universal property

TheInfoList

In
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, a branch of
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a universal property is an important property which is satisfied by a universal morphism (see
Formal Definition Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (form Form is the shape, visual appearance, or :wikt:configuration, configuration of an object. In a wider sense, the form is the ...
). Universal morphisms can also be thought of more abstractly as initial or terminal objects of a
comma category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
(see Connection with comma categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated. Universal properties may be used in other areas of mathematics implicitly, but the abstract and more precise definition of it can be studied in category theory. This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all
free objectIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s,
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and
direct sum The direct sum is an operation from abstract algebra, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...
,
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
,
free latticeIn mathematics, in the area of order theory, a free lattice is the free object corresponding to a Lattice (order), lattice. As free objects, they have the universal property. Formal definition Any set (mathematics), set ''X'' may be used to generate ...
,
Grothendieck groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
,
Dedekind–MacNeille completion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
,
product topology Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produc ...
,
Stone–Čech compactificationIn the mathematical discipline of general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic defin ...
,
tensor product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

,
inverse limitIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and
direct limit In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
,
kernel Kernel may refer to: Computing * Kernel (operating system) The kernel is a computer program at the core of a computer's operating system that has complete control over everything in the system. It is the "portion of the operating system co ...
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 ...

,
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
, pushout and equalizer.

# 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 over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', i ...
of a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
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 $F: C \to D$ be a functor between categories $C$ and $D$. In what follows, let $X$ be an object of $D$, while $A$ and $A\text{'}$ are objects of $C$. Thus, the functor $F$ maps $A$, $A\text{'}$ and $h$ in $C$ to $F\left(A\right)$, $F\left(A\text{'}\right)$ and $F\left(h\right)$ in $D$. A universal morphism from $X$ to $F$ is a unique pair $\left(A, u: X \to F\left(A\right)\right)$ in $D$ which has the following property, commonly referred to as a universal property. For any morphism of the form $f: X \to F\left(A\text{'}\right)$ in $D$, there exists a ''unique'' morphism $h: A \to A\text{'}$ in $C$ such that the following diagram : We can dualize this categorical concept. A universal morphism from $F$ to $X$ is a unique pair $\left(A, u: F\left(A\right) \to X\right)$ that satisfies the following universal property. For any morphism of the form $f: F\left(A\text{'}\right) \to X$ in $D$, there exists a ''unique'' morphism $h: A\text{'} \to A$ in $C$ 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 $\left(A, u\right)$ 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 $F: C \to D$ be a functor and $X$ an object of $D$. Then recall that the comma category $\left(X \downarrow F\right)$ is the category where * Objects are pairs of the form $\left(B, f: X \to F\left(B\right)\right)$, where $B$ is an object in $C$ * A morphism from $\left(B, f: X \to F\left(B\right)\right)$ to $\left(B\text{'}, f\text{'}: X \to F\left(B\text{'}\right)\right)$ is given by a morphism $h: B \to B\text{'}$ in $C$ such that the diagram commutes: Now suppose that the object $\left(A, u: X \to F\left(A\right)\right)$ in $\left(X \downarrow F\right)$ is initial. Then for every object $\left(A\text{'}, f: X \to F\left(A\text{'}\right)\right)$, there exists a unique morphism $h: A \to A\text{'}$ 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 $X$ to $F$. Therefore, we see that a universal morphism from $X$ to $F$ is equivalent to an initial object in the comma category $\left(X \downarrow F\right)$. Conversely, recall that the comma category $\left(F \downarrow X\right)$ is the category where *Objects are pairs of the form $\left(B, f: F\left(B\right) \to X\right)$ where $B$ is an object in $C$ *A morphism from $\left(B, f:F\left(B\right) \to X\right)$ to $\left(B\text{'}, f\text{'}:F\left(B\text{'}\right) \to X\right)$ is given by a morphism $h: B \to B\text{'}$ in $C$ such that the diagram commutes: Suppose $\left(A, u:F\left(A\right) \to X\right)$ is a terminal object in $\left(F \downarrow X\right)$. Then for every object $\left(A\text{'}, f: F\left(A\text{'}\right) \to X\right)$, there exists a unique morphism $h: A\text{'} \to A$ 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 $F$ to $X$. Hence, a universal morphism from $F$ to $X$ corresponds with a terminal object in the comma category $\left(F \downarrow X\right)$.

