Brown Representability and the Eilenberg-watts Theorem in Homotopical Algebra

It is well-known that every homology functor on the stable homotopy category is representable, so of the form E∗(X) = π∗(E ∧ X) for some spectrum E. However, Christensen, Keller, and Neeman [CKN01] have exhibited simple triangulated categories, such as the derived category of k[x, y] for sufficiently large fields k, for which not every homology functor is representable. In this paper, we show that this failure of Brown representability does not happen on the model category level. That is, we show that a homology theory is representable if and only if it lifts to a well-behaved functor on the model category level. We also show that, for a reasonable model category M, every functor that has the same formal properties as a functor of the form X 7→ X⊗E for some cofibrant E is naturally weakly equivalent to a functor of that form. This is closely related to the Eilenberg-Watts theorem in algebra, which proves that every functor with the same formal properties as the tensor product with a fixed object is isomorphic to such a functor.