Product (category theory)
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Motivation
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 {Xi | i ∈ I} be an indexed family of objects in C. The product of the set {Xi} is an object X together with a collection of morphisms πi : X → Xi (called projections) which satisfy a universal property: for any object Y and any collection of morphisms fi : Y → Xi, there exists a unique morphism f : Y → X such that for all i ∈ I it is the case that fi = πi f. That is, the following diagram commutes (for all i):
CategoricalProduct-01.png
Image:CategoricalProduct-01.png
If the family of objects consists of only two members X, Y, the product is usually written X×Y, and the diagram takes a form along the lines of:
Binaryproductdiagram.png
Image:Binaryproductdiagram.png
The unique arrow h making this diagram commute is notated <f,g>.
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 {Xi} needs to have a product, but if it does, then the product is unique in a strong sense: if πi : X → Xi and π'i : X' → Xi are two products of the family {Xi}, then (by the definition of products) there exists a unique isomorphism f : X → X' such that πi = 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 CI → C. The product of the family {Xi} is then often denoted by ∏i Xi, and the maps πi are known as the natural projections. We have a natural isomorphism
- <math>\operatorname{Mor}_C\left(Y,\prod_{i\in I}X_i\right) \simeq \prod_{i\in I}\operatorname{Mor}_C(Y,X_i)<math>
(where MorC(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 X1,...,Xn is often denoted by X1×...×Xn. 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
- <math>X\times (Y \times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z<math>
- <math>X\times 1 \simeq 1\times X \simeq X<math>
- <math>X\times Y \simeq Y\times X<math>
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