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Stream: learning: questions

Topic: "dualizing" geometric shapes using arrows/diagrams?

David Egolf (Apr 26 2024 at 18:23):

I was showing a friend of mine some cool pictures from John Baez's recent blog post. I've mentioned the concept of duality to him: that in category theory we can often get new constructions from old ones just by "turning arrows around". So, he sometimes asks me about the dual version of various concepts/constructions when I mention a concept/construction to him.

In this particular case, he asked me what the dual of one of those pictures (specifically, that of the "horosphere") would be. I've never thought before about trying to "turn arrows around" to somehow get some new dual construction from a geometric shape! To be able to "turn arrows around" and obtain "co-geometric shapes", I think one would first need to be able to express the concept of a particular geometric shape of interest in terms of arrows.

Is there some way to define a notion of, for example, a sphere using some diagram or (more generally) a bunch of arrows satisfying some conditions in some appropriate category? And if we can do this somehow, what is a "co-sphere": what concept do we obtain by turning all the arrows around?

David Egolf (Apr 26 2024 at 18:24):

The first idea that comes to mind for me is to start by considering simplices, and then thinking about "co-simplices". But I don't know much about simplices yet!

I do see this on the nLab: a simplicial object in a category $C$ is a functor $: \Delta^{\mathrm{op}} \to C$, while a co-simplicial object is a functor $:\Delta \to C$. (Here, $\Delta$ is the simplex category). I wonder if there is some way to express the definition of a simplicial object in terms of a bunch of commutative diagrams, and then to recover the notion of co-simplicial object by turning around all the arrows in those diagrams. Maybe this can be done just by expanding out (in the form of many commutative diagrams) what it means to have a functor $: \Delta^{\mathrm{op}} \to C$?

David Egolf (Apr 26 2024 at 18:40):

Anyways, I realize this is a fairly vague/broad question. Any thoughts are welcome!

Peva Blanchard (Apr 26 2024 at 21:25):

I haven't read John's post, so I'm not sure if I'll answer the question the way you expect.

There is a general slogan saying that the dual of geometry is algebra, and vice-versa. It's an extremely vast thing to unpack. See for instance:

• Stone duality
• Gelfand duality
• and the most abstract form I know, Isbell duality

More can be found on this nlab page

David Egolf (Apr 26 2024 at 21:57):

Oh my, I don't think I've properly encountered that idea before! That's exciting! Thanks for sharing those links.

Peva Blanchard (Apr 26 2024 at 22:16):

I'll just add few images and intuitions.

Take any "shape", e.g., the circle $S$ for instance. For now, think of it as a geometric shape, more or less a collection of points, glued together with some topology (we could add additional structures, like measuring arc lengths, etc.)

You can also consider $C(S)$ the set of continuous real-valued functions on $S$. It turns out this set has quite a lot of structure: you can add, multiply, scale, etc. functions together. I.e., we have a $\mathbb{R}$-algebra.

Now, take a point $p$ in $S$, and try to picture how it interacts with the continuous functions. The very natural thing to do when you have a point $p$ and a function $f$ is to evaluate the function $f$ at $p$. So a point is like a machine that takes a function $f$ and outputs a real value, namely the value $f(p)$ of $f$ at $p$.

$$p : C(S) \rightarrow \mathbb{R}$$

Moreover, this "machine" has nice properties: it preserves addition, multiplication and scaling. I.e., it is a $\mathbb{R}$-algebra homomorphism. So you see already that the geometric notion of a "point" is turned into the algebraic notion of "algebra homomorphism".

Now, we can even go a bit further. This algebra homomorphism $p: C(S) \rightarrow \mathbb{R}$ is equivalently described by its kernel $Ker~p = \{ f | f(p) = 0 \}$. This kernel has special properties: it is a maximal ideal of the algebra $C(S)$. The original algebra homomorphism $p$ is just the quotient map $C(S) \rightarrow C(S)/Ker~p \cong \mathbb{R}$.

So now the geometric notion of a "point" has been turned into the algebraic notion of maximal ideal of the algebra $C(S)$. One may wonder then if, starting from an arbitrary algebra $A$, we can recover the points by considering the maximal ideals of $A$.

Now, just to finish with the business of reversing the arrows. What does it mean to pick a point in $S$? The category theorist will say that this is exhibiting a morphism $1 \rightarrow S$ in the relevant category (e.g., topological spaces), where the object $1$ represents the "1-point shape".

What would be the algebra associated with $1$? It is the algebra of continuous functions on the 1-point space, i.e., it is the algebra $\mathbb{R}$.

If I sum up:

• on the "geometry" side, a point is a morphism $1 \rightarrow S$
• on the "algebra" side, a point is a algebra homomorphism $C(S) \rightarrow C(1) \cong \mathbb{R}$, equivalently described as the quotient map associated with a maximal ideal of $C(S)$.

caution: I have been extremely hand-wavy here.

