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Stream: community: general

Topic: Beyond dualities

Stelios Tsampas (Apr 05 2020 at 13:56):

There is a tragic irony in life: there are things for which we know not of their definition, but we learn about them through perception. We then try to come up with principles through our empirical evidence in a vain effort to explain things. True enough, the less we observe the less accurate our principles are. And this is deeply ironic because no matter what, our observations can never be complete. I.e. the Yoneda lemma, in its full generality, is beyond our grasp. Now, the irony being tragic depends on the existence of someone with complete access to observations, but let's not touch on that.

Before I got into CT I always thought math only care about well-defined objects and their properties. In an observationally unclear world, math could be this sanctum of well-foundedness that can give some semblance of order, of security, I thought. I couldn't have been more wrong. Definition vs observation, composition vs decomposition, structure vs destruction, this is not just a duality, it's an eternal conflict!

Strangely enough, I've come to appreciate and embrace this seemingly fundamental duality. Which is perhaps why I'm so into algebras and coalgebras, monads and comonads, distributive laws of one over the other. Distributive laws prove that observation and structure can co-exist, that behavior may arise compositionally through structure. But, just like in real life, you can never know if there's order hidden in the chaos.

Understanding this duality has been one of the most important lessons I could ever have, one that changed my viewpoint on Life, the Universe and Everything.

Fabrizio Genovese (Apr 05 2020 at 14:38):

Did you read Yoshihiro Maruyama's PhD thesis? It's fundamentally about dualities, also from a philosophical standpoint :slight_smile:

Stelios Tsampas (Apr 05 2020 at 14:45):

Fabrizio Genovese said:

Did you read Yoshihiro Maruyama's PhD thesis? It's fundamentally about dualities, also from a philosophical standpoint :)

No, I haven't... But I've seen that mentioned before, maybe from you! Have you?

Stelios Tsampas (Apr 05 2020 at 14:48):

I've entertained the idea (late at night, under the influence of... things) that duality is actually a jail we're all imprisoned in. That maybe there's something hidden in plain sight, like a "trinity" or "n-ity" that we just can't see. Like a conceptual fourth dimension of sorts.

Fabrizio Genovese (Apr 05 2020 at 14:48):

Not from me, anyway you can find it here: http://www.cs.ox.ac.uk/people/bob.coecke/Yoshi.pdf

Stelios Tsampas (Apr 05 2020 at 14:49):

Thanks!

Matteo Capucci (he/him) (Apr 05 2020 at 15:10):

Stelios Tsampas said:

I've entertained the idea (late at night, under the influence of... things) that duality is actually a jail we're all imprisoned in. That maybe there's something hidden in plain sight, like a "trinity" or "n-ity" that we just can't see. Like a conceptual fourth dimension of sorts.

I think something along Lawvere's notion of Aufhebung, itself based on the notion of levels of a topos, might be an interesting expansion of the dualistic point of view.

John Baez (Apr 05 2020 at 19:45):

Rota vaguely opined that eventually math is going to move on to take ternary operations seriously. The same might be said for trialities. The one really important triality I've put a lot of time into is the triality in 8 dimensions, which underlies the octonions and ultimately superstring theory. But this seems so exotic and specialized compared to dualities!

Todd Trimble (Apr 05 2020 at 20:48):

John Baez said:

Rota vaguely opined that eventually math is going to move on to take ternary operations seriously. The same might be said for trialities. The one really important triality I've put a lot of time into is the triality in 8 dimensions, which underlies the octonions and ultimately superstring theory. But this seems so exotic and specialized compared to dualities!

Might be fun to try to think of a few "convincing" examples of ternary operations (meaning, "ones to take seriously"). How convincing are heaps?

James Wood (Apr 05 2020 at 20:52):

“quantum heap” is an amazing name. 🤣

Oscar Cunningham (Apr 05 2020 at 20:54):

A lot of 'binary' operations are actually n-ary operations for all natural n. Thinking of ternary operations is difficult, but so is thinking of binary operations that aren't associative!

James Wood (Apr 05 2020 at 20:58):

Yeah, I wanted to say something like this. A monoid is not really about its binary and nullary operators, but rather its ability to handle lists of inputs in a uniform way (in the sense that it doesn't look to see how many elements the list has, or anything like that).

Evan Patterson (Apr 05 2020 at 21:01):

One ternary relation that is pretty convincing (and might actually be related to heaps) is betweenness: the relation of a point belonging to the line segment between two others. IIRC, you can axiomatize affine space using just this relation.

Stelios Tsampas (Apr 05 2020 at 21:02):

John Baez said:

Rota vaguely opined that eventually math is going to move on to take ternary operations seriously. The same might be said for trialities. The one really important triality I've put a lot of time into is the triality in 8 dimensions, which underlies the octonions and ultimately superstring theory. But this seems so exotic and specialized compared to dualities!

So that stuff actually exists. Huh.

Todd Trimble (Apr 05 2020 at 21:05):

James Wood said:

Yeah, I wanted to say something like this. A monoid is not really about its binary and nullary operators, but rather its ability to handle lists of inputs in a uniform way (in the sense that it doesn't look to see how many elements the list has, or anything like that).

