Representable Functor - Examples

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.
• Forgetful functors to Set are very often representable. In particular, a forgetful functor is represented by (A, u) 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 (Z, 1).
  • The forgetful functor Ring → Set on the category of rings is represented by (Z, x), the polynomial ring in one variable with integer coefficients.
  • The forgetful functor Vect → Set on the category of real vector spaces is represented by (R, 1).
  • 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 (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 nth 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.