In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a universal property is a property that characterizes
up to an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s from the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s from the integers, of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s from the rational numbers, and of
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
s 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 mean of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as
initial or terminal objects of a
comma category (see , below).
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
(all these constructions can be defined by universal properties).
Other objects that can be defined by universals properties include: all
free objects,
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
s 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 product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s,
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is slightly painful to actually construct, but using its universal property makes it much easier to deal with.
* Universal properties define objects uniquely up to a unique
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. 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.
[See for example, Polcino & Sehgal (2002), p. 133. exercise 1, about the universal property of group rings.]
* 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 (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.
Let
be a functor between categories
and
. In what follows, let
be an object of
, while
and
are objects of
, and
is a morphism in
.
Thus, the functor
maps
,
and
in
to
,
and
in
.
A universal morphism from
to
is a unique pair
in
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 (i.e. one where morphisms are seen as objects in their own right).
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
(assumed to be
unital and
associative). Let
:
:
-Alg →
-Vect
be the
forgetful functor which assigns to each algebra its underlying vector space.
Given any
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over
we can construct the
tensor algebra . The tensor algebra is characterized by the fact:
:“Any linear map from
to an algebra
can be uniquely extended to an
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
from
to
.”
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
(see the section below on
relation to adjoint functors).
Products
A
categorical product can be characterized by a universal construction. For concreteness, one may consider the
Cartesian product in
Set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
in
Grp, or the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
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
.
Limits and colimits
Categorical products are a particular kind of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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
to
(i.e.
for each
in
).
Given a functor
(thought of as an object in
), 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
: 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
The dual statements are also equivalent:
*
is a universal morphism from
to
*
is a
terminal object of the comma category
*
is a representation of
Relation to adjoint functors
Suppose
is a universal morphism from
to
and
is a universal morphism from
to
.
By the universal property of universal morphisms, given any morphism
there exists a unique morphism
such that the following diagram commutes:
If ''every'' object
of
admits a universal morphism to
, then the assignment
and
defines a functor
. The maps
then define a
natural transformation from
(the identity functor on
) to
. The functors
are then a pair of
adjoint functors, with
left-adjoint to
and
right-adjoint to
.
Similar statements apply to the dual situation of terminal morphisms from
. If such morphisms exist for every
in
one obtains a functor
which is right-adjoint to
(so
is left-adjoint to
).
Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let
and
be a pair of adjoint functors with unit
and co-unit
(see the article on
adjoint functors for the definitions). Then we have a universal morphism for each object in
and
:
*For each object
in
,
is a universal morphism from
to
. That is, for all
there exists a unique
for which the following diagrams commute.
*For each object
in
,
is a universal morphism from
to
. That is, for all
there exists a unique
for which the following diagrams commute.
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
(equivalently, every object of
).
History
Universal properties of various topological constructions were presented by
Pierre Samuel in 1948. They were later used extensively by
Bourbaki. The closely related concept of adjoint functors was introduced independently by
Daniel Kan in 1958.
See also
*
Free object
*
Natural transformation
*
Adjoint functor
*
Monad (category theory)
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is ...
*
Variety of algebras
*
Cartesian closed category
Notes
References
*
Paul Cohn, ''Universal Algebra'' (1981), D.Reidel Publishing, Holland. .
*
* Borceux, F. ''Handbook of Categorical Algebra: vol 1 Basic category theory'' (1994) Cambridge University Press, (Encyclopedia of Mathematics and its Applications)
* N. Bourbaki, ''Livre II : Algèbre'' (1970), Hermann, .
* Milies, César Polcino; Sehgal, Sudarshan K.. ''An introduction to group rings''. Algebras and applications, Volume 1. Springer, 2002.
* Jacobson. Basic Algebra II. Dover. 2009.
External links
nLab a wiki project on mathematics, physics and philosophy with emphasis on the ''n''-categorical point of view
*
André JoyalCatLab a wiki project dedicated to the exposition of categorical mathematics
* formal introduction to category theory.
* J. Adamek, H. Herrlich, G. Stecker
Abstract and Concrete Categories-The Joy of Cats*
Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
:
Category Theory—by Jean-Pierre Marquis. Extensive bibliography.
List of academic conferences on category theory* Baez, John, 1996
An informal introduction to higher order categories.
WildCatsis a category theory package for
Mathematica. Manipulation and visualization of objects,
morphisms, categories,
functors,
natural transformations,
universal properties.
The catsters a YouTube channel about category theory.
Video archiveof recorded talks relevant to categories, logic and the foundations of physics.
Interactive Web pagewhich generates examples of categorical constructions in the category of finite sets.
{{DEFAULTSORT:Universal Property
Category theory
Property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...