Representable functor

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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.

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.

1 Definition
2 Universal elements
3 Uniqueness
4 Examples
5 Relation to universal morphisms and adjoints

[edit] 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).

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

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.

[edit] 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

$u = \Phi_A(\mathrm{id}_A).\,$

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

$\Phi_X(f) = (Ff)(u)\,$

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.

[edit] 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

$\Phi_1^{-1}\circ\Phi_2 = \mathrm{Hom}(\varphi,-)$

as natural isomorphisms from Hom(A₂,–) to Hom(A₁,–). This fact follows easily from Yoneda's lemma.

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

$(F\varphi)u_1 = u_2.$

[edit] 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.

[edit] 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 (A,φ) is a universal morphism from X to G if and only if (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 if and only if 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

$\Phi_{X,Y}\colon \mathrm{Hom}_{\mathcal D}(FX,Y) \to \mathrm{Hom}_{\mathcal C}(X,GY)$

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 if and only if (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 if and only if Hom_D(F–,Y) is representable for all Y in D.