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.
Eilenberg-Watts Theorem
Given unital rings $R$ and $S$ and an $R$-$S$-bimodule $N$, the tensor product functor
between the categories of modules is additive and cocontinuous. Conversely, if $F \colon Mod_R \to Mod_S$ is additive and cocontinuous, then it is naturally isomorphic to tensoring with a bimodule.
This theorem was more or less simultaneously proved in (Watts) and (Eilenberg).
Given an additive 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 $R$.
There are various equivalent ways to state the hypotheses of the theorem:
The theorem is stated in the last form, 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 additive functors between their categories of modules is an equivalence of categories.
Rings can be seen as one-object $V$-enriched categories where $V = Ab$ is the category of abelian groups made symmetric monoidal with the usual tensor product of abelian groups. Similarly, bimodules between rings are the same as $V$-enriched profunctors between one-object $V$-enriched categories. The category of right modules of a ring $R$ is the category of $V$-enriched presheaves on the corresponding one-object $V$-enriched category. Thus, we can ask if the Eilenberg-Watts theorem generalizes to $V$-enriched categories. And indeed it does!
Suppose that $V$ is a Benabou cosmos, i.e. a complete and cocomplete symmetric monoidal closed category. Then there is a symmetric monoidal bicategory $V Mod$ where:
There is also a symmetric monoidal bicategory $V Cocont$ where:
Then the following is surely true, though a reference would be helpful:
Given a Benabou cosmos $V$, the symmetric monoidal bicategories $V Prof$ and $V Cocont$ are equivalent.
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 June 5, 2023 at 18:03:58. See the history of this page for a list of all contributions to it.