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. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. Euclid, detail from The School of Athens by Raphael. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... 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. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Partial plot of a function f. ...

From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. In mathematics, an upper set is a subset Y of a given set X such that, for all elements x and y, if x is less than or equal to y and x is an element of Y, then y is also in Y. More formally, An upper set is... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In group theory, Cayleys theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a... Group theory is that branch of mathematics concerned with the study of groups. ...

Definition

Let C be a locally small category and let Set be the category of sets. For each object A of C let Hom(A,–) be the hom-functor which maps objects X to the set Hom(A,X). This is a glossary of properties and concepts in category theory in mathematics. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, specifically in category theory, Hom-sets, i. ...

A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,–) for some object A of C. A representation of F is a pair (A, Φ) where Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... -1...

Φ : Hom(A,–) → F

is a natural isomorphism.

A contravariant functor G : C → Set is said to representable if it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C. For functors in computer science, see the function object article. ...

Universal elements

According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A). Given a natural transformation Φ : Hom(A,–) → F the corresponding element of u ∈ F(A) is given by In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ...

Conversely, given any element u ∈ F(A) we may define a natural transformation Φ : Hom(A,–) → F via

where f is an element of Hom(A,X). In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism. This leads to the following definition:

A universal element of a functor F : C → Set is a pair (A,u) consisting of an object A of C and an element u ∈ F(A) such that for every pair (X,v) with v ∈ F(X) there exists a unique morphism f : A → X with (Ff)u = v. A universal element may be viewed as a universal morphism from the one-point set {•} to the functor F or as an initial object in the category of elements of F.

The natural transformation induced by an element u ∈ F(A) is an isomorphism if and only if (A,u) is a universal element of F. We therefore conclude that representations of F are in one-to-one correspondence with universal elements of F. For this reason, it is common to refer to universal elements (A,u) as representations. In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In category theory, for every presheaf the category of elements of is the category defined as follows: its objects are pairs where is an object of and , its morphisms are the morphisms of such that . ...

Uniqueness

Representations of functors are unique up to a unique isomorphism. That is, if (A₁,Φ₁) and (A₂,Φ₂) represent the same functor, then there exists a unique isomorphism φ : A₁ → A₂ such that

as natural isomorphisms from Hom(A₂,–) to Hom(A₁,–). This fact follows easily from Yoneda's lemma. In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ...

Stated in terms of universal elements: if (A₁,u₁) and (A₂,u₂) represent the same functor, then there exists a unique isomorphism φ : A₁ → A₂ such that

Examples

• Consider the contravariant functor P : Set → Set which maps each set to its power set and each function to its inverse image map. To represent this functor we need a pair (A,u) where A is a set and u is a subset of A, i.e. an element of P(A), such that for all sets X, the hom-set Hom(X,A) is isomorphic to P(X) via Φ_X(f) = (Pf)u = f^–1(u). Take A = {0,1} and u = {1}. Given a subset S ⊆ X the corresponding function from X to A is the characteristic function of S.

• Let F : Grp → Set be the forgetful functor on the category of groups. To represent this functor we need a pair (A, u) where A is group and u ∈ A, so that for all groups G, Hom(A,G) is isomorphic to |G| via f : A → G maps to f(u). We can take A to be an infinite cyclic group generated by an element u. Any group homomorphism from A to G is uniquely determined by the image of u (which is arbitrary).

• In a similar vein to the previous example, let F : Ring → Set be the forgetful functor on the category of rings. Here we can take A = Z[x], to be the polynomial ring in one variable with integer coefficients, and u = x.

• A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a G-set. The unique hom-functor Hom(•,–) from G to Set corresponds to the canonical G-set G with the action of left multiplication. Standard arguments from group theory show that a functor from G to Set is representable if and only if the corresponding G-set is simply transitive (i.e. a G-torsor). Choosing a representation amounts to choosing an identity for the group structure.

• Let C be the category of CW-complexes with morphisms given by homotopy classes of continuous functions. For each natural number n there is a contravariant functor Hⁿ : C → Ab which assigns each CW-complex its n^th cohomology group (with integer coefficients). Composing this with the forgetful functor we have a contravariant functor from C to Set. Brown's representability theorem in algebraic topology says that this functor is represented by a CW-complex K(Z,n) called an Eilenberg-Mac Lane space.

In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... A forgetful functor is a type of functor in mathematics. ... In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... A forgetful functor is a type of functor in mathematics. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... A forgetful functor is a type of functor in mathematics. ... In mathematics, Browns representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Sets, to be a representable functor. ... In mathematics, an Eilenberg-MacLane space is a special kind of topological space that is important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomological operations. ...

Relation to universal morphisms and adjoints

The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors. In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

Let G : D → C be a functor and let X be an object of C. Then (A,φ) is a universal morphism from X to G iff (A,φ) is a representation of the functor Hom_C(X,G–) from D to Set. It follows that G has a left-adjoint F iff Hom_C(X,G–) is representable for all X in C. The natural isomorphism Φ_X : Hom_D(FX,–) → Hom_C(X,G–) yields the adjointness; that is IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

is a bijection for all X and Y.

The dual statements are also true. Let F : C → D be a functor and let Y be an object of D. Then (A,φ) is a universal morphism from F to Y iff (A,φ) is a representation of the functor Hom_D(F–,Y) from C to Set. It follows that F has a right-adjoint G iff Hom_D(F–,Y) is representable for all Y in D.