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In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The proof which you find in Rotman relies on the existence of a generator for $\text{Mod}_R$, the existence free resolutions, and the five lemma, all three things which are particular to the abelian category of modules over a ring.

I'm now wondering whether this theorem will hold for a general category $C$, enriched over $\mathcal{V}.$ The module category of $C$ is the functor category $\hom(C,\mathcal{V})$, which is also the category of presheaves over $C$, which enjoys the nice property of being the free cocompletion of $C$. So a cocontinuous functor out of $\text{Mod}_C$ is completely determined by its values on $C$. Hence the $\mathcal{V}$-category of cocontinuous functors $\text{Mod}_C\to\text{Mod}_D$ is isomorphic to the functors $C\to\text{Mod}_D,$ and by the hom-tensor adjunction, this is functors $C^\text{op}\otimes D\to\mathcal{V}$, also known as bimodules (or spans, correspondences, profunctors, distributors, it has a lot of names).

Perhaps even more easily seen, by the coYoneda lemma, every presheaf is a colimit of representable presheaves, which just explicitly says that so cocontinuous functor $F\cong -\otimes_C F(Y)$, where $Y$ is the Yoneda embedding.

So my question is, is this a correct proof of the Eilenberg-Watts theorem? Are the assumptions of a generator, the existence of free resolutions, and the validity of the five lemma really not necessary for this result?

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    $\begingroup$ The existence of a generator (in a sufficiently strong sense) is essential – as you say, $[\mathcal{C}, \mathcal{V}]$ is freely generated by the representables. The rest are just details. $\endgroup$
    – Zhen Lin
    Mar 8, 2014 at 2:17
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    $\begingroup$ @ZhenLin: Yes, the representables are generators of the presheaf category. But I still don't see how to proceed by analogy with the ring module case without some addition lemmas. In an arbitrary category with generator $P$, can we write every object as a colimit of a constant functor at $P$? I guess that would be sufficient to give me what I want, but it seems stronger than the definition of generator I know (either $\hom(P,-)$ is injective on arrows or every object receives an epimorphism from $\coprod P$) $\endgroup$
    – ziggurism
    Mar 8, 2014 at 16:52
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    $\begingroup$ If you have a dense generator (in the enriched sense), then every object is a weighted colimit of the generator; and vice versa if you state the colimit condition carefully enough. But we do need the enrichment here. For instance, $\mathbf{Ab}$ is densely generated by $\mathbb{Z}$ as an additive category but not as an ordinary category. $\endgroup$
    – Zhen Lin
    Mar 8, 2014 at 17:11
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    $\begingroup$ Joe, as to your specific question about the enriched category setting: the argument looks good to me! Which leaves the question: how are the enriched-category argument and the homological-algebra argument related? I think there's a lot lurking under the hood in the statement that the $\mathcal{V}$-presheaf category is a free $\mathcal{V}$-cocompletion. Also, maybe having free resolutions is analogous to the presentation of presheaves as colimits of representables (~free objects), which is related to their density. But I have no idea where something analogous to the five lemma would show up... $\endgroup$
    – Tim Campion ♦
    Mar 9, 2014 at 18:52
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    $\begingroup$ Thanks for the tip about dense generators, Zhen. That cleared everything up for me. So Eilenberg-Watts lemma is just another statement of the Yoneda lemma, basically? That's neat. $\endgroup$
    – ziggurism
    Mar 10, 2014 at 0:37

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My advisor Mark Hovey recently updated a paper from 2009 in which he proved the Eilenberg-Watts Theorem in a very general case, namely for model categories. The arxiv version is here. Note that there's a big difference between version 1 and version 2, and in particular version 2 is much shorter. I recommend version 1 for those of a more algebraic dint.

At the Oregon WCATSS13 conference, we also discussed this result for $\infty$-categories. If you want more information, I recommend emailing the organizers, as they have access to the abstracts and notes from most of the talks. In particular, we explicitly related Hovey's results to the results in Schwede and Shipley's paper Stable Model Categories are Categories of Modules. So this seems very related to your second paragraph, especially in situations arising from homotopy theory.

Hope that helps!

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  • $\begingroup$ To be clear on what the generalization is: here, rings are generalized to monoids in a symmetric monoidal model category, modules and bimodules are as you'd expect, and cocontinuous functors become Quillen functors (I suppose the model category structure lifts from the base category to the module category in some standard way). To get an Eilenberg-Watts theorem, further technical conditions are required. $\endgroup$
    – Tim Campion ♦
    Mar 9, 2014 at 19:08
  • $\begingroup$ Yes, that's exactly right. The paper which puts a model structure on the module category is Algebras and Modules in Monoidal Model Categories, by Schwede-Shipley. Also, Mark Hovey's preprint Monoidal Model Categories if you want to see what happens when the Schwede-Shipley hypotheses are weakened. $\endgroup$
    – David White
    Mar 9, 2014 at 22:54

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