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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeMay 3rd 2010
    • (edited May 3rd 2010)

    Eric wanted to know about closed functors, so we started a page. Probably somebody has written about these before, so references would be nice, if anybody knows them. (Google gives some hits that look promising, but I can’t read them now.)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2010

    Maybe it’s time to add a definition to closed category.

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeMay 3rd 2010

    We could - shock, horror! - ask on MO if anyone’s heard about the concept.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2010

    well, the basic idea is simple enough and in each given case probably easily fine-tuned. But I don’t know what precisely Eilenberg-Kelly considered in their article referenced at closed category, since I haven’t seen that.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 4th 2010

    I can pretty much guarantee that anyone on MO is just going to refer to the Eilenberg-Kelly article in the La Jolla conference volume. (They won’t go to the bother of drawing up the commutative diagrams.) Anyway, I’m pretty sure we can do this ourselves. I may get around to it later tonight (US East Coast time).

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2010

    I have the Eilenberg-Kelly article, which defines closed functors and closed natural transformations. I can put them up later tonight (US Central time).

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 4th 2010

    Cool - thanks, Mike!

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2010

    Definitions are in place at closed category, closed functor, and closed natural transformation.

    • CommentRowNumber9.
    • CommentAuthorEric
    • CommentTimeMay 4th 2010

    Thanks! Is that paper old enough that we can make it available?

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 4th 2010
    Yes, thanks. Added remarks to closed category and closed functor.

    Eric, not sure what you mean by "old enough". The conference took place in 1965 IIRC, and the proceedings are dated 1966. Both authors are deceased, but the publisher may still hold the rights -- don't really know. I think by now the paper is mainly of historical significance.
    • CommentRowNumber11.
    • CommentAuthorEric
    • CommentTimeMay 4th 2010

    Todd: I don’t know much about rights, etc, but I imagine that a journal can only “hold the rights” to a publication for a certain period of time until it becomes public domain. After that point, I think we can probably distribute it freely. Not sure though. I’d love to have a look at the paper even if it is only of historical significance at this point. I love reading old papers :)

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2010

    Thanks to Mike and Todd!

    Hm, interesting that every closed category embeds as a full subcategory of a symmetric monoidal one. Is there an analogous statement for monoidal categories? (Let me see, is that obvious?….)

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2010
    • (edited May 4th 2010)

    I gave the LaPlaza theorem its own Properties-subsection.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 4th 2010
    Urs #12: not symmetric monoidal, just monoidal. Also, every monoidal category embeds in a monoidal (bi)closed category, by Day convolution.
    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2010

    not symmetric monoidal, just monoidal.

    Hm, did I say symmetric? Apparently I did.

    also every monoidal category embeds in a monoidal (bi)closed category, by Day convolution.

    Ah, of course. I knew that! Thanks for reminding me.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeMay 4th 2010

    According to wikipedia, the paper is probably not yet in the public domain (and if Congress continues extending the duration of copyright, it may never be).

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2010

    Following the general rule that everything useful being said here should be archived in one way or other on the nLab, I created a section Properties at monoidal category and made explicit the statement that every small monoidal category embeds monoidally into a closed monoidal one.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2010

    I started also a section Properties at Day convolution, but have to interrupt now and take care of something else….

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