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Evan Patterson (Mar 31 2024 at 21:17):The triangulators of a lax biadjunction have to satisfy two coherence axioms, which go back at least to Gray's famous book. I noticed that on the nLab and the n-Category Cafe, these coherence axioms have been called the "swallowtail identities."
Who introduced this terminology? Is there a standard print reference for it or did it just arise organically in the online CT community?
Cole Comfort (Mar 31 2024 at 21:37):I have no strong evidence for this, I strongly suspect that this kind of surface was given the name before it was also used for coherence theorems. Here is a differential geometry paper from 1988 which refers to the shape as a swallowtail. I also found a paper Braids of algebraic functions and cohomologies of swallowtails from 1968, but it is written in Russian, so I don't know if they are talking about the same thing.
Evan Patterson (Mar 31 2024 at 21:45):Interesting references. Thanks!
Cole Comfort (Mar 31 2024 at 21:47):I found the same russian author wrote a paper in 1971 in english. He says swallowtails are semi-cubical parabola. The 2-dimensional version is drawn in figure 3.
https://iopscience.iop.org/article/10.1070/RM1971v026n02ABEH003827/pdf
edit: it actually seems like this is what we would call a "dove tail."
Cole Comfort (Mar 31 2024 at 21:53):My guess is that maybe the more symmetric "dovetail" shape came first, and then perhaps the "swallowtail" shape came later? Maybe we need a historian to find out.
David Corfield (Apr 01 2024 at 07:53):@John Baez will surely know. Here I am asking him about it.
He wrote a paper with Langford on 2-tangles in 1998. This borrows from Carter and Saito's work on knotted surfaces which points back to Thom's work on singularities.
John Baez (Apr 02 2024 at 16:38):@Evan Patterson - there's a dictionary, still poorly understood, relating catastrophes as classified by Rene Thom and identities that are important for n-categories with duals. One of the catastrophes on Thom's list is the swallowtail - I'm linking to Wikipedia, which has a discussion of it, and various pictures:
Swallowtail catastrophe animation
It's pretty hard to understand at first, but it's supposed to remind you of a swallow crossing the two parts of its tail. If you want to understand this stuff you have to start with simpler catastrophes like the fold catastrophe and the cusp catastrophe.
I always thought Thom made up the name "swallowtail", but I could be wrong.
John Baez (Apr 02 2024 at 16:42):As you mentioned, it shows up naturally as an identity for pseudoadjunctions (or - I'll take you word for it - even lax biadjunctions), which in turn are important for understanding monoidal 2-categories with duals for objects.
John Baez (Apr 02 2024 at 16:46):Carter, Saito and Rieger discussed the swallowtail in their paper giving a purely combinatorial description of 2-tangles in 4d space (that is, roughly, 2d surfaces embedded in a 4d cube):
Here is their picture of the swallowtail:
John Baez (Apr 02 2024 at 16:48):They wrote this paper at my request because Laurel Langford and I wanted to translate this combinatorial description into category theory and describe the braided monoidal 2-category whose 2-morphisms are 2-tangles in 4d space.
John Baez (Apr 02 2024 at 16:53):So, Laurel and I called the corresponding (already known) identity obeyed by pseudoadjunctions the swallowtail identity in our paper:
I just learned that Salvador Dali did a series of paintings based on Rene Thom's classification of catastrophes, and Dali's very last painting is called The Swallow's Tail.
Daniel Teixeira (Apr 02 2024 at 16:55):John Baez said:
there's a dictionary, still poorly understood, relating catastrophes as classified by Rene Thom and identities that are important for n-categories with duals
I've heard people say this a few times, but never in print, nevermind the catastrophes in higher dimensions (in fact I think I haven't seen any mention further than the swallowtail in 3d and the butterfly in 4d, although I think the cusp is the triangle identity and the fold may be the cap/cup
Do you know if anyone tried to write anything on that correspondence?
John Baez (Apr 02 2024 at 17:02):I don't know a good discussion of this. I think it's implicit here:
but I think one would need to be pretty knowledgeable about singularity theory to extract from this nice result a correspondence between catastrophes (or singularities) and axioms for k-tuply monoidal n-categories with duals.
Someone should do it! I'm particularly curious about when happens when the classification of catastrophes ceases to be discrete - when things get complicated enough they come in parametrized families ('moduli spaces').
John Baez (Apr 02 2024 at 17:04):But really any sort of simple exposition of this stuff would be helpful. I basically put my work into trying to understand 1d tangles in 2d, 3d, and 4d space and 2d tangles in 3d, 4d, 5d and 6d space.
John Baez (Apr 02 2024 at 17:04):Those are the first two columns of the periodic table.
John Baez (Apr 02 2024 at 17:06):Yes, the fold catastrophe corresponds to what category theorists call the cap and cup: the creation or annihilation of a pair consisting of an object and its dual .
Then the cusp corresponds to the two triangle identities: the creation or annihilation of a pair consisting of a cap and a cup.
John Baez (Apr 02 2024 at 17:10):Then comes the swallowtail.
