Brown represetability and the Eilenberg-Watts theorem in homotopical algebra
Mark Hovey
It is well-known that every homology functor on the stable homotopy
category is representable, so that the homology of X is the homotopy
of the smash product of E and X for some spectrum E. However,
Christensen, Keller, and Neeman 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 that is tensoring with E for some
cofibrant E is naturally weakly equivalent to such a functor. 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.