![Universal morphism definition](https://upload.wikimedia.org/wikipedia/commons/d/df/Universal_morphism_definition.svg)
In
mathematics, more specifically in
category theory, 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 rat ...
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
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
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s 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
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
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of by a
prime ideal can be identified with the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
of the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
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 object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between eleme ...
s,
direct products and
direct sums,
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 lattices,
Grothendieck group,
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 boun ...
,
completion of a ring In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), f ...
,
Dedekind–MacNeille completion
In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed ...
,
product topologies,
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
,
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
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
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 learn ...
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 nam ...
s,
quotient groups,
quotient vector space
In linear algebra, the quotient of a vector space ''V'' by a subspace ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a quotient space and is denoted ''V''/''N'' (read "''V'' mod ''N''" or "''V'' by ' ...
s, 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
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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
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:
![Universal morphism definition](https://upload.wikimedia.org/wikipedia/commons/d/df/Universal_morphism_definition.svg)
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:
![Universal definition dualized](https://upload.wikimedia.org/wikipedia/commons/e/ef/Universal_definition_dualized.svg)
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:
![Definition of a morphism in a comma category](https://upload.wikimedia.org/wikipedia/commons/f/f4/Definition_of_a_morphism_in_a_comma_category.svg)
Now suppose that the object
in
is initial. Then
for every object
, there exists a unique morphism
such that the following diagram commutes.
![Connection between universal diagrams and comma categories](https://upload.wikimedia.org/wikipedia/commons/0/08/Connection_between_universal_diagrams_and_comma_categories.svg)
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:
![Definition of a morphism in a comma category 1](https://upload.wikimedia.org/wikipedia/commons/c/c4/Definition_of_a_morphism_in_a_comma_category_1.svg)
Suppose
is a terminal object in
. Then for every object
,
there exists a unique morphism
such that the following diagrams commute.
![Connection between comma category and universal properties](https://upload.wikimedia.org/wikipedia/commons/8/8d/Connection_between_comma_category_and_universal_properties.svg)
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). Let
:
:
-Alg →
-Vect
be the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
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
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
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
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-s ...
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
In the mathematical field of category theory, the product of two categories ''C'' and ''D'', denoted and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifu ...
and define the
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct al ...
:
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 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 is ...
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 t ...
. The ''
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct al ...
''
:
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
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
of the
comma category
*
is a
representation of
The dual statements are also equivalent:
*
is a universal morphism from
to
*
is a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
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:
![Connection between universal elements inducing a functor](https://upload.wikimedia.org/wikipedia/commons/e/e9/Connection_between_universal_elements_inducing_a_functor.svg)
If ''every'' object
of
admits a universal morphism to
, then the assignment
and
defines a functor
. The maps
then define a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
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 kno ...
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 kno ...
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 morphisms appear as the unit and counit of adjunctions](https://upload.wikimedia.org/wikipedia/commons/1/1c/Universal_morphisms_appear_as_the_unit_and_counit_of_adjunctions.svg)
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
Pierre Samuel (12 September 1921 – 23 August 2009) was a French mathematician, known for his work in commutative algebra and its applications to algebraic geometry. The two-volume work ''Commutative Algebra'' that he wrote with Oscar Zariski ...
in 1948. They were later used extensively by
Bourbaki. The closely related concept of adjoint functors was introduced independently by
Daniel Kan
Daniel Marinus Kan (or simply Dan Kan) (August 4, 1927 – August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having authored or coauthored sever ...
in 1958.
See also
*
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between eleme ...
*
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
*
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 kno ...
*
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 a ...
*
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 math ...
Notes
References
*
Paul Cohn
Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutat ...
, ''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,
morphisms, categories,
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s,
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s,
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 ...