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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!

Eric M Downes (May 21 2024 at 16:13):

Aside from the standard universal algebra approach, it might be interesting to view ternary products as actions.

I mention this because of the unfortunately named planar ternary rings used to parametrize the construction of projective planes. Folks here know about them, but I hadn't seen them mentioned yet.

This approach seems a little useful for PTRs as it allows us to separate the possibly non-associative product from the simpler operation;
$t:X^3\to X$ being equivalent to a more familiar system, with constraints
$\tau:Y\to X^X; ~\iota:X\hookrightarrow Y, \pi:Y\twoheadrightarrow X$
$\iota\circ\pi=\iota, ~\pi\circ\iota=1_X, ~\forall z:Z\to X, \exists! u:Z\to Y; z=\pi\circ u$

Sometimes ternary relationships are irreducibly complex (like mod 2 addition parity-bit) but often there's a section we understand better. So, for instance, if non-associativity needs to be quarantined, you can take the part of the ternary relation you understand best, enforce that structure by placing $Y\in\mathsf{C}$ some proper category, and make $X\in\mathsf{Set}$, then take the Moore closure under composition of the image of $Y$ in $End(X)$. Then you have a functor category $\sf Mon^C$, with a bit of extra contravariant info... that is, we are back in boring old duality land.

My point is just that we do have some ways of decomposing these things into a more familiar language, at the cost of combinatorics -- instead of just left and right actions we now have to care about three different actions! I agree it would be great to have a systematic treatment that got specific enough... maybe the above will prove useful eventually.

Eric M Downes (May 21 2024 at 16:23):

(and the moore closure thing I mentioned is exactly one way we study Lie algebras; we look at the adjoint monoids! Even though I've rarely heard lie algebraists say the forbidden word "monoid"... @Stelios Tsampas can you refactor the jacobi identity John mentioned into two functions $\lambda_x(y)=xy, \rho_x(y)=yx$? You may find an interesting relationship with the more familiar commutator!)

Eric M Downes (May 21 2024 at 17:18):

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.

One random observation about the cross-ratio, that suggests it is really a generalized trace operation, like the Euler characteristic.

Start with your usual venn diagram having two sets $X, Y$. Recall that if you write inclusion-exclusion for the measures of sets, you get $p(X\cup Y)+p(X\cap Y)=p(X)+p(Y)$

Draw a horizontal line through the middle of the venn diagram

The cross ratio is the ratio of directed (signed) line segments, and we have some line segments to pick from; like we could pick $\overline{ac}, \overline{bc}, \overline{bd}, \overline{ad}$. Assert that the line segment lengths are related by kind of measure $p:\mathsf{Set}\to\mathbb{R}_+$, and naturally we want $p(X)\sim|\overline{ac}|, ~p(X\cap Y)\sim|\overline{bc}|$, etc.

so if $p(\ell)=\log|\ell|$ (surprisal, the expectation value of which is Shannon entropy) you have a specific cross-ratio(1)

$\log|ad|+\log|bc|-\log|ac|-\log|bd|=0$
$\frac{|ad|\cdot|bc|}{|ac|\cdot|bd|}=1$

So something about the choices above gives you a specific set of cross ratios, while a slightly more general relationship would give you the fully general cross-ratio
$\frac{|ad|\cdot|bc|}{|ac|\cdot|bd|}={\rm const.}\in\mathbb{R}\cup\set\infty$

Does anyone know if projective geometry is a triangulated category? All I'm familiar with is Anders Kock's characterization of three collinear points in a Desaurgesian projective plane as a morphism.

(1) -- why a cross-ratio of $1$... its a rare as a cross-ratio, due to doubled-up points iirc, though I think it might instead be interpreted that inclusion-exclusion with unsigned measures is the image of the cross ratio under a trivial homomorphism; once you lose the orientation of the lines its like you are trying to take a quotient of $S_3=S_4/V_4$ by $C_2$ and as we know $C_2$ isn't normal in that group, so somehow you end up instead with the constant hom... this is all speculation. A more careful analysis is required if indeed it means anything at all.)

