Generalized object in 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:
- For every object
and every pair of morphisms
there exists a unique morphism
such that the following diagram commutes:
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. This has the following meaning: let
be another cartesian product, there exists a unique isomorphism
such that
and
.
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:
- For every object
and every
-indexed family of morphisms
there exists a unique morphism
such that the following diagrams commute for all
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:
- Existence of
is guaranteed by existence of the operation 
- Commutativity of the diagrams above is guaranteed by the equality: for all
and all

- Uniqueness of
is guaranteed by the equality: for all
[1]
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
with a family of functions
the universal arrow
is defined by
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
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∞ norm.[2]
- 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).
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 (that is,
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
[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 a bifunctor. For
we should find a morphism
We choose
This operation on morphisms is called Cartesian product of morphisms.[4] 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[4]
(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.