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 of Cayley's theorem in group theory.
Definition
Let C be a locally small category and let Set be the category of sets. For each object A of C let Hom be the hom functor that maps object X to the set Hom.A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom for some object A of C. A representation of F is a pair where
is a natural isomorphism.
A contravariant functor G from C to Set is the same thing as a functor G : Cop → Set and is commonly called a presheaf. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom for some object A of C.
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 byConversely, 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
- 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 ΦX = 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 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 nth 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–Mac Lane space.
- 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.
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 thatas 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 HomC from D to Set. It follows that G has a left-adjoint F if and only if HomC is representable for all X in C. The natural isomorphism ΦX : HomD → HomC 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 HomD from C to Set. It follows that F has a right-adjoint G if and only if HomD is representable for all Y in D.