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Paolo Perrone (Mar 31 2020 at 14:55):Hello all. This is the official topic of discussion for Todd Trimble's talk, "Geometry of regular relational calculus".
Date and time: Thursday, April 2, 12 noon EDT (Boston time).
Streaming link here: https://youtu.be/gcYMrzQZilM
Nathanael Arkor (Mar 31 2020 at 20:51):*April 2
Paolo Perrone (Mar 31 2020 at 21:02):Thanks, corrected.
Alexander Kurz (Apr 01 2020 at 06:14):Could we have one place here on the list with all talks? It is so easy to miss one ... and is there a way to participate a bit later? I have to teach at that time ...
Paolo Perrone (Apr 01 2020 at 12:34):@Alexander Kurz for now, the list is at brendanfong.com/seminar.html.
If you arrive later you should still be able to see the streaming, with up to 2 hours of delay. Worst case, you can watch the YouTube video we upload afterwards.
Paolo Perrone (Apr 02 2020 at 15:53):Hello all, and welcome!
Paolo Perrone (Apr 02 2020 at 16:00):we start in 30 seconds :)
Reid Barton (Apr 02 2020 at 16:01)::ear:
Marc Gotliboym (Apr 02 2020 at 16:01):We can hear
Brendan Fong (Apr 02 2020 at 16:01):I can hear!
Tomáš Jakl (Apr 02 2020 at 16:01):I can hear!
Luc Chabassier (Apr 02 2020 at 16:01):I can hear too
Paul-André Melliès (Apr 02 2020 at 16:01):That works for me!
Max New (Apr 02 2020 at 16:12):Doesn't functions =~ adjoint relation use axiom of choice?
Reid Barton (Apr 02 2020 at 16:13):well, that might depend on your notion of function
Alex Kavvos (Apr 02 2020 at 16:16):Max New said:
Doesn't functions =~ adjoint relation use axiom of choice?
Not if you don't want to recapture the function!
Pawel Sobocinski (Apr 02 2020 at 16:17):All this stuff is a lot more readable with string diagrams, imo
David Spivak (Apr 02 2020 at 16:21):Is there a lag for others?
Matt Cuffaro (he/him) (Apr 02 2020 at 16:21):yes
Alex Kavvos (Apr 02 2020 at 16:21):Yes; it's buffering bigly.
Sam Tenka (Apr 02 2020 at 16:21):I thinm so
Matt Cuffaro (he/him) (Apr 02 2020 at 16:21):are the slides available somewhere?
Rich Hilliard (Apr 02 2020 at 16:21):yes
Sam Tenka (Apr 02 2020 at 16:22):@Brian Pinsky i think this is the channel about the current talk
Brian Pinsky (Apr 02 2020 at 16:24):Okay. Yeah, I'm getting lag sometimes. I was wondering if the pdf was online now so I could have it open and look through it during the talk (sometimes it lags and I miss things)
Reid Barton (Apr 02 2020 at 16:26):There was briefly a message on the screen which I think meant that the speaker's internet connection was acting up.
Paolo Perrone (Apr 02 2020 at 16:26):I think it's my connection. Sorry about that -- I guess the whole of Boston is working online these days :/
Reid Barton (Apr 02 2020 at 16:26):Oh yeah it could be the host's, too
Sam Tenka (Apr 02 2020 at 16:28):Ooh is this use of "module" related to modules as representations?
David Spivak (Apr 02 2020 at 16:29):looks like "co-design", boolean profunctors
Paolo Perrone (Apr 02 2020 at 16:30):Yep! :)
Max New (Apr 02 2020 at 16:31):Are cartesian bicategories equivalent to proarrow equipments that are thin in the 2-cells and have a cartesian product?
Alex Kavvos (Apr 02 2020 at 16:31):David Spivak said:
looks like "co-design", boolean profunctors
that's an interesting term, "co-design." where can I read more about it?
David Spivak (Apr 02 2020 at 16:31):Alex Kavvos said:
David Spivak said:
looks like "co-design", boolean profunctors
that's an interesting term, "co-design." where can I read more about it?
7 sketches chapter 4, or Andrea Censi "A mathematical model of co-design"
Pawel Sobocinski (Apr 02 2020 at 16:32):Cartesian Bicategories II, the stuff of my nightmares ;)
Alex Kavvos (Apr 02 2020 at 16:32):I thought it was everyone's nightmares...
