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.
ADITTYA CHAUDHURI (Jul 09 2020 at 19:47):I have some basic background in 2-Group Gauge theory. To be precise I have some basic familiarity with the following topics in 2 group Gauge Theory:
.Principal 2-bundles over Manifold/ Loop 2 space.(Both Strict and Weak) and their local description.
. Cocycle description of Non abelian Gerbes over a manifold using Crossed Modules.
. Higher Connection(Higher Parallel Transport) using Path 2-Groupoid.
. Lie groupoids.
. Basic notions of Stacks and Gerbes.
.Non abelian Cohomology.
. Cech 2-Grpoupids.
.Delooping Groupoids
. Weak 2-categories.
.Category of Generalised smooth spaces.
.... and some other topics commonly used in 2-Group Gauge Theory.
My plan is to learn about the theory of Principal Infinity Bundles , infinity Gerbes and the corresponding theory of connections.
For that I first want to understand the content of the paper Principal ∞-bundles – General theory by Thomas Nikolaus, Urs Schreiber, Danny Stevenson https://arxiv.org/pdf/1207.0248.pdf
But due to my lack of sufficient background both in Topos theory and category theory understanding the material of this paper will be quite a long road for me! But yes I want to enjoy the journey of transition from 2-Group Gauge Theory to category theoretic Gauge Theory.
It would be really a great pleasure if someone joins me in this exciting journey. I am also open to any alternative route for the transition from 2 group Gauge theory to - Gauge Theory.
A humble request to all Experts in the area:
Please guide us in a right direction!
Thank You
ADITTYA CHAUDHURI (Jul 09 2020 at 19:50):An account of Introductory Infinity Chern-Weil Theory is given in https://ncatlab.org/nlab/show/infinity-Chern-Weil+theory+introduction
Arthur Parzygnat (Jul 09 2020 at 21:18):You can dive right into this zoo, but I highly recommend complementing it with Dan Freed and Michael Hopkin's slightly more readable (but not as expansive) paper "Chern--Weil forms and abstract homotopy theory" https://arxiv.org/abs/1301.5959 (or the published version here https://www.ams.org/journals/bull/2013-50-03/S0273-0979-2013-01415-0/), which requires much less background than the paper you mentioned. I think if you can understand Freed--Hopkins' paper, you have a better chance of reading the paper you mentioned. I do admit though, I worked through that paper with my old advisor as a grad student, so it would have been hard going through that on my own. But I remember it was a really fun read!
Arthur Parzygnat (Jul 09 2020 at 21:18):Btw, we spent a whole semester going through that paper so... if you want to know it cold, prepare to spend a lot of time on it.
John Baez (Jul 09 2020 at 21:22):So the easier paper took a semester to read?
Arthur Parzygnat (Jul 09 2020 at 21:23):John Baez said:
So the easier paper took a semester to read?
:sweat_smile: Well, I also had to learn about sheaves and stuff. So we took digressions to cover some basics here and there.
ADITTYA CHAUDHURI (Jul 10 2020 at 04:27):@Arthur Parzygnat Thank you very much!! Also Thanks a lot for the reference (Freed-Hopkin's Paper)
Daniel Plácido (Jul 21 2020 at 13:26):I'm nowhere close to your knowledge on 2-Group Gauge Theory, but currently I'm studying Schreiber's Differential cohomology on..., which ocasionally hits higher Chern-Weil Theory. It would be interesting to keep in touch sometime.
My unfortunate remark is that in Brazil there are at most a handful of mathematicians or students interested in $\infty$-categories. We're trying to make that number grow.
ADITTYA CHAUDHURI (Jul 21 2020 at 14:02):@Daniel Plácido Thanks! It would be really great !! I really want to understand the content of https://arxiv.org/abs/1310.7930 some day. It would be really nice if you kindly tell me how did you start reading it? I mean .. did you first learn Topos theory and (infinity, 1) category before starting to read that paper? or learning about Topos and Infinity category simultaneously while reading that paper?
