Advanced undergraduateCategory TheoryPlanned

Representable functors

Functors naturally isomorphic to Hom(A, −) or Hom(−, A): the forgetful functor on groups (represented by ℤ), the power-set functor (represented by a two-element set), and representing objects as solutions of universal problems.

math.cat.representable-functors

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Builds on

Natural transformations

Required prerequisites, all the way down: 17

You will learn to

Show that the forgetful functor Grp → Set is represented by ℤ, and that the contravariant power-set functor on Set is represented by {0, 1}.
Prove that a representing object, together with its universal element, is unique up to unique isomorphism.
Express products, free objects and quotients as representing objects of suitable functors.

Definition

A functor F : C → Set is representable if F ≅ Hom(A, −) for some object A; a functor F : Cᵒᵖ → Set is representable if F ≅ Hom(−, A).
A universal element of F : C → Set is a pair (A, x) with x ∈ F(A) such that for every object B and y ∈ F(B) there is a unique f : A → B with F(f)(x) = y.

Theorem

The forgetful functor Grp → Set is represented by ℤ, with universal element 1.
The contravariant power-set functor is isomorphic to Hom(−, {0, 1}): subsets correspond to characteristic functions.
Representations of F correspond to universal elements of F.

Common misconceptions

Thinking every functor to Set is representable.
Confusing covariant with contravariant representability.

Leads to

The Yoneda lemma

Optional discoveries

Algebraic geometry Grothendieck's functor of points views a scheme X through the functor Hom(−, X); many moduli problems ask whether a given functor is representable.
Elementary topos theory In a topos, the functor sending an object to its set of subobjects is represented by the subobject classifier Ω, generalising the representation of power sets by {0, 1}.