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nLab > Latest Changes: internal profunctor

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  1.  
    • CommentRowNumber1.
    • CommentAuthorFinnLawler
    • CommentTimeJan 23rd 2010

    Created internal profunctor, which also describes an idea I saw somewhere about internal diagrams in fibrations over the base category. I added what I think are two examples, and asked a generic 'Help!' question. It might be better off on a page of its own, though.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 24th 2010

    As I am currently at IP which id banned from updating the nlab, I will leave here a note, that the stuff which you saw "somewhere" is in the reference of Maclane and Pare from 1980s, JPAA. I planned to write something more about it, than it is written in their paper, but this has to wait for some time...

    • CommentRowNumber3.
    • CommentAuthorFinnLawler
    • CommentTimeJan 27th 2010

    Thanks for that, Zoran. I didn't mean to be coy about the reference -- it's just that the ones I've seen (not Mac Lane & Paré) weren't at all comprehensive. One was a page of the Elephant that Google Books let me see, the other was a brief mention in a 1979 Cahiers paper by Bunge and Paré, which refers to Paré & Schumacher in LNM 661. I should be able to get at that one, but not the JPAA paper, because the TCD library's subscription doesn't go back far enough.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 27th 2010

    I should have a scan somewhere...I will find it for you, with few days of delay (remind me if nothing in a week).

    • CommentRowNumber5.
    • CommentAuthorFinnLawler
    • CommentTimeJan 29th 2010

    I expanded on Mike's remark at internal profunctor for the case of diagrams in a strict indexed category, but I can't quite figure it out for the pseudo case. Any suggestions?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJan 29th 2010

    Maybe the Yoneda lemma for bicategories? I'm not sure exactly where you're stuck.

    • CommentRowNumber7.
    • CommentAuthorFinnLawler
    • CommentTimeJan 29th 2010
    • (edited Jan 29th 2010)

    I should have explained the problem -- in the strict case you can treat both S(-,C) and E as internal categories in presheaves and apply Yoneda 'levelwise', as in the entry. I don't think this works if E is only pseudo. Also, in that case the argument is not AFAICT a simple application of bicategorical Yoneda, because S(-,C) is not the hom-functor of a bicategory. In other words, the identity on C doesn't live in S.

    You can still apply 1-categorical Yoneda and get (P,\phi), but I don't see how to turn those back into a pseudonatural \alpha : S(-,C) \to E -- the inverse construction only gives you presheaves S(-,C_0) \to E_0 = \mathrm{ob }\circ E and ditto for arrows.

    It's late and my brain is a bit fried, so I'm probably missing something stupidly obvious. I'll try again after sleep.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 30th 2010

    Ah. But you can treat S as a bicategory with trivial 2-cells, and C_0 and C_1 as objects of that bicategory, and then bicategorical Yoneda tells you that transformations S(-,C_i) \to E are the same as objects in E(C_i). The structure making the first one an indexed functor should be the same as the structure making the second an internal diagram.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2010

    I still did not find the file for Saunders Maclane, Robert Paré, Coherence for bicategories and indexed categories* JPAA 1985 but will look this week at all file systems and backup CDs with my files.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2010

    I have found it and sent a copy to F.L.

    • CommentRowNumber11.
    • CommentAuthorFinnLawler
    • CommentTimeFeb 3rd 2010

    @ Mike:

    Thanks for your replies. I think I've got it now, though the only real difference is that [S^{op},\mathrm{Set}] is embedded into [S^{op},\mathrm{Cat}]. I may just have been overcomplicating things in my head before. However, it's possible that, being but a Bear of Very Little Brain, I've made a silly mistake, so a quick professional glance would be helpful.

    @ Zoran:

    Thanks for that article! I'll give it a look and try to distil anything I find onto the lab.

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