# 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 $C$ be the
category of vector spacesIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
$K$-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 grassl ...
$K$ and let $D$ be the category of
algebras In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
$K$-Alg over $K$ (assumed to be unital and
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
). Let :$U$ : $K$-Alg → $K$-Vect be the
forgetful functorIn 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 signat ...
which assigns to each algebra its underlying vector space. Given any
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
$V$ over $K$ 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 over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', i ...
$T\left(V\right)$. The tensor algebra is characterized by the fact: :“Any linear map from $V$ to an algebra $A$ can be uniquely extended to an
algebra homomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
from $T\left(V\right)$ to $A$.” This statement is an initial property of the tensor algebra since it expresses the fact that the pair $\left(T\left(V\right),i\right)$, where $i:V \to U\left(T\left(V\right)\right)$ is the inclusion map, is a universal morphism from the vector space $V$ to the functor $U$. Since this construction works for any vector space $V$, we conclude that $T$ is a functor from $K$-Vect to $K$-Alg. This means that $T$ is ''left adjoint'' to the forgetful functor $U$ (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 (category theory), Set, the
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
in Grp (category theory), Grp, or the
product topology Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produc ...
in Top (category theory), Top, where products exist. Let $X$ and $Y$ be objects of a category $C$ with finite products. The product of $X$ and $Y$ is an object $X$ × $Y$ together with two morphisms :$\pi_1$ : $X \times Y \to X$ :$\pi_2$ : $X \times Y \to Y$ such that for any other object $Z$ of $C$ and morphisms $f: Z \to X$ and $g: Z \to Y$ there exists a unique morphism $h: Z \to X \times Y$ such that $f = \pi_1 \circ h$ and $g = \pi_2 \circ h$. To understand this characterization as a universal property, take the category $D$ to be the product category $C \times C$ and define the diagonal functor : $\Delta: C \to C \times C$ by $\Delta\left(X\right) = \left(X, X\right)$ and $\Delta\left(f: X \to Y\right) = \left(f, f\right)$. Then $\left(X \times Y, \left(\pi_1, \pi_2\right)\right)$ is a universal morphism from $\Delta$ to the object $\left(X, Y\right)$ of $C \times C$: if $\left(f, g\right)$ is any morphism from $\left(Z, Z\right)$ to $\left(X, Y\right)$, then it must equal a morphism $\Delta\left(h: Z \to X \times Y\right) = \left(h,h\right)$ from $\Delta\left(Z\right) = \left(Z, Z\right)$ to $\Delta\left(X \times Y\right) = \left(X \times Y, X \times Y\right)$ followed by $\left(\pi_1, \pi_2\right)$.

## Limits and colimits

Categorical products are a particular kind of limit (category theory), limit in category theory. One can generalize the above example to arbitrary limits and colimits. Let $J$ and $C$ be categories with $J$ a small category, small index category and let $C^J$ be the corresponding functor category. The ''diagonal functor'' :$\Delta: C \to C^J$ is the functor that maps each object $N$ in $C$ to the constant functor $\Delta\left(N\right): J \to C$ to $N$ (i.e. $\Delta\left(N\right)\left(X\right) = N$ for each $X$ in $J$). Given a functor $F: J \to C$ (thought of as an object in $C^J$), the ''limit'' of $F$, if it exists, is nothing but a universal morphism from $\Delta$ to $F$. Dually, the ''colimit'' of $F$ is a universal morphism from $F$ to $\Delta$.

