The Eilenberg-Watts theorem in homotopical algebra
Mark Hovey
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 T_{B} that takes A to A tensor B is an M-functor and
a left adjoint. The same is true if B is an E-E'-bimodule, where E
and E' are monoids in M, and T_{B} takes an E-module A to A tensored
over E with B. Define a closed symmetric monoidal category
(resp. model category) to be left self-contained (resp. homotopically
left self-contained) if every functor F from E-modules to E'-modules
that is an M-functor and a left adjoint (resp. and a left Quillen
functor) is naturally isomorphic (resp. naturally weakly equivalent)
to T_{B} for some B. The classical Eilenberg-Watts theorem in algebra
then just says that the category of abelian groups is left
self-contained, so we are generalizing that theorem.