Brown's representability theorem

In mathematics, Brown's representability theorem in

and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that 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.

Brown representability theorem for CW complexes

The representability theorem for CW complexes, due to Edgar H. Brown,^[2] is the following. Suppose that:

1. The functor F maps
   coproducts (i.e. wedge sums
   ) in Hotc to products in Set: ${\displaystyle F(\vee _{\alpha }X_{\alpha })\cong \prod _{\alpha }F(X_{\alpha }),}$
2. The functor F maps
   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.

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

Taking F(X) to be the

However, the theorem is false without the restriction to connected pointed spaces, and an analogous statement for unpointed spaces is also false.[3]

A similar statement does, however, hold for