John Baez (Apr 26 2024 at 22:17):

There are many forms of duality and it takes quite a while to go through them all. The duality between algebra and geometry is slightly more manageable. The basic idea is that if you have a space, then there's a commutative ring of functions on that space... and conversely, any commutative ring can be thought of as the ring of functions on some space. Stone duality and Gelfand duality are aspects of this thought.

@Peva Blanchard just expanded on this quite a bit. Another way to put it is that the opposite of a category of "spaces" tends to act like a category of commutative rings. For example in something we'd call a category of spaces, products tend to distribute over coproducts... but in the opposite of such a category, coproducts tend to distribute over products.

Though everyone call this the "duality between algebra and geometry", I'd prefer to call it the duality between commutative rings and spaces, or something like that, since "algebra" is very general, as is "geometry". E.g. groups are part of algebra, but not the kind that's being discussed here.

John Baez (Apr 26 2024 at 22:24):

However, "commutative rings" would need to be interpreted very broadly to make my description right, e.g. we should at least include sheaves of commutative rings.

Peva Blanchard (Apr 26 2024 at 22:39):

And this leads to one of the first thing that blew my mind when I started learning maths.

As John said, there is a duality between commutative rings and spaces: what you do on one side, you can do it on the other. But then, we know that there are more general rings than commutative rings, namely, non-commutative rings (e.g., matrix rings). One can then ask what kind of space would correspond to non-commutative rings, which is the topic of non-commutative geometry.

Peva Blanchard (Apr 26 2024 at 22:39):

ps: just a mini anecdote. I started learning undergrad maths in Lycée Thiers in Marseilles, France (it's called classe préparatoire), and on my first year, Alain Connes visited the school (he studied there too, and was invited for some celebration). He's a Fields medalist, especially well-known for his work on non-commutative geometry. It was the first time I met a real mathematician, and he made a strong impression on me. So much that I wanted to learn about non-commutative geometry ... it was very naive of me ^^' but it was also the first time I was amazed by how deep mathematics could be.

Jorge Soto-Andrade (Apr 27 2024 at 03:01):

David Egolf said:

I was showing a friend of mine some cool pictures from John Baez's recent blog post. I've mentioned the concept of duality to him: that in category theory we can often get new constructions from old ones just by "turning arrows around". So, he sometimes asks me about the dual version of various concepts/constructions when I mention a concept/construction to him.

In this particular case, he asked me what the dual of one of those pictures (specifically, that of the "horosphere") would be. I've never thought before about trying to "turn arrows around" to somehow get some new dual construction from a geometric shape! To be able to "turn arrows around" and obtain "co-geometric shapes", I think one would first need to be able to express the concept of a particular geometric shape of interest in terms of arrows.

Is there some way to define a notion of, for example, a sphere using some diagram or (more generally) a bunch of arrows satisfying some conditions in some appropriate category? And if we can do this somehow, what is a "co-sphere": what concept do we obtain by turning all the arrows around?

Nice question, David. And nice answers. In a somewhat different vein, you can look at the duals of regular polyhedra (geometric shapes by excellence...). For instance, the dual of the cube is the octahedron, which you can inscribe in the cube in a natural way, and also the other way around. You can iterate then, and the sky is the limit... Of course, you have here simplicial structures (0-objects = vertices, 1-objects = edges, 2-objects=faces and "duality" means that you have a kind of functorial mapping from one simplicial set to the other (which you could fathom as 2 - categories, I guess) which exchanges 0-objects and 2-objects.
Also horospheres connect nicely with this polyhedral geometry, I would claim (cf. next message).

Jorge Soto-Andrade (Apr 27 2024 at 03:14):

Re horospheres and polyhedral. geometry: In a nutshell, horospheres live in the Poincare Upper Half Plane , to begin with. Now, you can construct a nice finite analogue $H$ of this hyperbolic space, which carries a sort of hyperbolic geometry given by a graph structure. You can identify horospheres there, defined a la Gelfand, as orbits of the unipotent subgroup under the classical homographic action of $SL_2$, better $PGL_2$.

Jorge Soto-Andrade (Apr 27 2024 at 03:32):

More precisely, the "right" object to consider is the double cover of the classical PUHP $\mathbb C - \mathbb R$ acted upon by $G = PGL(2,\mathbb R)$ because it generalizes naturally to any local or finite field. You recover the classical PUHP as the quotient of this space by Galois (conjugation). For a finite field $\mathbb F_q$ you take then $\mathbb H = \mathbb F_{q^2} - \mathbb F_q$ acted upon homographically by $G = PGL(2,\mathbb F_q)$. You can quotient by the action of the Frobenius if you wish. Moreover you have a nice analogue of the hyperbolic distance, which classifies 2-point orbits under $G$, to wit $D(z,w) = \frac{N(z-w)}{ N(z-\bar w)}$ for $z,w \in \mathbb H$. Here $\bar w = w^q$, the Frobenius map. This is not well done in the literature because people usually restrict to odd $q$, choose an analogue of $i$ and try to mimic real and imaginary parts. No need for that if you do your Galois algebra well. Next, you can define a graph structure on $\mathbb H$ by connecting points which lie at a given fixed "distance" $d$. The graph $\Gamma_d$ so obtained has $$q^2 - q$ nodes and it is regular with valence $q+1$. So you get regular graphs with 6, 12, 20 nodes .... Does this ring a polyhedral bell to you? Notice that for $q = 3$ you get a regular graph with 6 vertices and 4 edges emanating from each vertex... The horocycles appear here as 3 - cycles ( 3 is the cardinality of the unipotent subgroups of $G$) and you can guess that... they will correspond to the faces of the octahedron! Does this make sense to you?
John will for sure have more than something to say here...
You can find details in this J. of Algebra paper https://www.researchgate.net/publication/243020845_Twisted_Spherical_Functions_on_the_Finite_Poincare_Upper_Half-Plane

