You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
John Baez (May 28 2020 at 17:19):I got so excited by Simon Willerton's talk yesterday that I wrote a quick explanation of Isbell duality on Twitter. I bet this explanation only works well for people who know that the category of presheaves on C is the free cocompletion of C; I say that but this marvelous fact takes a while to sink in!
I'm finally ready to think about Isbell duality. "Dualities" are important because they show you two different-looking things are secretly two views of the same thing - or at least closely linked. I'll explain Isbell duality; you can see if you're ready for it. :upside_down: (1/n) https://twitter.com/johncarlosbaez/status/1266029338190680064/photo/1
- John Carlos Baez (@johncarlosbaez)
Peiyuan Zhu (Jun 09 2022 at 21:07):Hmm I think this is useful to the information geometry problem that I was thinking about. I guess it was just another place where Legendre transform arises.
Peiyuan Zhu (Jun 11 2022 at 05:09):These slides are super cool! https://math.ucr.edu/home/baez/mathematical/ACTUCR/Willerton_Legendre_Transform.pdf
Simon Burton (Jun 11 2022 at 22:57):We need a bot that revives old threads like this, just so we can be reminded of cool stuff that is hiding here in this zulip! Maybe Peiyuan can be this bot. Every new bit of CT that I learn sheds light on every other part of CT, because all the concepts are so highly connected to each other..
John Baez (Jun 12 2022 at 17:19):I'm a bit frustrated because while the nLab hints that [[Isbell duality]] unifies a lot of other dualities, it doesn't seem to explain in detail how it does this, and I don't see how.
Mike Shulman (Jun 12 2022 at 23:41):Perhaps the idea is similar to the way it works for Chu constructions?
Morgan Rogers (he/him) (Jun 15 2022 at 14:28):@Guillaume GEOFFROY this topic could be interesting to you :grinning_face_with_smiling_eyes:
Guillaume GEOFFROY (Jun 15 2022 at 15:38):Morgan Rogers (he/him) said:
Guillaume GEOFFROY this topic could be interesting to you :grinning_face_with_smiling_eyes:
Indeed
Guillaume GEOFFROY (Jun 15 2022 at 15:45):Currently, I am writing a thesis proposal which tackle linear logic and Isbell duality. However I don't have much references about it.
More precisely, I am looking for references about linear logic as internal logic of monoidal categories and its extension to the internal logic of bicategories. About Isbell duality, I just need some basical references and some applications to logic through Lawvere theories.
Can you help me please?
John Baez (Jun 15 2022 at 15:47):Did you try the references at [[Isbell duality]]?
John Baez (Jun 15 2022 at 15:48):(If you type any phrase in double square brackets here, it becomes a link to the nLab.)
Guillaume GEOFFROY (Jun 15 2022 at 16:15):John Baez said:
Did you try the references at [[Isbell duality]]?
Thank you a lot to try and help me.
Yes. I thought of the reference Isbell duality from Michael Bart, John F. Kennison and R. Raphael. For lunear logic, I have found Profunctor seman'ntics in linear logic, which seems to be to computation-oriented. However, I can't know if a paper shows correctly what have been done in these areas since I don't know much of them. It is why I m looking for help.
John Baez (Jun 15 2022 at 16:26):Yes, I hope someone here can help you. It's also good to use usual technique of scholarship: look at a bunch of modern papers, read their bibliographies, look at papers in those bibliographies that seem interesting, read their bibliographies, and repeat until you feel you've found everything. This is how I dig into a new branch of math if I don't know people who can help me.
John Baez (Jun 15 2022 at 16:27):I don't really understand how various other dualities can be seen as special cases of (enriched) Isbell duality. I would like to learn about that someday.
Morgan Rogers (he/him) (Jun 15 2022 at 16:28):The challenge is always getting to the point where you feel you've reached the most up-to-date papers on a subject..!
John Baez (Jun 15 2022 at 16:35):Yes. Luckily a lot of modern papers are on the arXiv... but sadly, not all. MathSciNet is a powerful tool for searching for math papers, but I guess it takes a while for papers to get into that database.
Mike Shulman (Jun 15 2022 at 17:50):Regarding categorical semantics of linear logic, there are also a bunch of references on that at the nLab: [[linear logic]].
John Baez (Jun 15 2022 at 18:08):For a while I was taking the term [[abelian variety]] and skimming the MathSciNet reviews of papers whose reviews contained this term. It was a really fun way to get acquainted with the literature on abelian varieties. So I'd recommend doing that for other subjects too.
Guillaume GEOFFROY (Jun 15 2022 at 23:21):John Baez said:
Yes. Luckily a lot of modern papers are on the arXiv... but sadly, not all. MathSciNet is a powerful tool for searching for math papers, but I guess it takes a while for papers to get into that database.
I thank you for your answers. I'll dig into papers. I didn't know MathSciNet and it's a great tool.
Guillaume GEOFFROY (Jun 15 2022 at 23:22):Morgan Rogers (he/him) said:
The challenge is always getting to the point where you feel you've reached the most up-to-date papers on a subject..!
I thank you too. You've showed me this helpful stream and your advice is useful.
Guillaume GEOFFROY (Jun 15 2022 at 23:25):Mike Shulman said:
Regarding categorical semantics of linear logic, there are also a bunch of references on that at the nLab: [[linear logic]].
Thank you.
I've seen them (not all yet).
John Baez (Jul 06 2022 at 19:09):Do any of you know an application of Isbell duality that would be exciting to "ordinary mathematicians" - not category theorists or logicians?
John Baez (Jul 06 2022 at 19:11):I've seen the -categorical version of Isbell duality gets applied to algebraic geometry, but I'd prefer something that uses the 1-categorical version.
John Baez (Jul 06 2022 at 19:12):Avery and Leinster's paper notes that the dual of a module of a ring is an example of enriched Isbell duality. That's very illuminating, but I'm afraid my readers may think this is using a sledgehammer to open a walnut.