Brown's representability theorem

In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor. That is, we are given

F: Hot^op → Set,

and certain necessary conditions for F to be of type Hom(—, C) with C a CW-complex can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.

Formulation

The basic form of the conditions is that finite colimits should be preserved; F being contravariant, this comes out that a finite colimit in Hot should become a finite limit in Set, arrows having been reversed. According to a combinatorial result in category theory, all finite colimits are built up from coproducts and pushouts (or just coequalisers). It is traditional in algebraic topology, therefore, to state the conditions in Brown's theorem as two axioms: the wedge axiom stating that a wedge sum (coproduct of pointed spaces) becomes under F a product of sets, and a Mayer-Vietoris axiom explaining the effect of F for patching, in terms of F(W) where W is created by the identification of spaces U and V.

Statement

The statement of Brown's representability theorem, proved by E. H. Brown, is then that F is a representable functor on Hot (up to equivalence of functors) if and only if the wedge axiom and Mayer-Vietoris axiom are satisfied by F.

For an example, we can take 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).

There is another version, applying the same ideas to the category of spectra.

References

• Robert M. Switzer (1975), Algebraic Topology — Homotopy and Homology pp. 152-7.