Alex Kavvos (Apr 02 2020 at 16:34):Max New said:
Are cartesian bicategories equivalent to proarrow equipments that are thin in the 2-cells and have a cartesian product?
That sounds plausible - but depends on what you mean by "proarrow equipment." Shulman's framed bicategories also have the vertical 1-cells that are missing here, but possibly represented by Map(B) (?).
Then again, there's Modulated Bicategories... (internal screaming).
Max New (Apr 02 2020 at 16:35):Oh right, for equivalence you'd also need the property that maps coincide with the vertical maps. Or you could drop that property...
Sam Tenka (Apr 02 2020 at 16:36):Yay!!! <3 string diagrams
Max New (Apr 02 2020 at 16:36):especially since that property fails for things like profunctors
Alex Kavvos (Apr 02 2020 at 16:37):Which one? The having a right adjoint?
Max New (Apr 02 2020 at 16:39):that adjoint proarrows are equivalent to vertical morphisms
Sam Tenka (Apr 02 2020 at 16:39):Carter (author of "how surfaces intersect in space") has a nice reidmeister-move type classification of how movies of surfaces (2+1 intrinsic dimensions) decompose into basic moves
Pawel Sobocinski (Apr 02 2020 at 16:40):Peirce had a cute name for these things, the "teridentity"
Alex Kavvos (Apr 02 2020 at 16:41):Pawel Sobocinski said:
Peirce had a cute name for these things, the "teridentity"
In the sense that they "force equality" along all three wires, presumably?
Pawel Sobocinski (Apr 02 2020 at 16:41):Yes, exactly
Joe Moeller (Apr 02 2020 at 16:44):It seems like everything in the basic list of intuitions about continuity are specifically being contradicted: you can erase a piece, you can poke a hole in something, you can rip a string in two...
Jules Hedges (Apr 02 2020 at 16:44):Could've sworn it's spelled "laxator" and not "laxitor"...
David Spivak (Apr 02 2020 at 16:45):all of these rules are captured in a single thing: the 2-category Cospan^co
Reid Barton (Apr 02 2020 at 16:45):Well already lets you turn one thing into two things.
Pawel Sobocinski (Apr 02 2020 at 16:46):Yes, it's the free cartesian bicategory (of relations) on one object... I wonder why the Frobenius laws and the special law are missing from Todd's talk
Pawel Sobocinski (Apr 02 2020 at 16:47):(If your Cospan^co is cospans of finite sets) :)
David Spivak (Apr 02 2020 at 16:47):(it is)
Pawel Sobocinski (Apr 02 2020 at 16:50):This would make Cartesian Bicategories II about 50 times more readable, I love it!
Jules Hedges (Apr 02 2020 at 16:52):It shouldn't be too surprising, in the sense that monoidal bicategories are 3-dimensional things
Jules Hedges (Apr 02 2020 at 16:52):n-categories need n-dimensional string diagrams
Jules Hedges (Apr 02 2020 at 16:53):or rather, monoidal -categories need -dimensional string diagrams
David Spivak (Apr 02 2020 at 16:53):blurry?
Pawel Sobocinski (Apr 02 2020 at 16:54):So I need to think of wires as pipes instead of ribbons?
Reid Barton (Apr 02 2020 at 16:58):I am not experiencing blurriness
Pawel Sobocinski (Apr 02 2020 at 16:58):So every Cartesian Bicategory II is discrete in the sense of Cartesian Bicategories I?
Pawel Sobocinski (Apr 02 2020 at 16:59):ah no... ok, you need to impose invertibility
David Spivak (Apr 02 2020 at 16:59):Reid Barton said:
I am not experiencing blurriness
that's good. it stopped being blurry for me too, but it was blurry for about 5 minutes. not sure why
Brendan Fong (Apr 02 2020 at 16:59):(it didn't get blurry for me either. I think YouTube temporarily reduced your resolution because of a poor connection)
Andre Knispel (Apr 02 2020 at 17:01):Happens to me sometimes as well these days, usually just manually selecting higher quality works for me
Paolo Perrone (Apr 02 2020 at 17:02):If it's blurry it's probably your side. Sorry about that!
(What happens if the connection has issues on our side is rather buffering - buffering is there exactly to prevent "pixeling" and "stuttering" effects.)
Matt Cuffaro (he/him) (Apr 02 2020 at 17:12)::clap:
Brendan Fong (Apr 02 2020 at 17:12)::clap:
Matt Cuffaro (he/him) (Apr 02 2020 at 17:12):volume
Pawel Sobocinski (Apr 02 2020 at 17:13):I have a question. So far this is all about Peirce's alpha -- do you have any intuition about negation in this higher dimensional setting?