Jorge Soto-Andrade (Apr 27 2024 at 03:41):

PS "Same" for Siegel instead of Poincare... You can see Siegel as (involutive) non-commutative Poincare...

Jorge Soto-Andrade (Apr 27 2024 at 04:06):

PPS. The horosphere's picture in John's blog reminded me of the aulonia hexagona (with a caveat...)
https://i2.wp.com/www.cosmic-core.org/wp-content/uploads/2019/01/aulonia-hexagona.jpg?resize=600%2C629&ssl=1

John Baez (Apr 27 2024 at 11:27):

Peva Blanchard said:

And this leads to one of the first thing that blew my mind when
As John said, there is a duality between commutative rings and spaces: what you do on one side, you can do it on the other. But then, we know that there are more general rings than commutative rings, namely, non-commutative rings (e.g., matrix rings). One can then ask what kind of space would correspond to non-commutative rings, which is the topic of non-commutative geometry.

I used to hope noncommutative geometry would be useful in quantum gravity, so I studied Connes' work a lot, but it didn't pan out for me and now I think we'd need to study the foundations of noncommutative geometry a lot more to put it to good use.

John Baez (Apr 27 2024 at 11:37):

E.g. sometìmes noncommutative geometry can be seen as geometry in a topos of G-sets.

David Egolf (Apr 27 2024 at 16:27):

Thanks to everyone for your very nice answers! I'll have to keep this thread in mind as a reference for when I want to focus on learning more about this topic.

David Egolf (Apr 27 2024 at 16:35):

I did want to make one brief comment at this time, though:

For instance, the dual of the cube is the octahedron, which you can inscribe in the cube in a natural way, and also the other way around.

In what way is the cube dual to the octahedron? (Edit: Ah, I think this article expands on this topic).

Doing a search, I found this nice image:
image.png

It looks like the octahedron's vertices poke out in the middle of the cube's faces - and vice versa! That's pretty cool. I wonder what got people thinking about this scenario...

John Baez (Apr 27 2024 at 18:37):

The Platonic solids come in 2 dual pairs and one loner, the tetrahedron, which is self-dual. People probably noticed this simply because it's so strikingly beautiful. Here's a picture by Kepler from 1619:

Later it became clear that the two members of a dual pair have the same symmetry group: if we count both rotations and reflections the symmetry group of the tetrahedron is $A_4$ (even permutations of the 4 vertices or 4 faces), that of the cube and octahedron is $S_4$ (all permutations of the cube's main diagonals), while that of the dodecahedron and icosahedron is $A_5$ (even permutations of the 5 cubes you can inscribe in a dodecahedron).

One way to think about it is that these polyhedra have vertices, edges and faces forming a poset, with e.g. a vertex being $\le$ some face if it lies on that face. The dual polyhedron gives the opposite poset! But any of these posets has the same group of automorphisms as its opposite.

Further explorations of these ideas and their higher-dimensional generalizations led to a general theory of symmetry groups of highly symmetrical polytopes, especially at the hands of Coxeter. I started a series of posts about this here:

• Symmetry and the fourth dimension

and while I never got to the fourth dimension, I hope it's a decent way to learn about duality of polyhedra and Coxeter theory.

John Baez (Apr 27 2024 at 18:39):

Things get very beautiful in 4 dimensions. For example in 4 dimensions we have a magnificent regular polytope with 600 tetrahedra as faces, and 120 vertices... but also its dual, with 120 dodecahedra as faces, and 600 vertices!

Interestingly, regular polytopes become much more dull in 5 or more dimensions.

John Baez (Apr 27 2024 at 18:40):

In every case, in every dimension, this sort of duality interchanges the high-dimensional and low-dimensional facets of the polytope. It's called [[Poincare duality]] and it eventually became part of topology.

John Baez (Apr 27 2024 at 18:42):

It doesn't require any sort of symmetry to work (unlike Coxeter theory). For example here's a graph in the plane in red, and its dual in blue. The dual of the dual is the original graph:

John Baez (Apr 27 2024 at 18:43):

In category theory the Poincare dual of the usual sort of commuting diagrams (or their higher-dimensional versions) are string diagrams.

David Egolf (Apr 27 2024 at 19:35):

Wow, awesome stuff! And thanks for the links - those look interesting!