# Properties

## Existence and uniqueness

Defining a quantity does not guarantee its existence. Given a functor $F: C \to D$ and an object $X$ of $C$, there may or may not exist a universal morphism from $X$ to $F$. If, however, a universal morphism $\left(A, u\right)$ does exist, then it is essentially unique. Specifically, it is unique up to a ''unique''
isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

: if $\left(A\text{'}, u\text{'}\right)$ is another pair, then there exists a unique isomorphism $k: A \to A\text{'}$ such that $u\text{'} = F\left(k\right) \circ u$. This is easily seen by substituting $\left(A, u\text{'}\right)$ in the definition of a universal morphism. It is the pair $\left(A, u\right)$ which is essentially unique in this fashion. The object $A$ itself is only unique up to isomorphism. Indeed, if $\left(A, u\right)$ is a universal morphism and $k: A \to A\text{'}$ is any isomorphism then the pair $\left(A\text{'}, u\text{'}\right)$, where $u\text{'} = F\left(k\right) \circ u$ is also a universal morphism.

## Equivalent formulations

The definition of a universal morphism can be rephrased in a variety of ways. Let $F: C \to D$ be a functor and let $X$ be an object of $D$. Then the following statements are equivalent: * $\left(A, u\right)$ is a universal morphism from $X$ to $F$ * $\left(A, u\right)$ is an initial object of the
comma category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$\left(X \downarrow F\right)$ * $\left(A, u\right)$ is a representable functor, representation of $\text_D\left(X, F\left(-\right)\right)$ The dual statements are also equivalent: * $\left(A, u\right)$ is a universal morphism from $F$ to $X$ * $\left(A, u\right)$ is a terminal object of the comma category $\left(F \downarrow X\right)$ * $\left(A, u\right)$ is a representation of $\text_D\left(F\left(-\right), X\right)$

Suppose $\left(A_1, u_1\right)$ is a universal morphism from $X_1$ to $F$ and $\left(A_2, u_2\right)$ is a universal morphism from $X_2$ to $F$. By the universal property of universal morphisms, given any morphism $h: X_1 \to X_2$ there exists a unique morphism $g: A_1 \to A_2$ such that the following diagram commutes: If ''every'' object $X_i$ of $D$ admits a universal morphism to $F$, then the assignment $X_i \mapsto A_i$ and $h \mapsto g$ defines a functor $G: D \to C$. The maps $u_i$ then define a natural transformation from $1_C$ (the identity functor on $C$) to $F\circ G$. The functors $\left(F, G\right)$ are then a pair of adjoint functors, with $G$ left-adjoint to $F$ and $F$ right-adjoint to $G$. Similar statements apply to the dual situation of terminal morphisms from $F$. If such morphisms exist for every $X$ in $C$ one obtains a functor $G: C \to D$ which is right-adjoint to $F$ (so $F$ is left-adjoint to $G$). Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let $F$ and $G$ be a pair of adjoint functors with unit $\eta$ and co-unit $\epsilon$ (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in $C$ and $D$: *For each object $X$ in $C$, $\left(F\left(X\right), \eta_X\right)$ is a universal morphism from $X$ to $G$. That is, for all $f: X \to G\left(Y\right)$ there exists a unique $g: F\left(X\right) \to Y$ for which the following diagrams commute. *For each object $Y$ in $D$, $\left(G\left(Y\right), \epsilon_Y\right)$ is a universal morphism from $F$ to $Y$. That is, for all $g: F\left(X\right) \to Y$ there exists a unique $f: X \to G\left(Y\right)$ 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 $C$ (equivalently, every object of $D$).

# History

Universal properties of various topological constructions were presented by Pierre Samuel in 1948. They were later used extensively by Nicolas Bourbaki, Bourbaki. The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.

* Free object *Natural transformation * Adjoint functor * Monad (category theory) * Variety of algebras * Cartesian closed category

# 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.

nLab
a wiki project on mathematics, physics and philosophy with emphasis on the ''n''-categorical point of view * André Joyal
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.
WildCats
is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories,
functor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, natural transformations, universal properties.
The catsters