In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. These properties are called universal properties. Universal properties are studied abstractly using the language of category theory. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Below, we will give a general treatment of universal properties. It is advisable to study several examples first: direct product and direct sum, free group, product topology, Stone-Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer. In mathematics, one can often define a direct product of objects already known, giving a new one. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of directed families of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... In category theory, a branch of mathematics, the pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. ... Equalizer can mean: Equalizer, an audio processing tool. ...

Formal definition

Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consists of a pair (A, φ) where A is an object of D and φ : X → U(A) is a morphism in C, such that the following universal property is satisfied: Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

• Whenever Y is an object of D and f : X → U(Y) is a morphism in C, then there exists a unique morphism g : A → Y such that the following diagram commutes:

The existence of the morphism g intuitively expresses the fact that A is "general enough", while the uniqueness of the morphism ensures that A is "not too general". In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Wikipedia does not have an article with this exact name. ...

One can also consider the categorical dual of the above definition by reversing all the arrows. A universal morphism from U to X consists of a pair (A, φ) where A is an object of D and φ : U(A) → X is a morphism in C, such that the following universal property is satisfied: In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

• Whenever Y is an object of D and f : U(Y) → X is a morphism in C, then there exists a unique morphism g : Y → A such that the following diagram commutes:

Note that some authors may call one of these constructions a universal morphism and the other one a co-universal morphism. Which is which depends on the author. In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Wikipedia does not have an article with this exact name. ...

Properties

Existence and uniqueness

Defining a quantity does not guarantee its existence. Given a functor U and an object X as above, there may or may not exist a universal morphism from X to U (or from U to X). If, however, a universal morphism (A, φ) does exists then it is unique up to a unique isomorphism. That is, if (A′, φ′) is another such pair then there exists a unique isomorphism g : A → A′ such that φ′ = U(g)φ. This is easily seen by substituting (A′, φ′) for (Y, f) in the definition of the universal property. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

Equivalent formulations

The definition of a universal morphism can be rephrased in a variety of ways. Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

• (A, φ) is a universal morphism from X to U
• (A, φ) is an initial object of the comma category (X ↓ U)
• (A, φ) is a representation of Hom_C(X, U—)

The dual statements are also equivalent: In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... A comma category is a construction in category theory, a branch of mathematics. ... In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

• (A, φ) is a universal morphism from U to X
• (A, φ) is a terminal object of the comma category (U ↓ X)
• (A, φ) is a representation of Hom_C(U—, X)

In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... A comma category is a construction in category theory, a branch of mathematics. ... In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

Relation to adjoint functors

Suppose (A₁, φ₁) is a universal morphism from X₁ to U and (A₂, φ₂) is a universal morphism from X₂ to U. By the universal property, given any morphism h : X₁ → X₂ there exists a unique morphism g : A₁ → A₂ such that the following diagram commutes:

If every object X_i of C admits a universal morphism to U, then the assignment X_i A_i and h g defines a functor V from C to D. The maps φ_i then define a natural transformation from 1_C (the identity functor on C) to U V. The functors (V, U) are then a pair of adjoint functors, with V left-adjoint to U and U right-adjoint to V. Wikipedia does not have an article with this exact name. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

Similar statements apply to the dual situation of morphisms from U. If such morphisms exist for every X in C one obtains a functor V : C → D which is right-adjoint to U (so U is left-adjoint to V).

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 η and co-unit ε (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in C and D: The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

• For each object X in C, (F(X), η_X) is a universal morphism from X to G. That is, for all f : X → G(Y) there exists a unique g : F(X) → Y for which the following diagrams commute.
• For each object Y in D, (G(Y), ε_Y) is a universal morphism from F to Y. That is, for all g : F(X) → Y there exists a unique f : X → G(Y) 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). Commutative diagram for adjoint functors File links The following pages link to this file: Universal property Adjoint functors Categories: GFDL images ...

Examples

We give a few worked 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 spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative). Let U be the forgetful functor which assigns to each algebra its underlying vector space. In mathematics, the category K_Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ... This article presents the essential definitions. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... A forgetful functor is a type of functor in mathematics. ...

Given any vector space V over K we can construct the tensor algebra T(V) of V. The universal property of the tensor algebra expresses the fact that the pair (T(V), i), where i : V → T(V) is the inclusion map, is a universal morphism from V to U. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... 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. ...

Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg. This functor is left-adjoint to the forgetful functor U.

Kernels

Suppose D is a category with zero morphisms (such as the category of groups) and f : X → Y is a morphism in D. A kernel of f is any morphism k: K → X such that In category theory, a zero morphism is a special kind of trivial morphism. ... In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ...

• f k is the zero morphism from K to Y;
• Given any morphism k′: K′ → X such that f k′ is the zero morphism, there is a unique morphism u: K′ → K such that k u = k′.

To understand this in the framework of the general setting above, we define the category C of morphisms in D. The objects of C are morphisms f : X → Y in D, and a morphism from f : X → Y to g : S → T is given by a pair (α,β) of morphisms α : X → S and β : Y → T such that βf = gα. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

Define a functor F : D → C that maps an object K of D to the zero morphism 0_KK : K → K and a morphism r : K → L to the pair (r,r).

Now, given a morphism f : X → Y in the category D (thought of as an object in the category C) and an object K of D, a morphism from F(K) to f is given by a pair (k,l) such that f k = l 0_KK = 0_KY, which is exactly what shows up in the universal property of kernels given above. The abstract “universal morphism from F to f ” is nothing but the universal property of a kernel.

Limits and colimits

Limits and colimits are important special cases of universal constructions. Let J and C be categories with J small (J is thought of as an index category) and let C^J be the corresponding functor category. The diagonal functor Δ : C → C^J is the functor that maps each object N in C to the constant functor Δ(N) : J → C to N (i.e. Δ(N)(X) = N for each X in J). In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...

Given a functor F : J → C (thought of as an object in C^J), the limit of F, if it exists, is nothing but a universal morphism from Δ to F. Dually, the colimit of F is a universal morphism from F to Δ.

What is it good for?

Once one recognizes a certain construction as given by a universal property, one gains several benefits:

• Universal properties define objects up to a unique isomorphism; one strategy to prove that two objects are isomorphic is therefore to show that they satisfy the same universal property.
• 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 construct is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details.
• If the universal construction can be carried out for every X in C, then we know that we obtain a functor from C to D. (So for example, forming kernels is functorial: every morphism (α,β) from the morphism f to the morphism g induces a morphism from the kernel of f to the kernel of g.)
• Furthermore, this functor is a right or left adjoint to U. But right adjoints commute with limits and left adjoints commute with colimits! (So we can for example immediately conclude that the kernel of a product of maps is equal to the product of the kernels.)

Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ... In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. ...

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. Pierre Samuel was a French mathematician, known for his work in commutative algebra and its applications to algebraic geometry. ... 1948 (MCMXLVIII) was a leap year starting on Thursday (the link is to a full 1948 calendar). ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... Daniel Marinus Kan is a mathematician, more specifically a homotopy theorist. ... 1958 (MCMLVIII) was a common year starting on Wednesday of the Gregorian calendar. ...

References

• Cohen, Paul M., Universal Algebra (1981), D.Reidel Publishing, Holland. ISBN 90-277-1213-1.
• Mac Lane, Saunders, Categories for the Working Mathematician 2nd ed. (1998), Graduate Texts in Mathematics 5. Springer. ISBN 0-387-98403-8.