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José Siqueira (May 17 2021 at 11:59):I thought it'd be a good idea to tell you what I've been up to as well!
I submitted a couple of papers last year, in collaboration with @Graham Manuell and @Peter Faul :
1) In https://arxiv.org/abs/2010.07926 we prove an extension of a classical theorem on collations of families of functors into a bifunctor, which turns out to also generalize the process of composing monads via a distributive law. This provides a notion of "distributive law" for lax functors.
2) In https://arxiv.org/abs/2012.04963, we show that Artin glueings of toposes correspond to a 2-categorical notion of adjoint split extensions in the 2-category of toposes, cartesian functors, and natural transformations. This is another instance of a statement for frames/locales carrying over to the topos level, although the process wasn't easy.
I've also just submitted my PhD thesis, which doesn't really have to do with those papers; it is a new attempt to categorify nonstandard analysis, going through the route of Internal Set Theory and using Tripos Theory as the main tool. The idea is to show that if you start with a model of ZFC (with the language including a st(x) symbol), then the satisfaction of the additional axioms of IST that make nonstandard proofs work (Idealization, Standardization, and Transfer) correspond to certain relationships holding between 3 triposes you can build. This then
suggests a notion of "tripos model of IST" which abstracts away what is necessary to make the machinery work, so that you can replicate the nonstandard proof methods in any such model. We can then describe what Standardization and Idealization (and not just Transfer) mean in contexts such as the one studied by A. Kock and C. J. Mikkelsen in their old paper on nonstandard extensions, for example.
Now that it is submitted, I'll probably think about how to structure some of the material (plus some extra stuff) into paper form and look for postdocs and collaborations. Please do get in touch if you think you can help with that :)
Nathanael Arkor (May 17 2021 at 15:43):2-dimensional bifunctor theorems and distributive laws is a really nice paper. It seems very much in the vein of Street's Two constructions on Lax functors; I think it's surprising, but very satisfying, how many constructions on monads can be generalised to constructions on lax functors.
Mike Shulman (May 17 2021 at 16:30):I'm curious what other generalizations of constructions on monads to lax functors you have in mind?
Nathanael Arkor (May 17 2021 at 16:33):E.g. every adjunction gives rise to a monad; the Kleisli and Eilenberg–Moore constructions; the monadicity theorem; the structure–semantics adjunction.
Mike Shulman (May 17 2021 at 16:34):The Kleisli and EM constructions are just certain 2-categorical limits and colimits, so it's not surprising that they can be generalized. I'm not sure what generalizations you are referring to for the others, can you give references?
Mike Shulman (May 17 2021 at 16:36):I ask because I recently noticed a way to interpret lax functors as literally being monads in an appropriate place, so I'm wondering how many of these generalizations can be viewed conversely as specializations of formal monad notions to lax functors from that viewpoint.
Nathanael Arkor (May 17 2021 at 16:38):The Kleisli and EM constructions are just certain 2-categorical limits and colimits, so it's not surprising that they can be generalized.
If you have the right way of looking at things, nothing is surprising – but I think phrasing them as lax colimits and limits in the first place is a surprising fact when one first encounters it.
Nathanael Arkor (May 17 2021 at 16:39):I'm not sure what generalizations you are referring to for the others, can you give references?
All the other generalisations I mentioned are in Two constructions on Lax functors.
Nathanael Arkor (May 17 2021 at 16:39):Mike Shulman said:
I ask because I recently noticed a way to interpret lax functors as literally being monads in an appropriate place, so I'm wondering how many of these generalizations can be viewed conversely as specializations of formal monad notions to lax functors from that viewpoint.
That sounds very interesting! Can you say any more?
Graham Manuell (May 17 2021 at 19:52):Mike Shulman said:
I ask because I recently noticed a way to interpret lax functors as literally being monads in an appropriate place, so I'm wondering how many of these generalizations can be viewed conversely as specializations of formal monad notions to lax functors from that viewpoint.
I'd also be interested to hear about this, because I thought about it a bit when we were writing the paper, but I couldn't see how to make it work. Later at a talk I gave, @Tom Leinster suggested trying to generalise the result that monoidal functors from C to D are monoids in the category of all functors from C to D with Day convolution as the tensor product. This suggests a rather weird generalisation to some kind of 2-category of maps between 2-categories that don't need to preserve the 1-dimensional structure, but I haven't had time to think exactly how this would work.
