Universal property


In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property.
Technically, a universal property is defined in terms of categories and functors by means of a universal morphism. Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category.
Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given a commutative ring, the field of fractions of the quotient ring of by a prime ideal can be identified with the residue field of the localization of at ; that is .
Other objects that can be defined by universal properties include: all free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces.

Motivation

Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.
  • The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construction is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly complicated to construct, but much easier to deal with by its universal property.
  • Universal properties define objects uniquely up to a unique isomorphism. Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property.
  • Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property.
  • Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance.

    Formal definition

To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions. Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples.
Let be a functor between categories and. In what follows, let be an object of, and be objects of, and be a morphism in.
Then, the functor maps, and in to, and in.
A universal morphism from to is a unique pair which has the following property, commonly referred to as a universal property:
For any morphism of the form
in, there exists a unique morphism in such that the following diagram commutes:
We can dualize this categorical concept. A universal morphism from to is a unique pair that satisfies the following universal property:
For any morphism of the form in, there exists a unique morphism in such that the following diagram commutes:
Note that in each definition, the arrows are reversed. Both definitions are necessary to describe universal constructions which appear in mathematics; but they also arise due to the inherent duality present in category theory.
In either case, we say that the pair which behaves as above satisfies a universal property.

Connection with comma categories

Universal morphisms can be described more concisely as initial and terminal objects in a comma category.
Let be a functor and an object of. Then recall that the comma category is the category where
  • Objects are pairs of the form, where is an object in
  • A morphism from to is given by a morphism in such that the diagram commutes:
Now suppose that the object in is initial. Then
for every object, there exists a unique morphism such that the following diagram commutes.
Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a universal morphism from to . Therefore, we see that a universal morphism from to is equivalent to an initial object in the comma category.
Conversely, recall that the comma category is the category where
  • Objects are pairs of the form where is an object in
  • A morphism from to is given by a morphism in such that the diagram commutes:
Suppose is a terminal object in. Then for every object,
there exists a unique morphism such that the following diagrams commute.
The diagram on the right side of the equality is the same diagram pictured when defining a universal morphism from to . Hence, a universal morphism from to corresponds with a terminal object in the comma category

Examples

Below are a few examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction.

Tensor algebras

Let be the category of vector spaces -Vect over a field and let be the category of algebras -Alg over . Let
be the forgetful functor which assigns to each algebra its underlying vector space.
Given any vector space over we can construct the tensor algebra. The tensor algebra is characterized by the fact:
This statement is an initial property of the tensor algebra since it expresses the fact that the pair, where is the inclusion map, is a universal morphism from the vector space to the functor.
Since this construction works for any vector space, we conclude that is a functor from -Vect to -Alg. This means that is left adjoint to the forgetful functor .

Products

A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist.
Let and be objects of a category with finite products. The product of and is an object × together with two morphisms
such that for any other object of and morphisms and there exists a unique morphism such that and.
To understand this characterization as a universal property, take the category to be the product category and define the diagonal functor
by and. Then is a universal morphism from to the object of : if is any morphism from to, then it must equal
a morphism from
to followed by. As a commutative diagram:
For the example of the Cartesian product in Set, the morphism comprises the two projections and. Given any set and functions the unique map such that the required diagram commutes is given by.

Limits and colimits

Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits.
Let and be categories with a small index category and let be the corresponding functor category. The diagonal functor
is the functor that maps each object in to the constant functor and each morphism in to the natural transformation in defined as, for every object of, the component
at. In other words, the natural transformation is the one defined by having constant component for every object of.
Given a functor , the limit of, if it exists, is nothing but a universal morphism from to. Dually, the colimit of is a universal morphism from to.

Properties

Existence and uniqueness

Defining a quantity does not guarantee its existence. Given a functor and an object of,
there may or may not exist a universal morphism from to. If, however, a universal morphism does exist, then it is essentially unique.
Specifically, it is unique up to a unique isomorphism: if is another pair, then there exists a unique isomorphism
such that.
This is easily seen by substituting in the definition of a universal morphism.
It is the pair which is essentially unique in this fashion. The object itself is only unique up to isomorphism. Indeed, if is a universal morphism and is any isomorphism then the pair, where is also a universal morphism.

Equivalent formulations

The definition of a universal morphism can be rephrased in a variety of ways. Let be a functor and let be an object of. Then the following statements are equivalent:
  • is a universal morphism from to
  • is an initial object of the comma category
  • is a representation of, where its components are defined by
for each object in
The dual statements are also equivalent:
  • is a universal morphism from to
  • is a terminal object of the comma category
  • is a representation of, where its components are defined by
for each object in