Max New (Apr 02 2020 at 17:13):What is the relationship between cartesian bicategories and proarrow equipments? They seem very similar, and RJ Wood has worked on both
Max New (Apr 02 2020 at 17:13):Is it just a matter of taste?
Pawel Sobocinski (Apr 02 2020 at 17:14):ah yeah... sorry, I meant beta without negation :)
Matt Cuffaro (he/him) (Apr 02 2020 at 17:14)::question: Is there any relationship between Peirce's "gamma" modal logic and the recent work in modal HoTT?
Rémy Tuyéras (Apr 02 2020 at 17:15):Looks like concatenations of cobordisms, as in Russian dolls. Could there be an operad structure hidden where the inputs would be all the 1-paths "going down" inside the 3-cobordisms? The composition could be "insertions" of cobordisms inside the 1-paths
Sam Tenka (Apr 02 2020 at 17:16):@Remy Tuyeras but (imagine two positive genus cobordisms) the stacking doesnt need to contract to a 1path, right?
Joe Moeller (Apr 02 2020 at 17:19):Is this "separation" related to "separation logic"? I don't know anything about the latter except that it's in Sarah's title for the UCR seminar :joy:
Pawel Sobocinski (Apr 02 2020 at 17:19):no
Sam Tenka (Apr 02 2020 at 17:19):Ooh i know separation logic
Sam Tenka (Apr 02 2020 at 17:19):And yeah, i think it's different
Joe Moeller (Apr 02 2020 at 17:19):Thanks!
Pawel Sobocinski (Apr 02 2020 at 17:20):Thanks a lot!
Max New (Apr 02 2020 at 17:21):yes
Max New (Apr 02 2020 at 17:21):proarrow equipment = framed bicategory
David Spivak (Apr 02 2020 at 17:21):yes
Rémy Tuyéras (Apr 02 2020 at 17:21):@Samuel C Tenka I am not sure if I understand what you mean by stacking. Are you asking about the onion layer structure that the operad structure would induce?
Paul-André Melliès (Apr 02 2020 at 17:23):Proarrow equipments do not study the properties of the monoidal product on the bicategory
Max New (Apr 02 2020 at 17:24):Thanks Todd, happy to talk on the nforum!
Paolo Perrone (Apr 02 2020 at 17:24):topological.musings@gmail
Max New (Apr 02 2020 at 17:24):have to run to a meeting now. I'll check back later
Sam Tenka (Apr 02 2020 at 17:24):@Remy Tuyeras i just meant that the operad composition interface would be richer than 1 paths, right? Since when we embed an inner in an outer, the induced map on fundamental groups might be nontrivial
Sam Tenka (Apr 02 2020 at 17:25):@Remy Tuyeras that said, I might be seriously misimagining! If so, I apologize!
Rémy Tuyéras (Apr 02 2020 at 17:30):@Samuel C Tenka I think I want to use 1-paths because of the directionality that Todd mentioned during the talk. The idea would be to grow the 1-path into a tube and this tube could take any positive genus cobordism that fits (in a large sense) inside this tube... now, is it the right way to think about it, I am not sure.
Sam Tenka (Apr 02 2020 at 17:31):@Remy Tuyeras i think the directionality is one dimension higher than this though... I'll draw a picture...
Max New (Apr 02 2020 at 17:31):@Paul-André Melliès Right I'm thinking proarrow equipment with a cartesian product and locally thin sounds very similar to cartesian bicategory
Sam Tenka (Apr 02 2020 at 17:32):15858487227195619456201901574903.jpg
Sam Tenka (Apr 02 2020 at 17:33):I think the both embeddings obey the higher d directionality, but the right one doesnt correspond to putting a small cobordism inside a grown path?
Sam Tenka (Apr 02 2020 at 17:33)::clap: #CSPeirce!
Paul-André Melliès (Apr 02 2020 at 17:34):Maybe one last question
Paolo Perrone (Apr 02 2020 at 17:34):okay, go ahead :)
Paul-André Melliès (Apr 02 2020 at 17:34):There is a nice relationship between Frobenius monoids and star-autonomous categories
Paul-André Melliès (Apr 02 2020 at 17:34):Do you see appearing it in your work?
