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• 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.