In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
such as the
Cartesian product of
sets, the
direct product of
groups or
rings, and the
product of
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s. 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: if
is another product, there exists a unique isomorphism
such that
and
.
The morphisms
and
are called the
canonical projections or projection morphisms; the letter
alliterates with projection. Given
and
the unique morphism
is called the product of morphisms
and
and may be denoted
,
, or
.
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
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
A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...
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
norm.
* 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
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
(that is,
is the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
) 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
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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
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, there is the canonical isomorphism
See also
*
Coproduct – the
dual of the product
*
Diagonal functor – the
left adjoint of the product functor.
*
*
*
*
*
References
*
* Chapter 5.
*
* Definition 2.1.1 in
External links
Interactive Web page which generates examples of products in the category of finite sets. Written b
Jocelyn Paine
*
{{DEFAULTSORT:Product (Category Theory)
Limits (category theory)