David Yetter (Apr 02 2020 at 17:35):The pictures you are drawing came up in a paper of mine with Louis Crane, "On Algebraic Structure Implicit in Topological Quantum
Field Theories'', and a related solo paper "Portrait of the Handle as a Hopf Algebra". We were limiting things to surfaces with a single boundary component and cobordisms of the same.
Brian Pinsky (Apr 02 2020 at 17:35):@sam I think the first picture you drew might be the same as just a cylinder through one side up to diffeomorphism; I think it's only different if you restrict yourself to isotopies
Paul-André Melliès (Apr 02 2020 at 17:36):A star-autonomous category is a Frobenius object in the bicategory of modules
Paul-André Melliès (Apr 02 2020 at 17:36):categorical modules
Paul-André Melliès (Apr 02 2020 at 17:36):yes!
Pawel Sobocinski (Apr 02 2020 at 17:37):Max New said:
Paul-André Melliès Right I'm thinking proarrow equipment with a cartesian product and locally thin sounds very similar to cartesian bicategory
You should investigate and write this up -- clarification between any of all the different categorical relational calculi is a really useful thing to do.
Paul-André Melliès (Apr 02 2020 at 17:37):Thank you for your talk, Todd, yes, let us all think about it!
David Yetter (Apr 02 2020 at 17:39):I should have @Samuel C Tenka my last post.
Your right picture is the identity cobordism.
Incidentally (and this might matter to @Todd Trimble 's work, when made into an excisive cobordism the commutativity uses a braiding, not a symmetry.
Sam Tenka (Apr 02 2020 at 17:40):Yep, it's identity. I was just trying to see how this would fit into Remy's framework of embedding inside 1paths
Paul-André Melliès (Apr 02 2020 at 17:41):Pawel Sobocinski said:
Max New said:
Paul-André Melliès Right I'm thinking proarrow equipment with a cartesian product and locally thin sounds very similar to cartesian bicategory
You should investigate and write this up -- clarification between any of all the different categorical relational calculi is a really useful thing to do.
I fully agree with Pawel, especially taking into account the reformulation of proarrow equipments as framed bicategories
Joachim Kock (Apr 02 2020 at 17:44):That was really great, Todd!
Rémy Tuyéras (Apr 02 2020 at 17:45):@Samuel C Tenka Yes, you are right regarding the right picture. Even though this is an identity, we can always add another genus next to it not not have an identity. You picture means that we have to break whatever we insert in the 1-cobordism (and not 1-path) before being able to break the hole.
Rémy Tuyéras (Apr 02 2020 at 17:46):@Samuel C Tenka Maybe, embeddings of pi_1 groups of the cobordisms could inform us about the temporality of the system
Paolo Perrone (Apr 02 2020 at 17:46):Thanks a lot Todd!
Sam Tenka (Apr 02 2020 at 17:52):@David Yetter Ooh, I haven't encountered enough math to know what "commutativity uses a braiding, not a symmetry" means (I am uninitiated in the mysteries of category theory!) --- where might be a good place for me to
learn these words?
Joachim Kock (Apr 02 2020 at 17:55):Regarding online-talk technology, I much prefer zoom over youtube streaming. With everybody inside the same platform there is a feeling of being part of an audience. Youtube streaming is a more lonesome experience. I think a zulip chat cannot quite compensate for that.
John Baez (Apr 02 2020 at 17:56):Samuel C Tenka said:
David Yetter Ooh, I haven't encountered enough math to know what "commutativity uses a braiding, not a symmetry" means (I am uninitiated in the mysteries of category theory!) --- where might be a good place for me to learn these words?
They're explained here, for starters:
https://en.wikipedia.org/wiki/Braided_monoidal_category
https://en.wikipedia.org/wiki/Symmetric_monoidal_category
Sam Tenka (Apr 02 2020 at 17:59):@John Baez ah great! yes, I've seen symmetric monoidal cats. Good to "know" what a braided monoidal cat is. Now to digest for a few months... I suppose the name comes from the corresponding string diagrams.
Thanks! :slight_smile:
John Baez (Apr 02 2020 at 18:04):Yes, braided monoidal categories burst into prominence in the 1990s when the Jones polynomial, an invariant of knots, revealed that besides groups, which have symmetric monoidal categories of representations, physics needs "quantum groups", which have braided monoidal categories of representations.
Sam Tenka (Apr 02 2020 at 18:06):@John Baez woah!! that's so cool! I've been developing a string diagram calculus for basic rep theory (on compact groups, just to better visualize orthogonality relations / frobenius reciprocity etc) --- it sounds like this has already been deeply considered! I'll check out what's been done on quantum groups, since they sound so cool.
