Brown's Representability Theorem - Brown Representability Theorem For CW Complexes

Brown Representability Theorem For CW Complexes

The representability theorem for CW complexes, due to E. H. Brown, is the following. Suppose that:

1. The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set:
2. The functor F maps homotopy pushouts in Hotc to weak pullbacks. This is often stated as a Mayer-Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F(U), v ∈ F(V) such that u and v restrict to the same element of F(U ∩ V), there is an element w ∈ F(W) restricting to u and v, respectively.

Then F is representable by some CW complex C, that is to say there is an isomorphism

F(Z) ≅ Hom_Hotc(Z, C)

for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the induced maps F(Y) → F(Z) and Hom_Hot(Y, C) → Hom_Hot(Z, C) are compatible with these isomorphisms.

The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.

The representing object C above can be shown to depend functorially on F: any natural transformation from F to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of Yoneda's lemma.

Taking F(X) to be the singular cohomology group Hi(X,A) with coefficients in a given abelian group A, for fixed i > 0; then the representing space for F is the Eilenberg-MacLane space K(A, i). This gives a means of showing the existence of Eilenberg-MacLane spaces.