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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and These properties are called universal properties. Universal properties are studied abstractly using the language of category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

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: direct product and direct sum, free group, free lattice, Grothendieck 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 objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Mathematics, in the area of Order theory, a free lattice is the Free object corresponding to a lattice. In Mathematics, the Grothendieck group construction in Abstract algebra constructs an Abelian group from a Commutative Monoid in the In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In the mathematical discipline of General topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects the precise In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a In Category theory, a branch of Mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square) is the In Mathematics, an equaliser, or equalizer, is a set of arguments where two or more functions have equal values

Contents 1 Motivation 2 Formal definition 3 Examples 3.1 Tensor algebras 3.2 Products 3.3 Limits and colimits 4 Properties 4.1 Existence and uniqueness 4.2 Equivalent formulations 4.3 Relation to adjoint functors 5 History 6 See also 7 References

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 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.
• Universal properties define objects 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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property.
• 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

Let U: D → C be a functor from a category D to a category C, and let X be an object of C. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships 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:

• 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 and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also

One can also consider the categorical dual of the above definition by reversing all the arrows. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the 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:

• 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, although in order to be consistent with the naming of limits and colimits the former construction should be named couniversal and the latter universal. 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

Examples

Below are 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). In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) 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 Let

U : K-Alg → K-Vect

be the forgetful functor which assigns to each algebra its underlying vector space. In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor.

Given any vector space V over K we can construct the tensor algebra T(V) of V. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra 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 from T(V) to A. A homomorphism between two algebras over a field K, A and B, is a map FA\rightarrow B such that for all k ”

This statement is a universal property of the tensor algebra since it expresses the fact that the pair (T(V), i), where i : V → T(V) 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 functor is left adjoint to the forgetful functor (see the section below on relation to adjoint functors).

Products

Categorical products can be characterized by a universal property. 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 For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural

Let X and Y be objects of a category D. The product of X and Y is an object X × Y together with two morphisms

π₁ : X × Y → X

π₂ : X × Y → Y

such that for any other object Z of D and morphisms f : Z → X and g : Z → Y there exists a unique morphism h : Z → X × Y such that f = π₁∘h and g = π₂∘h.

To understand this characterization as a universal property we take the category C to be the product category D × D and define the diagonal functor

Δ : D → D × D

by Δ(X) = (X, X) and Δ(f : X → Y) = (f, f). In the mathematical field of Category theory, the product of two categories C and D, denoted C × D and called a product category In Category theory, for any object a in any category C where the product a × a exists there exists the diagonal morphism Then (X × Y, (π₁, π₂)) is a universal morphism from Δ to the object (X, Y) of D × D. This is just a restatement of the above since the pair (f, g) represents an (arbitrary) morphism from Δ(Z) to (X, Y).

Limits and colimits

Categorical products are a particular kind of limit in category theory. 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 One can generalize the above example to arbitrary limits and colimits.

Let J and C be categories with a J small index category and let C^J be the corresponding functor category. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Category theory, a branch of mathematics a diagram is the categorical analogue of a Indexed family in Set theory. In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor 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. In Category theory, for any object a in any category C where the product a × a exists there exists the diagonal morphism e. Δ(N)(X) = N for each X in J).

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.

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 essentially unique. Specifically, it is unique up to a unique isomorphism: if (A′, φ′) is another such pair, then there exists a unique isomorphism k: A → A′ such that φ′ = U(k)φ. 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 Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This is easily seen by substituting (A′, φ′) for (Y, f) in the definition of the universal property.

It is the pair (A, φ) which is essentially unique in this fashion. The object A itself is only unique up to isomorphism. Indeed, if (A, φ) is a universal morphism and k: A → A′ is any isomorphism then the pair (A′, φ′), where φ′ = U(k)φ, is also a universal morphism.

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:

• (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)

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. A comma category (a special case being a slice 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 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 and 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 UV. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal The functors (V, U) are then a pair of adjoint functors, with V left-adjoint to U and U right-adjoint to V.

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:

• 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).

History

Universal properties of various topological constructions were presented by Pierre Samuel in 1948. Pierre Samuel (born 12 September 1921 in Paris) is a French mathematician known for his work in Commutative algebra and its applications Year 1948 ( MCMXLVIII) was a Leap year starting on Thursday (link will display the 1948 calendar of the Gregorian calendar. They were later used extensively by Bourbaki. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958. Daniel Marinus Kan is a Mathematician working in Homotopy theory. Year 1958 ( MCMLVIII) was a Common year starting on Wednesday (link will display full calendar of the Gregorian calendar.

See also

• Free object
• Monad (category theory)
• Variety of algebras
• Cartesian closed category

References

• Cohen, Paul M. In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities 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 , 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.
• Borceux, F. Handbook of Categorical Algebra: vol 1 Basic category theory (1994) Cambridge University Press, (Encyclopedia of Mathematics and its Applications) ISBN 0-521-44178-1
• N. Bourbaki, Livre II : Algèbre (1970), Hermann, ISBN 0201006391.

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