So, today I am going to talk about the Brown representability theorem. This is a central fact in algebraic topology, proved in the 1950s by Edgar Brown in a paper in the Annals. It states that under mild conditions a contravariant functor on the homotopy category of pointed CW complexes is representable. As we saw yesterday, this guarantees the existence of Eilenberg-Maclane spaces. More importantly, it guarantees—at least for CW complexes—things like universal bundles. I will have more to say about these applications in the future. First, let us try to understand the result itself.
So let be a contravariant functor to the category of pointed sets. We require that satisfy two axioms below. First, like any representable functor, it should send coproducts to products. Since the coproduct in is the wedge sum, we require that (for ranging over some index set)
under the canonical map that arises by taking the product (over ) of the maps .
Second, we require that satisfy the following Mayer-Vietoris axiom. If for subcomplexes , then if and “glue together,” i.e. become the same element of , they both come from a fixed element of . This is a sheaf-theoretic condition.
The first observation is that any representable functor must satisfy the coproduct axiom and the Mayer-Vietoris (i.e. sheafish) axiom. The coproduct axiom is automatic for any category.
The sheaf axiom is less trivial. Let as before. Suppose that is represented by a pointed complex . Given “gluable” elements and , by assumption the restrictions are equivalent (i.e. homotopic). This does not immediately mean we can glue the maps. However, by the homotopy extension property for CW pairs, we can homotope to some such that become equal (not merely homotopic) on . Together these define a map that becomes equivalent to .
These axioms are not too difficult to check in many interesting cases. For instance, they are true for singular cohomology. This is why the following is highly important:
Theorem 1 (Brown) If is a contravariant functor satisfying the coproduct and Mayer-Vietoris axioms, then is representable.
This is the result that I would like to begin to prove today.