John Baez (Apr 02 2020 at 18:11):If you like string diagrams for compact group representations, I suggest Cvitanovic's free book Group Theory. For my story of how string diagrams became important in physics, try A prehistory of n-categorical physics.
Todd Trimble (Apr 02 2020 at 20:48):Paul-André Melliès said:
There is a nice relationship between Frobenius monoids and star-autonomous categories
Thanks again, Paul-André, for your attention. I was a little slow on the uptake. I think it might well be just the fact that a Frobenius bicategory (as I called them, faute de mieux) is compact closed in the bicategorical sense, and compact closed is a (somewhat degenerate) case of *-autonomous.
Paolo Perrone (Apr 03 2020 at 01:49):Hey all! The recording of today's talk is here, https://youtu.be/RonyrB0kLew
The HD version should be ready in a few minutes too :)
Simon Burton (Apr 03 2020 at 20:12):I very much enjoyed this talk by Todd. The characterization of functions as relations with a right adjoint is so cool.. The corresponding zig-zag equation looks like a categorification of the highschool definition of a function, it's a graph that doesn't bend back over itself. (This may be a silly thing to say...)
Mike Stay (Apr 03 2020 at 22:03):Sam T (naive student) said:
David Yetter Ooh, I haven't encountered enough math to know what "commutativity uses a braiding, not a symmetry" means (I am uninitiated in the mysteries of category theory!) --- where might be a good place for me to
learn these words?
This summary goes from mere categories up to Monoidal, Braided Monoidal, and Symmetric Monoidal Natural Transformations
http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf
Max New (Apr 14 2020 at 13:43):@Todd Trimble, @Paul-André Melliès, @Pawel Sobocinski and anyone else that is interested, I have continued the discussion of the relationship between cartesian bicategories and proarrow equipments on the nforum here: https://nforum.ncatlab.org/discussion/2865/cartesian-bicategories/#Item_14 . If people are interested, we could coordinate there or here on splitting up some of the work to verify the relationship that we conjecture there
Simon Burton (Jul 12 2021 at 10:53):Sam Tenka said:
... I've been developing a string diagram calculus for basic rep theory (on compact groups, just to better visualize orthogonality relations / frobenius reciprocity etc) --- it sounds like this has already been deeply considered! I'll check out what's been done on quantum groups, since they sound so cool.
I'm interested in string diagrams for representation theory & would be keen to see any notes that you have on this.
Tobias Fritz (Jul 12 2021 at 10:58):(Deleted, as I'm just repeating what John already said!)
Simon Burton (Jul 12 2021 at 11:10):Thanks Tobias, yes I did look at the birdtrack book. For some reason I find it to be a very opaque book to read. I think its written for a different audience, used to traditional physics Lie groups notation, ie. index notation.... There's no mention of Frobenius reciprocity, even the statement of Schur's lemma is a total pain to read, and its not clear what it applies to .
John Baez (Jul 12 2021 at 14:22):It's difficult to read in certain respects, even for me, though I'm used to traditional physics notation so that part is fine. But what makes it great is that it develops the theory of compact simple Lie groups, including the exceptional groups, starting from their categories of representations, which are described using string diagrams. It wouldn't be the place to start learning about Frobenius reciprocity, Schur's lemma, or other general concepts in group representation theory. But if you want to know what's going on with representations of , , and there's really no other book that has this information.
John Baez (Jul 12 2021 at 14:23):I have at times wanted to translate that information into a more category-theoretic presentation.
John Baez (Jul 12 2021 at 14:23):But it would take a fair amount of work!
Simon Burton (Jul 12 2021 at 15:22):Oh yes, you are right, the birdtracks book has a lot of interesting calculations in it, if I pretend that I understand representation theory and QFT..!
John Baez (Jul 12 2021 at 15:58):What frustrates me is that the book seems to contain a classification of compact simple Lie groups based on string diagrams rather than the usual Dynkin diagram stuff. But it's presented so informally that it's quite hard to state it as a theorem with a proof.
John Baez (Jul 12 2021 at 15:59):I'd like to do that sometime.
Simon Burton (Jul 14 2021 at 12:04):That sounds good to me.
The reason I started posting here in this topic was because I'm applying some of this "relational bicategory" machinery to group theory / representation theory. The idea is to be able to vary the ambient category, to see what pops out. And Rel is a kind of half-way-house between Set and Vec.