There are a lot of forms of the Brown representability theorem, which all basically assert that a functor on a suitable homotopy category which plays well with arbitrary coproducts and satisfies a weak condition on push-outs, is representable.

The form proved by Brown was the following. Let ${\mathbf{hCW}}$ be the homotopy category of pointed CW complexes

Theorem 1 (Brown, c. 1950) Let ${F: \mathbf{hCW} \rightarrow \mathbf{Sets}}$ be a contravariant functor such that ${F}$ sends coproducts to products. Suppose that if ${(X_1, X_2, A) \subset X}$ is a proper triad—i.e., that ${(X_1, A)}$ and ${(X_2, A)}$ were CW pairs with ${X_1 \cap X_2 = A, X_1 \cup X_2 = X}$—then the map

$\displaystyle F(X) \rightarrow F(X_1) \times_{F(A)} F(X_2)$

is surjective. Then ${F}$ is representable. (more…)

I’ve been a bit busy and consequently behind on this blog, but I thought I would advertise some notes I wrote, covering chapter 1 of “Higher Topos Theory.” I’m giving a talk on them at a seminar tomorrow, and I wrote them mostly to prepare, but I tried to be detailed.