symmetric monoidal (∞,1)-category of spectra
Conversely, if is right exact and preserves small coproducts, then it is naturally isomorphic to tensoring with a bimodule.
Given a cocontinuous functor , the reconstructed --bimodule is given as follows:
the underlying right -module is , where is regarded as a right module over itself in the canonical way;
the left -module structure on is given for and by
where denotes the right -module homomorphism given by left multiplication by .
The theorem holds for nonunital rings as well, but then reconstructs as where is the extension of by adjoining the unit element (the tensor product is still over the original ). If is a flat functor then is a flat module over .
In the statement of the theorem we can replace “additive, right exact and preserves direct sum” by “additive and left adjoint”.
In this form the theorem is stated for instance in (Hovey, theorem 0.1).
The standard Eilenberg-Watts theorem is a statement about monoids and their actions in Ab. More generally one may ask for generalizations of the theorem to other internalization contexts, and in particular to homotopy theory. See the introduction of (Hovey).
In particular Eilenberg-Watts theorems hold true in the homotopy theory following model categories (see at model structure on modules over an algebra over an operad)
model structure on spectra (symmetric, orthogonal, -modules).
This is the main theorem in (Hovey).
More generally, we have the first half of the Eilenberg-Watts theorem in (∞,1)-category theory:
For a monoidal (∞,1)-category with geometric realization of simplicial objects in an (∞,1)-category such that the tensor product preserves this in each variable, then for all A-∞ algebra in , the tensor product of ∞-bimodules
preserves (∞,1)-colimits separately in each argument.
This is (Lurie, cor. 220.127.116.11).
The original articles are
A generalized statement in which the codomain is not assumed to be a category of modules is discussed in