Smyth and Wolf wisely avoid trying to continue this sort of dictionary to arbitrarily high dimensions, because it gets more and more complicated; instead they take a more 'synthetic' approach. But I would love to see someone work out more of this dictionary! It's so cool how singularities of projections of smoothly embedded surfaces in higher-dimensional space are related to duality in n-categories.
Evan Patterson (Apr 02 2024 at 19:13):Thanks for the context, John. Very helpful. And such an interesting connection between ideas (not that I understand catastrophe theory).
John Baez (Apr 03 2024 at 00:06):Basically the idea is that as you gradually morph a string diagram, or a higher-dimensional surface diagram, there are certain moments when a dramatic qualitative change occurs: a "catastrophe".
For example, suppose you pull a string straight:
Time goes from front to back in this picture. In front our string has a zig-zag, at back it does not. The "catastrophe" happens the moment the zig-zag disappears. This catastrophe gives the famous "triangle equation" or "zigzag equation" in the definition of adjunction.
But when we categorify and look at pseudoadunctions, or lax adjunctions, intead of an equation this catastrophe gives a 2-morphism! Or if you prefer, the whole picture is a surface diagram of a 2-morphism.
And this 2-morphism then obeys its own equation, the swallowtail identity, which comes from a catastrophe one dimension higher. We can only draw this in 4 dimensions, which we can do using a "movie" where time passes, and then a "meta-movie" where there's a "before" and "after" movie:
Evan Patterson (Apr 03 2024 at 00:15):Thanks, that's a very evocative way to see it.
The earliest reference I know that states the swallowtail identities in algebraic form (not with that name) is Gray's 1974 book on formal category theory, where he defines a lax adjunction. I wonder how he came up those coherence equations? By just looking for pasteable cells?
John Baez (Apr 03 2024 at 00:21):Thanks, I'm not sure I ever knew the historical origin of the swallowtail identities. John Gray was so far ahead of his time! I sort of doubt he was secretly using string diagrams, but he must have known some identities should hold, so maybe he just started looking for them... or maybe he did something smarter.
John Baez (Apr 03 2024 at 00:22):One cool thing proved by Nick Gurski - maybe you know it, but if you don't I'd better tell you! - is that if you start with a "flawed" pseudoadjunction where the swallowtail identities don't hold, you can always fiddle around and "improve" it to get one where they do hold.
John Baez (Apr 03 2024 at 00:22):This changed my attitude a bit.
Evan Patterson (Apr 03 2024 at 00:26):I read that somewhere, probably on the nLab. But it wasn't clear to me whether that worked for all flavors of "lax adjunctions" or just the biadjunctions/pseudoadjunctions. I had to construct a lax adjunction, so I figured I had better check the coherence laws, which I was able to do.
John Baez (Apr 03 2024 at 00:35):Impressive. I've only seen it stated for biadjunctions, not "lax" ones. Vicary and his team have given a string diagram proof in Globular. I sure don't have the energy to see if the proof generalizes to lax adjunctions!
John Baez (Apr 03 2024 at 00:36):I don't even have the energy to understand what's going on in the Globular proof.
Evan Patterson (Apr 03 2024 at 00:38):That sort of thing seemed scary to me. It felt easier to just bite the bullet and check that my lax biadjucntion was already coherent, which is probably good to know anyway.
Dylan Braithwaite (Apr 06 2024 at 09:51):John Baez said:
I don't even have the energy to understand what's going on in the Globular proof.
Jamie gave a couple of lectures about using homotopy.io at MGS last year, and his last lecture had a graphical sketch of the proof, I think with the intention that you could spend some time formalising it in the proof assistant if you wanted.
But it's nice for getting an intuitive picture of what's going on in the proof.
94a745d2-3817-4f97-9a9e-0a9fca234bb7.png
I still don't know what it would mean for lax adjunctions though!
John Baez (Apr 06 2024 at 17:53):Thanks for posting that slide: it's nice.
John Baez (Apr 06 2024 at 17:54):I don't string diagrams are optimal for reasoning about lax adjunctions.
Dylan Braithwaite (Apr 06 2024 at 18:41):John Baez said:
I don't string diagrams are optimal for reasoning about lax adjunctions.
Reasoning about them intuitively, maybe not, but I think you could still write down the definition of a lax adjunction in Globular/homotopy.io based on the string diagrammatic form there and slide them around, to see what you can prove.
Looking at the diagrams naively I'd guess that the same proof wouldn't work for lax adjunctions since the modified cusp 3-cell uses both the original (non-coherent) cusp 3-cell, as well as its inverse
Daniel Teixeira (Apr 19 2024 at 03:51):Dylan Braithwaite said:
Looking at the diagrams naively I'd guess that the same proof wouldn't work for lax adjunctions since the modified cusp 3-cell uses both the original (non-coherent) cusp 3-cell, as well as its inverse
It's generally surprising how many things break with lax adjunctions in 3-cats. I say surprising because when we go from isomorphisms / equivalences to their "lax version" aka adjunctions we still have nice properties (e.g. uniqueness of right adjoints), but when laxify one dimension further you no longer have that.