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nLab > Latest Changes: Eilenberg-Watts theorem

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2013

    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?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2013
    • (edited Feb 4th 2013)

    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…

  3. Fixed link to Hovey’s paper.

    diff, v12, current

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeJun 5th 2023

    Clarified various ways of stating the hypotheses. I believe the Eilenberg-Watts theorem requires the hypothesis of additivity. (E.g. Eilenberg states it that way.)

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTimeJun 5th 2023

    Sketched out some remarks indicating how the Eilenberg-Watts theorem is a special case of a result on enriched profunctors. It would be great to have a reference for this result!

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeJun 5th 2023

    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].

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    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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