Universal property

Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them.

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.

Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below).

For example, given a commutative ring R, the field of fractions of the quotient ring of R by a prime ideal p can be identified with the residue field of the localization of R at p; 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.

Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.

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 (see Examples below).

Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples.

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.

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

is equivalent to an initial object in the comma category

corresponds with a terminal object in the comma category

The reader can construct numerous other examples by consulting the articles mentioned in the introduction.

Let be the forgetful functor which assigns to each algebra its underlying vector space.

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

is the inclusion map, is a universal morphism from the vector space

As a commutative diagram: For the example of the Cartesian product in Set, the morphism

the unique map such that the required diagram commutes is given by

[3] Categorical products are a particular kind of limit in category theory.

One can generalize the above example to arbitrary limits and colimits.

In other words, the natural transformation is the one defined by having constant component

The definition of a universal morphism can be rephrased in a variety of ways.

Similar statements apply to the dual situation of terminal morphisms from

Indeed, all pairs of adjoint functors arise from universal constructions in this manner.

: Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of

Universal properties of various topological constructions were presented by Pierre Samuel in 1948.

The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.

The typical diagram of the definition of a universal morphism.
The typical diagram of the definition of a universal morphism.
The typical diagram of the definition of a universal morphism.
The most important arrow here is '"`UNIQ--postMath-00000024-QINU`"' which establishes the universal property.
The most important arrow here is which establishes the universal property.
A morphism in the comma category is given by the morphism '"`UNIQ--postMath-00000031-QINU`"' which also makes the diagram commute.
A morphism in the comma category is given by the morphism which also makes the diagram commute.
This demonstrates the connection between a universal diagram being an initial object in a comma category.
This demonstrates the connection between a universal diagram being an initial object in a comma category.
This simply demonstrates the definition of a morphism in a comma category.
This simply demonstrates the definition of a morphism in a comma category.
This shows that a terminal object in a specific comma category corresponds to a universal morphism.
This shows that a terminal object in a specific comma category corresponds to a universal morphism.
Commutative diagram showing how products have a universal property.
Commutative diagram showing how products have a universal property.
Universal morphisms can behave like a natural transformation between functors under suitable conditions.
Universal morphisms can behave like a natural transformation between functors under suitable conditions.
The unit and counit of an adjunction, which are natural transformations between functors, are an important example of universal morphisms.
The unit and counit of an adjunction, which are natural transformations between functors, are an important example of universal morphisms.