symmetric monoidal (∞,1)-category of spectra
The Eilenberg-Watts theorem identifies colimit-preserving functors between categories of modules with the operations of forming tensor products with bimodules.
Given unital rings $R$ and $S$ and an $R$-$S$-bimodule $N$, the tensor product functor
between the categories of modules is right exact and preserves small coproducts.
Conversely, if $F \colon Mod_R \to Mod_S$ is right exact and preserves small coproducts, then it is naturally isomorphic to tensoring with a bimodule.
This theorem was more or less simultaneously proved in (Watts) and (Eilenberg).
Given a cocontinuous functor $F \colon Mod_R \to Mod_S$, the reconstructed $R$-$S$-bimodule is given as follows:
the underlying right $S$-module is $F(R)$, where $R$ is regarded as a right module over itself in the canonical way;
the left $R$-module structure on $F(R)$ is given for $r \in R$ and $n \in N$ by
where $r \cdot (-) \colon R \to R$ denotes the right $R$-module homomorphism given by left multiplication by $r$.
The theorem holds for nonunital rings as well, but then $B$ reconstructs as $F(R_1)$ where $R_1$ is the extension of $R$ by adjoining the unit element (the tensor product is still over the original $R$). If $F$ is a flat functor then $F(R_1)$ is a flat module over $R$.
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).
In fact both bimodules and intertwiners as well as functors and natural transformations form a category. In more detail the theorem is:
For $R$ and $S$ two rings, the functor
from the category of bimodules to that of colimit-preserving functors between their categories of modules is an equivalence of categories.
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, $\mathbb{S}$-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 $(\mathcal{C}, \otimes)$ a monoidal (∞,1)-category with geometric realization of simplicial objects in an (∞,1)-category such that the tensor product $\otimes$ preserves this in each variable, then for all A-∞ algebra $A,B,C$ in $\mathcal{C}$, the tensor product of ∞-bimodules
preserves (∞,1)-colimits separately in each argument.
This is (Lurie, cor. 4.3.5.15).
The original articles are
A generalized statement in which the codomain is not assumed to be a category of modules is discussed in
Generalization to homotopy theory/higher algebra is discussed in
and
Last revised on December 16, 2022 at 16:52:59. See the history of this page for a list of all contributions to it.