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nLab > Latest Changes: internal profunctor

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- CommentRowNumber1.
- CommentAuthorFinnLawler
- CommentTimeJan 23rd 2010
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Author: FinnLawler
Format: MarkdownCreated [[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.

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
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Author: zskoda
Format: MarkdownAs 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...

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
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Author: FinnLawler
Format: MarkdownThanks 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.

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

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
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Author: FinnLawler
Format: MarkdownI 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?

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
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Author: Mike Shulman
Format: MarkdownMaybe the [[Yoneda lemma for bicategories]]? I'm not sure exactly where you're stuck.

Maybe the Yoneda lemma for bicategories? I'm not sure exactly where you're stuck.
- CommentRowNumber7.
- CommentAuthorFinnLawler
- CommentTimeJan 29th 2010
- (edited Jan 29th 2010)
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Author: FinnLawler
Format: MarkdownI should have explained the problem -- in the strict case you can treat both <latex>S(-,C)</latex> and <latex>E</latex> as internal categories in presheaves and apply Yoneda 'levelwise', as in the entry. I don't think this works if <latex>E</latex> is only pseudo. Also, in that case the argument is not AFAICT a simple application of bicategorical Yoneda, because <latex>S(-,C)</latex> is not the hom-functor of a bicategory. In other words, the identity on <latex>C</latex> doesn't live in <latex>S</latex>. You can still apply 1-categorical Yoneda and get <latex>(P,\phi)</latex>, but I don't see how to turn those back into a pseudonatural <latex>\alpha : S(-,C) \to E</latex> -- the inverse construction only gives you presheaves <latex>S(-,C_0) \to E_0 = \mathrm{ob }\circ E</latex> 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.

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

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
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Author: zskoda
Format: MarkdownI 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 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
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Author: zskoda
Format: MarkdownI have found it and sent a copy to F.L.

I have found it and sent a copy to F.L.
- CommentRowNumber11.
- CommentAuthorFinnLawler
- CommentTimeFeb 3rd 2010
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Author: FinnLawler
Format: Markdown@ Mike: Thanks for your replies. I think I've got it [[internal profunctor|now]], though the only real difference is that <latex>[S^{op},\mathrm{Set}]</latex> is embedded into <latex>[S^{op},\mathrm{Cat}]</latex>. 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.

@ 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.
- CommentRowNumber12.
- CommentAuthorncfavier
- CommentTimeMay 12th 2023
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Author: ncfavier
Format: MarkdownItexFixed the direction of the bimodule defining an internal presheaf from 1 ⇸ C to C ⇸ 1. <a href="https://ncatlab.org/nlab/revision/diff/internal+profunctor/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/internal+profunctor/8">v8</a>, <a href="https://ncatlab.org/nlab/show/internal+profunctor">current</a>

Fixed the direction of the bimodule defining an internal presheaf from 1 ⇸ C to C ⇸ 1.

diff, v8, current
- CommentRowNumber13.
- CommentAuthorncfavier
- CommentTimeMay 12th 2023
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Author: ncfavier
Format: MarkdownItexWait, a profunctor C ⇸ D is a bimodule D ⇸ C in Span? I am so confused.

Wait, a profunctor C ⇸ D is a bimodule D ⇸ C in Span? I am so confused.
- CommentRowNumber14.
- CommentAuthorvarkor
- CommentTimeMay 12th 2023
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Author: varkor
Format: MarkdownItex@ncfavier: as far as I'm aware, the standard convention on the nLab is that a profunctor $C ⇸ D$ is a functor $D^{op} \times C \to \mathrm{Set}$.

@ncfavier: as far as I’m aware, the standard convention on the nLab is that a profunctor $C ⇸ D$ is a functor $D^{op} \times C \to \mathrm{Set}$ .
- CommentRowNumber15.
- CommentAuthorncfavier
- CommentTimeMay 12th 2023
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Author: ncfavier
Format: MarkdownItexBut 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?

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?
- CommentRowNumber16.
- CommentAuthorncfavier
- CommentTimeMay 12th 2023
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Author: ncfavier
Format: MarkdownItexI 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?

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?
- CommentRowNumber17.
- CommentAuthorBryceClarke
- CommentTimeMay 17th 2024
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Author: BryceClarke
Format: MarkdownItexAdded many additional details and diagrams. Will add some more later. <a href="https://ncatlab.org/nlab/revision/diff/internal+profunctor/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/internal+profunctor/9">v9</a>, <a href="https://ncatlab.org/nlab/show/internal+profunctor">current</a>

Added many additional details and diagrams. Will add some more later.

diff, v9, current

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nLab > Latest Changes: internal profunctor