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Keith Elliott Peterson (Jan 09 2025 at 06:43):The nlab page (https://ncatlab.org/nlab/show/sylleptic+monoidal+2-category) is rather sparse.
What does it mean for a transformation to be "in-between" a symmetry and a braiding? What kind of generalized string (surface maybe?) diagrams do they form? What dimension does it naturally occur in (if we consider braidings to occur in 3D space, and symmetries to occur in 4D space)? To my brain used to 3D Euclidean space, the idea throws me a loop. Can a syllepsis be visualized?
Generalizing up, is there a kind of monoidal 4-category that has transforms that are in between a syllepsis and a braiding, or are they in between being a syllepsis and a symmetry? Is this an open question?
John Baez (Jan 09 2025 at 08:15):Larry Breen, James Dolan and I thought about up these ideas as part of the "periodic table of n-categories". You can definitely visualize what's going on if you work at it.
The reason braided monoidal categories are relevant to 3d space is that in 3d space one piece of string can cross another in two different ways: over or under. These are the braiding
and the inverse braiding
In 4d space this distinction goes away because there's enough room to continuously deform one of the crossings to the other. Thus, we get symmetric monoidal categories, where
However, a 2-categorical viewpoint goes deeper. In 4d space there are two fundamentally different ways to continuously deform one of the crossings to the other: we can push one of the strings a bit 'up' into the 4th dimension, or 'down'. Either deformation gives an 2-isomorphism
If we pick one of these, we can call it the syllepsis.
John Baez (Jan 09 2025 at 08:19):In 4d space the syllepsis and the 'inverse syllepsis' are different. When we go two 4d space there is a 3-isomorphism between them... but in fact there are two. And so on: the pattern keeps repeating itself. This is a way to think about why the periodic table has the zig-zag shape it does:
John Baez (Jan 09 2025 at 08:20):This chart shows just 3 columns, but there are infinitely many columns and the pattern keeps going on.
Kevin Carlson (Jan 09 2025 at 18:15):In the last paragraph did you mean a 3-isomorphism when we go to 5d space?
Keith Elliott Peterson (Jan 09 2025 at 20:29):John Baez said:
Larry Breen, James Dolan and I thought about up these ideas as part of the "periodic table of n-categories". You can definitely visualize what's going on if you work at it.
[...]
If we pick one of these, we can call it the syllepsis.
Why was the name syllepsis chosen?
John Baez (Jan 09 2025 at 21:32):Kevin Carlson said:
In the last paragraph did you mean a 3-isomorphism when we go to 5d space?
Yes, I seem to have been falling asleep: "When we go two 4d space" should have been "When we go to 5d space".
John Baez (Jan 09 2025 at 21:42):Keith Elliott Peterson said:
Why was the name syllepsis chosen?
In our original paper introducing the periodic table of n-categories, James and I called them "weakly involutory 2-categories". But "involution" means too many things, so later Ross Street, who has a way with words, called them "sylleptic monoidal 2-categories".
Syllepsis is a real word that has something to do with the number 2. It comes from the Greek word syllēpsis, which means "a putting together". But I don't think either of these really justify its new use in math. Probably the fact that it sounds like "symmetric" is the key here: in the category column of the periodic table we have
monoidal, braided, symmetric
while in the 2-category column we have
monoidal, braided, sylleptic, symmetric.
I'm glad to see Wikipedia has an article distinguishing syllepsis from zeugma. You wouldn't want to get those mixed up! :crazy:
David Michael Roberts (Jan 10 2025 at 04:02):I wouldn't be surprised if Ross consulted with Max Kelly on the Greek...
Kevin Carlson (Jan 10 2025 at 18:21):Is there a name for the new kind of monoidal tricategory that appears between sylleptic and symmetric yet? Perhaps it should be zeugmatic, to be rather impishly indifferent to the actual meaning of the words.
Mike Shulman (Jan 10 2025 at 18:25):At that point I'd probably just start saying "".
John Baez (Jan 10 2025 at 18:36):I would say "quadruply monoidal", but even this doesn't scale well. However, I do like the idea of using "zeugma" in mathematics: it's as scary as "syzygy" and "plethysm". :smiling_imp:
Keith Elliott Peterson (Jan 11 2025 at 21:19):I think I will just stick with 4-tuply monoidal 3-category (assuming a "zeugma monoidal 3-category" is just 4-tuply monoidal 3-category, and a sylleptic monoidal 3-category is a 3-tuply monoidal 3-category).
You can't make me learn Greek!
John Baez (Jan 11 2025 at 21:24):But you should at least say "zeugmatic" if you're going to use "zeugma" as an adjective.
David Michael Roberts (Jan 12 2025 at 12:01):"Zeugmoidal" :upside_down:
Kevin Carlson (Jan 13 2025 at 00:04):Gorgeous!
Keith Elliott Peterson (Jan 23 2025 at 02:49):From Funny linguistic fact, metathetic would be a good term, since it is about a kind of non-symmetric swapping.