Representable functor

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.
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 Yoneda's representability theorem generalizes Cayley's theorem in group theory.

Definition

Let be a locally small category and let be the category of sets. For each object of let be the hom functor that maps object to the set.
A functor is said to be representable if it is naturally isomorphic to for some object of. A representation of is a pair where
is a natural isomorphism.
A contravariant functor from to is the same thing as a functor and is commonly called a presheaf. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor for some object of.

Universal elements

According to Yoneda's lemma, natural transformations from Hom to F are in one-to-one correspondence with the elements of F. Given a natural transformation Φ : Hom → F the corresponding element u ∈ F is given by
Conversely, given any element u ∈ F we may define a natural transformation Φ : Hom → F via
where f is an element of Hom. 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 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 is an isomorphism if and only if 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 as representations.

Examples

The functor represented by a scheme A can sometimes describe families of geometric objects. For example, vector bundles of rank k over a given algebraic variety or scheme X correspond to algebraic morphisms where A is the Grassmannian of k-planes in a high-dimensional space. Also certain types of subschemes are represented by Hilbert schemes.
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. 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 called an Eilenberg–MacLane space.
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 where A is a set and u is a subset of A, i.e. an element of P, such that for all sets X, the hom-set Hom is isomorphic to P via Φ_X = u = f⁻¹. Take A = and u =. Given a subset S ⊆ X the corresponding function from X to A is the characteristic function of S.
Forgetful functors to Set are very often representable. In particular, a forgetful functor is represented by whenever A is a free object over a singleton set with generator u.
* The forgetful functor Grp → Set on the category of groups is represented by.
* The forgetful functor Ring → Set on the category of rings is represented by, the polynomial ring in one variable with integer coefficients.
* The forgetful functor Vect → Set on the category of real vector spaces is represented by.
* The forgetful functor Top → Set on the category of topological spaces is represented by any singleton topological space with its unique element.
A group G can be considered a category 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. Choosing a representation amounts to choosing an identity for the heap.
Let R be a commutative ring with identity, and let R-'Mod be the category of R''-modules. If M and N are unitary modules over R, there is a covariant functor B: R'-Mod → Set which assigns to each R''-module P the set of R-bilinear maps M × N → P and to each R-module homomorphism f : P → Q the function B : B → B which sends each bilinear map g : M × N → P to the bilinear map f∘''g : M'' × N→''Q. The functor B'' is represented by the R-module M ⊗_R N.

Analogy: Representable functionals

Consider a linear functional on a complex Hilbert space H, i.e. a linear function. The Riesz representation theorem states that if F is continuous, then there exists a unique element which represents F in the sense that F is equal to the inner product functional, that is for.
For example, the continuous linear functionals on the square-integrable function space are all representable in the form for a unique function. The theory of distributions considers more general continuous functionals on the space of test functions. Such a distribution functional is not necessarily representable by a function, but it may be considered intuitively as a generalized function. For instance, the Dirac delta function is the distribution defined by for each test function, and may be thought of as "represented" by an infinitely tall and thin bump function near.
Thus, a function may be determined not by its values, but by its effect on other functions via the inner product. Analogously, an object A in a category may be characterized not by its internal features, but by its functor of points, i.e. its relation to other objects via morphisms. Just as non-representable functionals are described by distributions, non-representable functors may be described by more complicated structures such as stacks.

Properties

Uniqueness

Representations of functors are unique up to a unique isomorphism. That is, if and represent the same functor, then there exists a unique isomorphism φ : A₁ → A₂ such that
as natural isomorphisms from Hom to Hom. This fact follows easily from Yoneda's lemma.
Stated in terms of universal elements: if and represent the same functor, then there exists a unique isomorphism φ : A₁ → A₂ such that

Preservation of limits

Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, representable functors preserve all limits. It follows that any functor which fails to preserve some limit is not representable.
Contravariant representable functors take colimits to limits.

Left adjoint

Any functor K : C → Set with a left adjoint F : Set → C is represented by where X = is a singleton set and η is the unit of the adjunction.
Conversely, if K is represented by a pair and all small copowers of A exist in C then K has a left adjoint F which sends each set I to the Ith copower of A.
Therefore, if C is a category with all small copowers, a functor K : C → Set is representable if and only if it has a left adjoint.

Relation to universal morphisms and adjoints

The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors.
Let G : D → C be a functor and let X be an object of C. Then is a universal morphism from X to G if and only if is a representation of the functor Hom_C from D to Set. It follows that G has a left-adjoint F if and only if Hom_C is representable for all X in C. The natural isomorphism Φ_X : Hom_D → Hom_C yields the adjointness; that is
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 is a universal morphism from F to Y if and only if is a representation of the functor Hom_D from C to Set. It follows that F has a right-adjoint G if and only if Hom_D is representable for all Y in D.