Of course I agree, but I think I mean the question in a loose "change hearts and minds" sense, that might get people to consider researching certain ternary operations as important.

Jules Hedges (Apr 05 2020 at 21:06):

Between a shape having an inside and an outside, and an arrow having a pointy end and a feathery end, it's hard to imagine trialities ever being as important as dualities

T Murrills (Apr 05 2020 at 21:14):

VERY vague thought here, but there is a third element hiding in all of these: in the first, the boundary of the shape itself, and in the second, the arrow itself. I feel like we might at some point move from “object-level manipulations” (e.g. take a binary product of these two things) to “statement-level manipulations” (e.g. look at the equality between the arguments of the binary product and its output as a ternary relationship, and treat that as “primitively” as we do the binary operation now)

Jules Hedges (Apr 05 2020 at 21:15):

There are some trichotomies eg. positive/negative/zero, inside/outside/boundary, parabolic/hyperbolic/elliptic... there was a mathoverflow question about this once

Todd Trimble (Apr 05 2020 at 21:22):

Actually, the cross-ratio is a classical sort of 4-ary operation on projective lines that seems to be of some interest.

Todd Trimble (Apr 05 2020 at 21:47):

Todd Trimble said:

Actually, the cross-ratio is a classical sort of 4-ary operation on projective lines that seems to be of some interest.

Hm, probably didn't say that right. Maybe better to consider a family of ternary operations $C_r$ (parametrized by points $r$ in the model $k \sqcup \{\infty\}$) where $C_r(x, y, z) = w$ means $[x, y; z , w] = r$. Then at least for projective transformations $\phi$, we have $\phi(C_r(x, y, z)) = C_r(\phi(x), \phi(y), \phi(z))$.

Dan Doel (Apr 05 2020 at 22:34):

I think there are plenty of non-binary operations to be had in computer science. For instance, binary trees have a node forming operation $\mathsf{Tree}\ A → A → \mathsf{Tree}\ A → \mathsf{Tree}\ A$.

Todd Trimble (Apr 05 2020 at 22:41):

What's the node forming operation?

Dan Doel (Apr 05 2020 at 22:42):

It forms a tree with the specified root value and two subtrees.

Dan Doel (Apr 05 2020 at 22:45):

I think there might be a name for this sort of structure for which trees are the free example, but I forget the name.

Dan Doel (Apr 05 2020 at 22:48):

Similar to monoids, but to glue two things together, you must put something in between ($M → A → M → M$), and trees aren't associative so it might have 'skew' in the name.

John Baez (Apr 05 2020 at 22:51):

Hi, @Todd Trimble! Great to see you here! Yes, heaps are a nice way to formalize "groups that have forgotten their origin", a nice alternative to torsors where you need to have a group around to define them.

Some other examples of ternary operations: James Dolan used to like explaining Lie triple systems, which are to symmetric spaces as Lie algebras are to Lie groups.

Also there are Jordan triple systems, which I forget the deep inner meaning of... I think they're like Jordan algebras that have "forgotten its 1". It's important to be able to change your mind about which element is the multiplicative unit in a Jordan algebra (sorry if that remark is obscure, I could explain it now that it's coming back to me).

Also there's the Freudenthal triple system, which is highly specialized: a way to talk about the structure possessed by the smallest irrep of $E_7$.

And generalizing the last two, there are Freudenthal-Kantor triple systems, which as you can see have quite noble ambitions. I don't understand them.

Todd Trimble (Apr 05 2020 at 22:54):

Kinda wanna ask, "what would Rota say?" to any one of these interesting examples.

Dan Doel (Apr 05 2020 at 23:07):

I guess the non-associativity is essential, because otherwise you could just say it's a free semigroup/monoid. The non-associativity means that the ternary operation encodes meaning that isn't obviously divided into an inclusion and a unital binary operation.

Stelios Tsampas (Apr 07 2020 at 19:24):

John Baez said:

Some other examples of ternary operations: James Dolan used to like explaining Lie triple systems, which are to symmetric spaces as Lie algebras are to Lie groups.

I took a look in the wikipedia page for Lie triple systems and I noticed that their second law, which looks like it involves a shifting of sorts, resembles what my idea of a triality would be: If the (informal) principle of duality manifests as "Take a construct represented in a commutative diagram and reverse the arrows to create its dual" in Category Theory, a possible triality would instead involve a shifting operation between three objects (they can be something like "source, target and observer"). Of course, I'd expect that after three shifts I'd be right where I started, much like $(\mathbb{C}^{op})^{op} \equiv \mathbb{C}^{op}$. Trippy...

Stelios Tsampas (Apr 07 2020 at 19:27):

But this sounds a lot like (co)slice categories so I don't know.

John Baez (Apr 07 2020 at 19:57):

This second law is a mutant version of the "Jacobi identity", which for a binary operation [-,-] says

[a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0

(we're summing over cyclic permutations). The Jacobi identity holds for Lie algebras.

This stuff is fun!