# Product category theory

In category theory, one defines **products** to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the 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.

## Definition

Let *C* be a category and let {*X _{i}* |

*i*∈

*I*} be an indexed family of objects in

*C*. The product of the set {

*X*} is an object

_{i}*X*together with a collection of morphisms

*π*:

_{i}*X*→

*X*(called

_{i}*projections*) which satisfy a universal property: for any object

*Y*and any collection of morphisms

*f*:

_{i}*Y*→

*X*, there exists a unique morphism

_{i}*f*:

*Y*→

*X*such that for all

*i*∈

*I*it is the case that

*f*=

_{i}*π*

_{i}*f*. That is, the following diagram commutes (for all

*i*):

If the family of objects consists of only two members the product is usually written *X*_{1}×*X*_{2}, and the diagram takes the form:

The unique arrow *f* making this diagram commute is sometimes denoted <*f*_{1},*f*_{2}>.

## Discussion

The product construction given above is actually a special case of a limit in category theory. The product can be defined as the limit of any discrete subcategory in *C*. Not every family {*X*_{i}} needs to have a product, but if it does, then the product is unique in a strong sense: if *π*_{i} : *X* → *X*_{i} and *π*’_{i} : *X*’ → *X*_{i} are two products of the family {*X*_{i}}, then (by the definition of products) there exists a unique isomorphism *f* : *X* → *X*’ such that *π*_{i} = *π*’_{i} *f* for each *i* in *I*.

An empty product (i.e. *I* is the empty set) is the same as a terminal object in *C*.

If *I* is a set such that all products for families indexed with *I* exist, then it is possible to choose the products in a compatible fashion so that the product turns into a functor *C*^{I} → *C*. The product of the family {*X*_{i}} is then often denoted by ∏_{i} *X*_{i}, and the maps π_{i} are known as the **natural projections**. We have a natural isomorphism

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Hom}_C\left(Y,\prod_{i\in I}X_i\right) \simeq \prod_{i\in I}\operatorname{Hom}_C(Y,X_i)}**

(where Hom_{C}(*U*,*V*) denotes the set of all morphisms from *U* to *V* in *C*, the left product is the one in *C* and the right is the cartesian product of sets).

If *I* is a finite set, say *I* = {1,...,*n*}, then the product of objects *X*_{1},...,*X*_{n} is often denoted by *X*_{1}×...×*X*_{n}.
Suppose all finite products exist in *C*, product functors have been chosen as above, and 1 denotes the terminal object of *C* corresponding to the empty product. We then have natural isomorphisms

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\times (Y \times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\times 1 \simeq 1\times X \simeq X}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\times Y \simeq Y\times X}**

These properties are formally similar to those of a commutative monoid.

## See also

- Coproduct – the dual of the product
- Limit and colimits
- Equalizer
- Inverse limit
- Cartesian closed category