Mike Shulman (May 17 2021 at 23:26):I'll send a note by email.
Mike Shulman (May 17 2021 at 23:30):Or I would, if I knew your email addresses. I have Nathanael's.
José Siqueira (May 18 2021 at 09:08):I'd love to hear about that too (jvp27@cam.ac.uk). I can forward it to Graham.
Chetan Vuppulury (May 18 2021 at 10:03):@Mike Shulman Could you also send it to me? chetanv2801@gmail.com
Mike Shulman (May 18 2021 at 14:35):Will do.
Nick Hu (May 18 2021 at 14:46):I'd also be interested in seeing this if you wouldn't mind Mike (nick.hu@cs.ox.ac.uk)
John Baez (May 18 2021 at 16:15):It may be easier to create a list of people who don't want to see it. I don't want to see it.
José Siqueira (Dec 08 2021 at 20:48):The final version of my thesis ("Tripos models of Internal Set Theory") has been approved. For now I plan to start some joint work with @Hugo Luiz Mariano and @Valeria de Paiva, and should stay in Cambridge for another term to lecture a course. No idea of what else the future will bring, but I'm open to suggestions!
Valeria de Paiva (Dec 08 2021 at 20:51):Congrats!!
Sam Staton (Dec 08 2021 at 21:01):@José Siqueira, congrats! Is there a PDF version of your thesis online? I couldn't find it.
José Siqueira (Dec 09 2021 at 11:15):Thanks! It seems that the deposited copy is not live yet, but it should be available to access from the repository soon.
Alexander Campbell (Dec 09 2021 at 12:50):@José Siqueira Congratulations, José!
Alexander Campbell (Dec 09 2021 at 12:56):@José Siqueira Does that make you PTJ’s final student?
José Siqueira (Dec 09 2021 at 13:25):Thanks! I suppose it does; the department doesn't seem keen on letting him have new students after his retirement.
Zhen Lin Low (Dec 09 2021 at 13:31):Congratulations! I didn't know Peter took a new student.
José Siqueira (Dec 09 2021 at 13:41):He actually took @Peter Faul as a student as well, who was my office mate (and collaborator on a couple of papers, along with @Graham Manuell ). Peter defended his thesis in April, I believe.
John Baez (Dec 09 2021 at 23:53):José Siqueira said:
Thanks! It seems that the deposited copy is not live yet, but it should be available to access from the repository soon.
Congratulations! It would be wise to put it on the arXiv, since that way more people will find it. I require all my students to put their theses on the arXiv.
Graham Manuell (Dec 10 2021 at 16:59):John Baez said:
It would be wise to put it on the arXiv, since that way more people will find it. I require all my students to put their theses on the arXiv.
This is a little off topic, but I was wondering if you have any advice about how to go about doing this. It seems to me that it would require an inordinate amount of effort to get my thesis into a form that would compile properly on the arXiv. I'd happily upload the resulting pdf, but this is against their policy (for reasons that I don't understand) and I'm worried they will ban me or something if I try to circumvent the restrictions.
Nathanael Arkor (Dec 10 2021 at 17:04):You can contact arXiv directly, and they may be willing to make an exception for the "no PDF" rule. I know others who have done this.
José Siqueira (Dec 10 2021 at 22:21):I have that exact problem with arXiv too. I'll try getting in touch with them.
Zhen Lin Low (Dec 10 2021 at 22:38):That's precisely the reason why I haven't uploaded my thesis to arXiv, but I figured if it's on dSpace then that's good enough...
Fabrizio Genovese (Dec 11 2021 at 01:45):Graham Manuell said:
John Baez said:
It would be wise to put it on the arXiv, since that way more people will find it. I require all my students to put their theses on the arXiv.
This is a little off topic, but I was wondering if you have any advice about how to go about doing this. It seems to me that it would require an inordinate amount of effort to get my thesis into a form that would compile properly on the arXiv. I'd happily upload the resulting pdf, but this is against their policy (for reasons that I don't understand) and I'm worried they will ban me or something if I try to circumvent the restrictions.
