Brown representability theorem

Statement

1. Let $F$ be a contravariant functor from the category of pointed connected CW complexes and pointed maps to the category of pointed sets. Assume that:

   • (homotopy) homotopic pointed maps induce the same map;
   • (wedge) for every family $(X_\alpha)$, the map $F(\bigvee_\alpha X_\alpha) \to \prod_\alpha F(X_\alpha)$ induced by the inclusions into the wedge sum is a bijection;
   • (Mayer–Vietoris) let $X = A \cup B$ with $A$, $B$ and $A \cap B$ connected subcomplexes containing the base point. If $a \in F(A)$ and $b \in F(B)$ restrict to the same element of $F(A \cap B)$, then some $x \in F(X)$ restricts to $a$ and to $b$.

   Then $F$ is representable. That is, there are a pointed connected CW complex $K$ and an element $u \in F(K)$ with the following property. For every $X$ the map $[X, K]_* \to F(X)$, $f \mapsto F(f)(u)$, is a bijection, where $[X, K]_*$ is the set of pointed homotopy classes of maps.

2. (Neeman's version) Let $\mathcal{T}$ be a compactly generated triangulated category with all small coproducts. Let $H \colon \mathcal{T}^{\mathrm{op}} \to \mathrm{Ab}$ be a cohomological functor that sends coproducts in $\mathcal{T}$ to products. Then $H \cong \Hom_{\mathcal{T}}(-, Y)$ for some object $Y$ of $\mathcal{T}$.