nLab Eilenberg-Watts theorem




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


For ordinary rings and modules

Eilenberg-Watts Theorem

Given unital rings RR and SS and an RR-SS-bimodule NN, the tensor product functor

() RN:Mod RMod S (-) \otimes_R N \;\colon\; Mod_R \to Mod_S

between the categories of modules is additive and cocontinuous. Conversely, if F:Mod RMod SF \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:Mod RMod SF \colon Mod_R \to Mod_S, the reconstructed RR-SS-bimodule is given as follows:

  • the underlying right SS-module is F(R)F(R), where RR is regarded as a right module over itself in the canonical way;

  • the left RR-module structure on F(R)F(R) is given for rRr \in R and nNn \in N by

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

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


The theorem holds for nonunital rings as well, but then BB reconstructs as F(R 1)F(R_1) where R 1R_1 is the extension of RR by adjoining the unit element (the tensor product is still over the original RR). If FF is a flat functor then F(R 1)F(R_1) is a flat module over RR.


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 RR and SS two rings, the functor

RMod SFunc coc(Mod R,Mod S) {}_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 VV-enriched categories where V=AbV = 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 VV-enriched profunctors between one-object VV-enriched categories. The category of right modules of a ring RR is the category of VV-enriched presheaves on the corresponding one-object VV-enriched category. Thus, we can ask if the Eilenberg-Watts theorem generalizes to VV-enriched categories. And indeed it does!

Suppose that VV is a Benabou cosmos, i.e. a complete and cocomplete symmetric monoidal closed category. Then there is a symmetric monoidal bicategory VModV Mod where:

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

There is also a symmetric monoidal bicategory VCocontV Cocont where:

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

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

Generalized Eilenberg-Watts’ Theorem

Given a Benabou cosmos VV, the symmetric monoidal bicategories VProfV Prof and VCocontV 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)

This is the main theorem in (Hovey).

More generally, we have the first half of the Eilenberg-Watts theorem in (∞,1)-category theory:


For (𝒞,)(\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,CA,B,C in 𝒞\mathcal{C}, the tensor product of ∞-bimodules

() B(): AMod B× BMod C AMod C (-) \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.


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)


Last revised on June 5, 2023 at 18:03:58. See the history of this page for a list of all contributions to it.