If your document is written in but you worry about packages incompatibilities, you can use this: https://github.com/djsutherland/arxiv-collector
Zhen Lin Low (Dec 11 2021 at 03:43):In my case it's written in LaTeX but with unicode-math (so requiring lualatex or xelatex) and (gasp) commercial fonts, so many hurdles to get it to compile...
José Siqueira (Dec 11 2021 at 11:34):At the very least, it seems to be live on Apollo now: https://www.repository.cam.ac.uk/handle/1810/331351
Zhen Lin Low (Dec 11 2021 at 13:16):That looks interesting! The proposal to replace the standard-element predicate with modified quantifiers seems clever. Is there a quick summary / presentation / talk I can look at?
José Siqueira (Dec 11 2021 at 13:29):I'm actually giving a quick talk on this at the Brazil-Colombia Logic Meeting next Thursday. Would you like me to email you the slides?
Zhen Lin Low (Dec 11 2021 at 13:31):Sure, thanks! You can email me at my @cantab.net address (discoverable on arXiv, by autoresponder at my old @cam.ac.uk or @dpmms address, etc.)
José Siqueira (Dec 11 2021 at 13:39):Done!
Zhen Lin Low (Dec 11 2021 at 13:39):Got it, thanks!
Graham Manuell (Dec 11 2021 at 15:24):Fabrizio Genovese said:
If your document is written in but you worry about packages incompatibilities, you can use this: https://github.com/djsutherland/arxiv-collector
Thanks. That looks like it could be generally useful. I'd have to play around a bit to see if this is sufficient.
John Baez (Dec 11 2021 at 17:49):Graham Manuell said:
John Baez said:
It would be wise to put it on the arXiv, since that way more people will find it. I require all my students to put their theses on the arXiv.
This is a little off topic, but I was wondering if you have any advice about how to go about doing this. It seems to me that it would require an inordinate amount of effort to get my thesis into a form that would compile properly on the arXiv.
If you're going into academia in mathematics, you need to start preparing documents in LaTeX in ways that compile on the arXiv: this is just a fact of life now. So if it helps, you can think of this effort as work that you probably should have already done, which you are doing now.
John Baez (Dec 11 2021 at 17:54):Most of my grad students write papers in LaTeX, publish them (or at least submit them to journals) - and then when it comes time to write their thesis, they take these papers and turn them into a thesis. By that time they're already used to the arXiv, so it doesn't take a vast amount of work to make their thesis arXiv-ready.
By the way, you can upload LaTeX packages in your arXiv submission.
Eduardo Ochs (Dec 12 2021 at 03:53):@Zhen Lin Low, I tried to use unicode-math and newunicodechar, but they were buggy for me and I found their innards too hard to understand... now I use this trick here, that is non-standard but only uses \ifluatex, \DeclareUnicodeChar and \catcode:
http://angg.twu.net/LATEX/2021composes-hash.tex.html
http://angg.twu.net/LATEX/2021composes-hash.tex.html#test
Zhen Lin Low (Dec 12 2021 at 04:22):@Eduardo Ochs That's not what I meant by I use unicode-math. I don't input Unicode mathematical symbols directly, but I do use Unicode mathematical fonts (specifically, STIX-Math) in the output, instead of traditional TeX fonts. I don't know what the situation is now, but 6 years ago when I was writing my thesis STIX-Math had excellent coverage of symbols in a single font, which helped visual consistency. Times Roman is not my favourite typeface but STIX-Math was compelling enough that I typeset the running text in Times Roman for consistency.
Eduardo Ochs (Dec 12 2021 at 04:28):Oops! Ok, then I can't help you... =(
Sam Staton (Dec 13 2021 at 12:18):Thanks for the link, José!
Recently I had a paper that wouldn't compile on arxiv. I went through the "contact us" link to get to technical support, and they immediately granted an exception to allow us to submit a pdf on its own. It's perhaps not ideal but I think it's much better than sweating over it for days or putting it off. (I think my recent issue was that we had used xetex or luatex for some reason... I understand John's point but I also understand why some people want to experiment with different/better typesetting tools.)