The Eilenberg-watts Theorem in Homotopical Algebra

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or smash product) with a fixed object is in fact such a tensor product, up to homotopy. The right half says any functor that looks like it could be Hom into a fixed object is so, up to homotopy. More precisely, suppose we have a closed symmetric monoidal category (resp. Quillen model category) M. Then the functor TN : M − → M that takes M to M ⊗ N is an M-functor and a left adjoint. The same is true if N is an E-E′-bimodule, where E and E′ are monoids in M, and TN : ModE −→ ModE′ is defined by TN (M) = M ⊗E N . Define a closed symmetric monoidal category (resp. model category) to be left self-contained (resp. homotopically left selfcontained) if every functor F : ModE −→ ModE′ that is anM-functor and a left adjoint (resp. and a left Quillen functor) is naturally isomorphic (resp. naturally weakly equivalent) to TN for some N . The classical Eilenberg-Watts theorem in algebra then just says that the category Ab of abelian groups is left self-contained, so we are generalizing that theorem.