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: Hotop → 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.

Brown representability theorem for CW complexes

The representability theorem for CW complexes, due to E. H. Brown, is the following: suppose the functor F maps any colimit in Hot to a limit in Set. Then F is representable by some CW complex C, that is to say there is an isomorphism

F(Z) ≅ HomHot(Z, C)

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

According to a combinatorial result in category theory, all small (that is to say, indexed over a set, as opposed to a proper class) colimits are built up from coproducts and pushouts (or just coequalisers). The traditional, equivalent statement of the theorem in algebraic topology is thus the following: suppose F satisfies the wedge axiom:

F(\vee_\alpha X_\alpha) \cong \prod_\alpha F(X_\alpha),

i.e., F converts any wedge sum (coproduct of pointed spaces) into a product of sets, and the Mayer-Vietoris axiom, requiring that 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. Under these two conditions, F is representable.

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

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.

Variants

A similar statement holds for spectra instead of CW complexes.

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.

A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman. Together with the preceding remark, it gives a criterion for a (covariant) functor F: C → D between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums, then F is a left adjoint. Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.

Jacob Lurie has proved a version of the Brown representability theorem for the homotopy category of a pointed quasicategory with a compact set of generators which are cogroup objects in the homotopy category. For instance, this applies to the homotopy category of (pointed) connected CW complexes, as well as to the unbounded derived category of a Grothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category).

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