Eric M Downes (May 22 2024 at 00:45):

Dan Doel said:

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.

I'm interested in what you mean by this. I think I disagree, though maybe the issue turns on what is "obvious".

First, you can have associative ops that don't decompose as an inclusion and a monoid; the truth table for the symmetric system $\{a,b,a+b\}; ~\forall a,b\in\mathbb{Z}/2\mathbb{Z}$ comes to mind -- we have a monoid but any of the two truth-table-columns are mutually perpendicular, so there is no inclusion from just two of the three that I can see. It's really the "rank" of the interaction that measures its essential (in)decomposability, not per se the associativity.

But also you can always view a non-associative system (such as a magma action) as a formal retract of its closure under composition, which is of course an associative system; $cl_\circ:{\sf Mag\to Mon}$. So, while not a free monoid action, there is exactly one (albiet non-distinct) per magma action, and often computable. I say "formal" as for very large systems its non-trivial to pick the correct retract among many candidates, but at least one is guaranteed to exist. (The retract needs to be a partial homomorphism, but cannot be a full hom due to non-associativity so sadly its not an adjoint either.)

In case that is all garbledegook, I am happy to break it out further; a nice example with a unique retract is to take the completion of the subtraction quasigroup of $\mathbb{Z}/n\mathbb{Z}$. (The left and right actions yield technically different but isomorphic groups in this case) The group you should obtain is the dihedral group $D_{2\cdot n}$; the behavior of the retract is governed by the need for a partial homomorphism that recovers the quasigroup.

But my point is that IMO the complexity of non-associativity, while it can be a burden, is separate from the complexity of a ternary relation.

Ibrahim Tencer (Jun 06 2024 at 15:27):

There is also the projective harmonic conjugate, a ternary operation that gives the fourth point such that the cross-ratio equals -1. But it can also be used in axiomatic synthetic geometry as a way to define the underlying field operations prior to the definition of the cross-ratio.

Carlos Zapata-Carratala (Jun 10 2024 at 10:39):

I had not seen this thread before! Just wanted to point out that "going beyond dualities" is one of the main themes of my research (here is my website: https://www.arity.science/). Some of my collaborators are on this forum actually @Jacques Carette @Joshua Grochow .

We have been investigating ternary algebras of cubic arrays, (semi)heaps and triality over the past couple of years. If anyone would like to chat about these topics, just sent me a DM!

Joshua Grochow (Jun 25 2024 at 22:02):

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!

Do you know where he opined this? Sounds like a piece I'd like to read...

John Baez (Jun 26 2024 at 05:41):

I don't think he wrote a whole piece on this, probably more like a couple of sentences. I don't remember where he wrote them: maybe somewhere in his books Discrete Thoughts and Indiscrete Thoughts. They're so much fun that even if you don't find that remark you'll enjoy yourself.

John Baez (Jun 26 2024 at 05:45):

Having just rewritten the nLab page on [[heaps]], ternary operations are on my mind! There's some kind of mental retooling required to work with axioms like

$t(t(a,b,c),d,e) = t(a,b,t(c,d,e))$

We're so used to binary operations that formulas like this seem painful at first - at least to me.

Carlos Zapata-Carratala (Jul 01 2024 at 23:05):

John Baez said:

Having just rewritten the nLab page on [[heaps]], ternary operations are on my mind! There's some kind of mental retooling required to work with axioms like

$t(t(a,b,c),d,e) = t(a,b,t(c,d,e))$

We're so used to binary operations that formulas like this seem painful at first - at least to me.

This is exactly my experience with (semi)heaps! In case it is useful at all, this paper tries to provide some clarity:

Biunit pairs in semiheaps and associated semigroups

In this paper we generalize the well-known correspondence between pointed heaps and groups (involuted semigroups) to pointed semiheaps and a certain class of semigroups equipped with a map (not necessarily a homomorphism, generalizing an involution that we call a switch) such that:

$$\phi (a \cdot \phi ( b ) \cdot c) = \phi (a) \cdot b \cdot \phi (c)$$

These are pretty fun but pesky algebraic creatures to play with!