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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.
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...
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.
I should have a scan somewhere...I will find it for you, with few days of delay (remind me if nothing in a week).
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?
Maybe the Yoneda lemma for bicategories? I'm not sure exactly where you're stuck.
I should have explained the problem -- in the strict case you can treat both and as internal categories in presheaves and apply Yoneda 'levelwise', as in the entry. I don't think this works if is only pseudo. Also, in that case the argument is not AFAICT a simple application of bicategorical Yoneda, because is not the hom-functor of a bicategory. In other words, the identity on doesn't live in .
You can still apply 1-categorical Yoneda and get , but I don't see how to turn those back into a pseudonatural -- the inverse construction only gives you presheaves 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.
Ah. But you can treat S as a bicategory with trivial 2-cells, and and as objects of that bicategory, and then bicategorical Yoneda tells you that transformations are the same as objects in . The structure making the first one an indexed functor should be the same as the structure making the second an internal diagram.
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.
I have found it and sent a copy to F.L.
@ Mike:
Thanks for your replies. I think I've got it now, though the only real difference is that is embedded into . 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.
Wait, a profunctor C ⇸ D is a bimodule D ⇸ C in Span? I am so confused.
@ncfavier: as far as I’m aware, the standard convention on the nLab is that a profunctor $C ⇸ D$ is a functor $D^{op} \times C \to \mathrm{Set}$.
But that would correspond to a bimodule D ⇸ C, unless you mean to tell me that the left leg of a span represents “destination” and the right leg “source”…
It also doesn’t reflect the direction of the heteromorphisms, from D to C.
If this choice of notation is well-accepted and recognised, doesn’t it a minima deserve a huge red warning at the top of the relevant pages?
I also don’t understand why this page claims that a bimodule C ⇸ 1 is equivalently a right C-module: the former has an underlying span C₀ → 1, while the latter has an underlying span C₀ → D₀, where D₀ is some set. So a right C-module is a functor $C^{op} \times D_0 \to \mathbf{Set}$, where D₀ is viewed as a discrete category.
Am I misunderstanding something?
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