nLab Eilenberg-Watts theorem

Idea

The Eilenberg-Watts theorem identifies colimit-preserving functors between categories of modules with the operations of forming tensor products with bimodules.

Statement

For ordinary rings and modules

Eilenberg-Watts Theorem

Given unital rings $R$ and $S$ and an $R$-$S$-bimodule $N$, the tensor product functor

$$(-) \otimes_R N \;\colon\; Mod_R \to Mod_S$$

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).

Remark

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

  $$r \cdot n \coloneqq F(r\cdot(-))n \,,$$

  where $r \cdot (-) \colon R \to R$ denotes the right $R$-module homomorphism given by left multiplication by $r$.

Remark

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 $R$.

Remark

There are various equivalent ways to state the hypotheses of the theorem:

• $F$ is additive and cocontinuous (i.e. preserves small colimits).
• $F$ is additive, right exact and preserves small coproducts.
• $F$ is additive and left adjoint.

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:

Theorem

For $R$ and $S$ two rings, the functor

$${}_R Mod_{S} \stackrel{\simeq}{\to} Func_{coc}(Mod_R, Mod_S)$$

from the category of bimodules to that of colimit-preserving additive functors between their categories of modules is an equivalence of categories.

For enriched 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:

• objects are small $V$-enriched categories,
• morphisms are $V$-enriched profunctors,
• 2-morphisms are $V$-enriched natural transformations between profunctors.

There is also a symmetric monoidal bicategory $V Cocont$ where:

• objects are the $V$-enriched presheaf categories $[C^{op},V]$ where $C$ ranges over all small $V$-enriched categories,
• morphisms are cocontinuous $V$-functors, i.e. $V$-functors preserving all $V$-enriched colimits,
• 2-morphisms are $V$-enriched natural transformations between cocontinuous $V$-functors.

Then the following is surely true, though a reference would be helpful:

Generalized Eilenberg-Watts’ Theorem

Given a Benabou cosmos $V$, the symmetric monoidal bicategories $V Prof$ and $V Cocont$ are equivalent.

For other internal monoids and internal modules

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).

For $\infty$-algebras and $\infty$-modules

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 topological spaces

• model structure on simplicial sets

• model structure on chain complexes

• 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:

Proposition

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

$$(-) \otimes_B (-) \;\colon\; {}_A Mod_{B} \times {}_B Mod_{C} \to {}_{A} Mod_{C}$$

preserves (∞,1)-colimits separately in each argument.

This is (Lurie, cor. 4.3.5.15).

Related concepts

• Morita equivalence

• additive functor, Tor

References

The original articles are

• Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8, MR0806.0832, doi

• Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24, 1960, 231–234 (1961), MR0125148

A generalized statement in which the codomain is not assumed to be a category of modules is discussed in

• A. Nyman, S. Paul Smith, A generalization of Watts’s Theorem: Right exact functors on module categories, Communications in Algebra 44:7 (2016) 3160-3170 arxiv/0806.0832 doi

Generalization to homotopy theory/higher algebra is discussed in

• Mark Hovey, The Eilenberg-Watts theorem in homotopical algebra (pdf)

and

• Jacob Lurie, Higher Algebra