In
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, cate ...
, a universal property is a property that characterizes
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
*Categories (Peirce)
* ...
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
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become obje ...
(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
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
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 sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
s,
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1 ...
s,
free lattice In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property.
Formal definition
Any set ''X'' may be used to generate the free semilattice ''FX''. Th ...
s,
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
,
completion of a metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
,
completion of a ring In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing c ...
,
Dedekind–MacNeille completion,
product topologies,
Stone–Čech compactification,
tensor products,
inverse limit and
direct limit,
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
s and
cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of .
Cokernels are dual to the kernels of category theory, hence the name: ...
s,
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
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of a
vector space 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. 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 ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
s.
* 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
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become obje ...
(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
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
-Vect over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and let
be the category of
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
-Alg over
(assumed to be
unital and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
). Let
:
:
-Alg →
-Vect
be the
forgetful functor which assigns to each algebra its underlying vector space.
Given any
vector space over
we can construct the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
. The tensor algebra is characterized by the fact:
:“Any linear map from
to an algebra
can be uniquely extended to an
algebra homomorphism 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
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 ring ...
can be characterized by a universal construction. For concreteness, one may consider the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
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
Top
A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect.
Once set in motion, a top will usually wobble for a few ...
, 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
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets i ...
and let
be the corresponding
functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in ...
. 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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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
In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become obje ...
*
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 functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
s, 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
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
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
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
*
Monad (category theory)
*
Variety of algebras
*
Cartesian closed category
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ma ...
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 Joyal
André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to jo ...
CatLab 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:
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,
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s, categories,
functors,
natural transformations,
universal properties
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 fr ...
.
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