ADITTYA CHAUDHURI (Jul 21 2020 at 14:05):@Daniel Plácido Unfortunately the situation(only a handful of mathematicians or students interested in infinity-categories ) is quite similar in India.
Tim Hosgood (Jul 21 2020 at 21:06):i’ve read a few sections of this paper and thought about how to understand them quite a bit, so would be interested in talking about it sometime too 🙂
John Baez (Jul 21 2020 at 21:13):Cool! By the way, @Daniel Plácido - this Zulip requires double dollar signs to make math work, so not $\infty$-categories but $$\infty$$-categories. The price of math has gone up.
ADITTYA CHAUDHURI (Jul 22 2020 at 02:41):@Tim Hosgood That would be great!
John Baez (Jul 24 2020 at 02:08):There may be some relevant talks here:
ADITTYA CHAUDHURI (Jul 24 2020 at 04:16):@John Baez Thank you Sir.
Emilio Minichiello (Feb 20 2021 at 01:02):Hey I'm super late to this but I'm planning on reading Principal Infinity Bundles for my oral exam, is anybody here interested in connecting?
Joe Moeller (Feb 20 2021 at 01:05):Is that this?
https://arxiv.org/abs/1207.0248
Emilio Minichiello (Feb 20 2021 at 01:06):Yup, and there's a sequel paper called Presentations and I believe a third paper was planned but hasn't been published?
David Michael Roberts (Feb 20 2021 at 03:37):I could ask Danny, if you want. I vaguely recall a third paper, too.
Emilio Minichiello (Feb 20 2021 at 03:40):@David Michael Roberts yeah that'd be great, thanks!
David Michael Roberts (Feb 20 2021 at 03:41):Yes, see here: https://ncatlab.org/schreiber/revision/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications/23 it looks like it might have been planned to extract stuff from Urs' dcct and publish it separately.
David Michael Roberts (Feb 20 2021 at 03:44):I suspect the other two just dropped off the project and Urs has since just published stuff on applications on his own and with Hisham Sati, not explicitly linked to this would-be trilogy. But I can check. Danny is very, very busy, though (like: no time for any research in the past six month!), so I don't know how quickly he'd respond, or want to discuss history in the corridor.
David Michael Roberts (Feb 20 2021 at 03:47):OK, so here, on 4th June 2014:
https://ncatlab.org/schreiber/revision/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications/36 the third part is still listed, and the first two papers had both been accepted, and even assigned DOIs.
But by the 14th August 2014, https://ncatlab.org/schreiber/revision/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications/36, the first two papers have appeared online, but the third, planned one, has vanished.
Emilio Minichiello (Feb 20 2021 at 03:47):Ah, I see, very interesting. Well glad to know that the results are contained in his book somewhere. I'll have to check that out. I wonder if there are other resources on similar material? I've seen the Freed Hopkins paper but it's not really at the level of the Principal Infinity Bundles papers.
David Michael Roberts (Feb 20 2021 at 03:50):It depends how example-conscious you want the applications to be. Pick almost anything Urs has done in the last 5 years....
David Michael Roberts (Feb 20 2021 at 03:52):The unfilled 'Applications' section of Urs' nLab page there survived until June 2019, so maybe that's a good cutoff time to look at work before then.
David Michael Roberts (Feb 20 2021 at 03:53):Actually, @Emilio Minichiello, one of Urs' lecture notes pdfs points to this page: https://ncatlab.org/nlab/show/twisted+smooth+cohomology+in+string+theory, where there are lots of application-y type things.
Emilio Minichiello (Feb 20 2021 at 03:55):Ah okay, thank you! My lack of background in physics makes Urs stuff a little bit harder for me, but I guess that's where the main motivation for this kind of stuff is
David Michael Roberts (Feb 20 2021 at 03:58):I don't think that page in particular (which is still very sketchy, but starts slowly), needs much or any physics. If you can read Dan Freed, you can read that stuff. The more recent Hyppthesis H material is advertised as being more connected with physics, but really it's just providing proofs (in mathematics!) of things that were claimed in the string/M-theory literature based on hunches and physical arguments from toy models and baby cases.
