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I had started to expand Eilenberg-Watts theorem a little. Stated it in more modern form as an equivalence of categories. Also started adding pointers to Hoyes homotopy-theoretical versions, but then I ran out of steam for the moment. I should come back to this later.
Is there more than Hovey’s article on the higher-algebra/homotopy-theoretic versions?
I see that the first half of Eilenberg-Watts in $\infty$-category theory is in Higher Algebra cor. 4.3.5.15 (now recorded in the entry here).
I wouldn’t be surprised if the second half/converse statement is also hidden there somewhere, but so far I don’t see it…
Is Profunctors, open maps and bisimulation (Cattani and Winskel, Mathematical Structures in Computer Science, 2005) of use.? This paper is mentioned on the [nLab page], see also [here].
Re. #6: that reference is also the one that came to mind for me, but they don’t prove the enriched version. I don’t know a reference for the enriched version, but it follows straightforwardly from the theory of 2-monads, since V-Prof is* the Kleisli bicategory of the free small cocompletion 2-monad T on V-CAT, whilst V-Cocont is* the restriction of the image of the embedding Kl(T) → EM(T), which is essentially fully faithful. (*With the small caveat that we must restrict to small V-categories.)
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