Lecture 13: Representable Functors and the Brown Representability Theorem

Let C be a category. A functor F : C → Sets is called representable if there exists an object B = BF in C with the property that there is a natural isomorphism of functors φ : C(−, BF ) −→ F. Thus, for every object X in C, there is an isomorphism φX from the set of arrows C(X,BF ) to the value F (X) of the functor. The naturality condition states that for any map f : Y → X in C, the identity F (f)(φX(α)) = φY (α ◦ f) holds, for any α : X → B. One usually expresses this in terms of a commutative diagram