nLab
Brown representability theorem

Idea

The classical Brown representability theorem says that certain functors on Top${}^{\mathrm{op}}$ that look like assigning cohomology groups to topological spaces – in that they satisfy the Eilenberg-Steenrod axioms – are representable in that they may be realized by the assignments of homotopy classes of maps into a spectrum.

For more background on this classical theorem see generalized (Eilenberg-Steenrod) cohomology

There are various generalizations of this result from the Top to more general model categories and triangulated categories

The Eilenberg-Steenrod axioms were written down in an effort to axiomatize the notion of cohomology on topological spaces by extrapolating crucial properties of ordinary integral cohomology. The classical Brown representability theorem and its generalizations show that these complicated axioms have a very simple repackaging. The theorem is one of the crucial ingredients that motivate the definition of cohomology in terms of maps into certain coefficient objects. This general notion of cohomology is described at cohomology.

Classical Brown representability

Axiom of sum: For any family $\{X_{\alpha }{\}}_{\alpha \in A}$ of pointed CW-complexes the morphism $(i_{\beta }{)}_{*}:F({\vee }_{\alpha }{X}_{\alpha })\to {\prod }_{\alpha }F(X_{\alpha })$ induced by the inclusion $i_{\beta }:X_{\beta }\to {\vee }_{\alpha }{X}_{\alpha }$ is a bijection.

Mayer-Vietoris axiom: For every triad $(X;A_1,A_2)$ of CW-spaces (with $A_1\cup A_2=X$) and any elements $x_1\in F(A_1)$, $x_2\in F(A_2)$, such that $x_1\mid A_1\cap A_2=x_2\mid A_1\cap A_2$ there is $y\in F(X)$ such that $y\mid A_1=x_1$ and $y\mid A_2=x_2$.

Brown representability theorem: A contravariant functor $F:{\mathrm{CW}}_{*}^{\mathrm{op}}\to {\mathrm{Set}}_{*}$ from pointed CW-complexes to pointed sets which satisfies the axiom of sum and axiom of Mayer-Vietoris is representable. In other words, there is a pointed CW-complex $(Y,y_0)$ and a universal element $u\in F(Y,y_0)$ such that $T_u:[-;Y,y_0]\to F$ is a natural equivalence.

Details

version for ordinary (model) categories

Theorem (Jardine)

Let $C$ be a simplicial model category such that

1. it has a zero object $*$;

2. there is a set $S$ of cofibrant compact object such that the weak equivalences in $C$ are precisely the $S$-local equivalences.

Let $F:C^{\mathrm{op}}\to {\mathrm{Set}}_{*}$ be a functor to the category of pointed sets on $C^{\mathrm{op}}$ such that

1. $F$ is a homotopical functor

2. $F(*)=*$

3. $F$ preserves small coproducts of cofibrant objects in that the induced maps

   $$F(\coprod _i{X}_i)\to \prod _i(F(X_i))$$F(\coprod_i X_i) \to \prod_i(F(X_i))

   are bijections

4. (Mayer-Vietoris property) For every pushout diagrams

   $$\begin{array}{ccc}A& \to & X\\ {\downarrow }^i& & \downarrow \\ B& \to & B\coprod _{A}X\end{array}$$\array{ A &\to& X \\ \downarrow^i && \downarrow \\ B &\to& B \coprod_A X }

   with all objects cofibrant and $i$ a cofibration the induced function

   $$F(B\coprod _{A}X)\to F(B){\times }_{F(A)}F(X)$$F(B \coprod_A X) \to F(B) \times_{F(A)} F(X)

   is a surjection.

Then $F$ is representable.

Remarks

Notice that

• the existence of the 0-object is the generalized analogue of working with pointed topological spaces;

• the condition that the value of $F$ on the point $*$ is trivial means that this is about reduced cohomology theory;

• that every representable functor has the given properties is immediate. The nontrivial statement is that these properties already characterize representable functors.

version for simplicially enriched (model) categories

References

The generalizaton of the Brown representability theorem from topological spaces to ∞-stacks – or rather to the standard models for ∞-stack (∞,1)-toposes in terms of the standard model structure on simplicial presheaves – is given in

• Jardine, Representability theorem for simplicial presheaves

warning: this is probably implicialy about reduced cohomology theory, as the functors considered always assign the trivial result to the terminal object (the point in the usual examples).