Emilio Minichiello (Feb 20 2021 at 04:01):That is really fascinating, I've never heard of Hypothesis H
David Michael Roberts (Mar 03 2021 at 06:10):@Emilio Minichiello I asked Danny, and he said he pulled out of the oo-bundle project due to lack of time to commit to it, and then I guess Urs just absorbed whatever was going to be in the third paper into other ongoing work.
Keyao Peng (May 10 2021 at 20:43):Interesting, I also want to find a point to get in Urs' differential cohesive topos theory. I think that would be a good starting point.
Emilio Minichiello (May 10 2021 at 20:48):David Michael Roberts said:
Emilio Minichiello I asked Danny, and he said he pulled out of the oo-bundle project due to lack of time to commit to it, and then I guess Urs just absorbed whatever was going to be in the third paper into other ongoing work.
Oh weird I didn't get a notification for this. Thanks for letting me know!
Emilio Minichiello (May 10 2021 at 20:50):Keyao Peng said:
Interesting, I also want to find a point to get in Urs' differential cohesive topos theory. I think that would be a good starting point.
Yeah so far the Principal Infinity Bundles papers have helped me start perusing his book, but going back and forth between quasicategory theory and model category theory has been the most difficult part for me
Keyao Peng (May 10 2021 at 21:00):Recently I read a paper about synthetic geometry of pde, pretty interesting https://arxiv.org/abs/1701.06238
Chetan Vuppulury (May 10 2021 at 21:32):Keyao Peng said:
Recently I read a paper about synthetic geometry of pde, pretty interesting https://arxiv.org/abs/1701.06238
Ah yes, that is a paper I've been meaning to read. He had an version of it in his book, I think. But someone told me it's a good idea to read this paper and understand it in -category land first.
Emilio Minichiello (May 10 2021 at 21:35):Keyao Peng said:
Recently I read a paper about synthetic geometry of pde, pretty interesting https://arxiv.org/abs/1701.06238
This looks great! I wish there was more of a community of people doing synthetic differential geometry.
Keyao Peng (May 11 2021 at 15:33):So really to say, would nlab community like to organize a reading group or seminar online to learn "Differential cohomology in a cohesive infinity-topos". I think one benefit of holding online is it can last for a long period so that we can have time to work on some prerequired things.
I recently joined one like that for "Exodromy". It's pretty nice, I learn a lot of higher topos from this. So I hope to have more.
John Baez (May 11 2021 at 15:53):If you want the nLab community to organize something, you should ask on the nForum, since that's where the nLab community is - and especially Urs Schreiber.
Keyao Peng (May 11 2021 at 16:05):Yeah, I mean all the people who interest nlab math. But maybe I can also ask on the nForum.
Morgan Rogers (he/him) (May 11 2021 at 16:11):... do you just mean the Category Theory community, perhaps?
John Baez (May 11 2021 at 16:13):If you want to get Urs Schreiber involved, the nForum is the place to go.
John Baez (May 11 2021 at 16:13):But there are some people here who might be interested too.
Keyao Peng (May 11 2021 at 16:14):Morgan Rogers (he/him) said:
... do you just mean the Category Theory community, perhaps?
of course it contains people here
Keyao Peng (May 11 2021 at 18:40):I also add a discussion on nForum
Tim Hosgood (May 11 2021 at 22:17):i’d be very interested in joining such a reading group 😊
ADITTYA CHAUDHURI (May 15 2021 at 20:55):Keyao Peng said:
So really to say, would nlab community like to organize a reading group or seminar online to learn "Differential cohomology in a cohesive infinity-topos". I think one benefit of holding online is it can last for a long period so that we can have time to work on some prerequired things.
I recently joined one like that for "Exodromy". It's pretty nice, I learn a lot of higher topos from this. So I hope to have more.
Wow!! It looks awesome!
ADITTYA CHAUDHURI (May 15 2021 at 21:01):Last year I attended a workshop on Infinity categories organised by MPIM. Professor C. Barwick gave talks on stratified homotopy theory and Exodromy that time.