les Publications CRM Montreal (publication PM023). Archived fromthe original(PDF) on 2016-03-04. Retrieved2016-03-21. Chapter 5.
• Mac Lane, Saunders (1998).Categories for the Working Mathematician.Graduate Texts in Mathematics5 (2nd ed.). Springer.ISBN 0-387-98403-8.
• Definition 2.1.1 inBorceux, Francis (1994).Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50–51, 53 [i.e. 52]. Vol. 1. Cambridge University Press. p. 39.ISBN 0-521-44178-1.

• Interactive Web page which generates examples of products in the category of finite sets. Written byJocelyn Paine.
• Product at thenLab

• Limits (category theory)

• Articles with short description
• Short description matches Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link