Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or 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.
Product of two objects.
Fix a category . Let and be objects of . A product of and is an object , typically denoted , equipped with a pair of morphisms , satisfying the following universal property:
Whether a product exists may depend on or on and . If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of "the" product.
The morphisms and are called the canonical projections or projection morphisms. Given and , , the unique morphism is called the product of morphisms and and is denoted . 
Product of an arbitrary family.
Instead of two objects, we can start with an arbitrary family of objects indexed by a set .
Given a family of objects, a product of the family is an object equipped with morphisms satisfying the following universal property: 
The product is denoted . If , then it is denoted and the product of morphisms is denoted .
Equational definition.
Alternatively, the product may be defined through equations. So, for example, for the binary product:
As a limit.
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 considered as a discrete category. The definition of the product then coincides with the definition of the limit, being a 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 universal property of limits, take as the discrete category with two objects, so that is simply the product category . The diagonal functor assigns to each object the ordered pair and to each morphism the pair . The product in is given by a universal morphism from the functor to the object in . This universal morphism consists of an object of and a morphism which contains projections.
Examples.
In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets the product is defined as
with the canonical projections
Given any set "Y" with a family of functions ,
the universal arrow is defined by := .
Other examples:
Discussion.
An example in which the product does not exist: In the category of fields, the product does not exist, since there is no field with homomorphisms to both and .
Another example: An empty product (i.e. 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 there are infinitely many morphisms so cannot be terminal.
If is a set such that all products for families indexed with exist, then one can treat each product as a functor . 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 we should find a morphism . We choose . This operation on morphisms is called Cartesian product of morphisms. Second, consider the general product functor. For families we should find a morphism . We choose the product of morphisms .
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 is a Cartesian category, product functors have been chosen as above, and denotes a terminal object of . We then have natural isomorphisms
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.
Distributivity.
For any objects , , and of a category with finite products and coproducts, there is a canonical morphism , where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):
The universal property of the product then guarantees a unique morphism 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, one has the canonical isomorphism