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Ben Kaminsky (Jun 23 2025 at 03:19):Hi all, I’m relatively new to category theory (I started learning earlier this year) but have been working on applying it to the semantics of programming languages (both statically and dynamically typed). Think the work of Dr. Mattias Felleisen (Typed Racket) on micro vs macro expressiveness.
I’m trying to model computational processes using enriched categories, where the enrichment is over a symmetric monoidal category of “geometric flows” (essentially, structured diffeomorphisms with chirality and torsion data). The objects are manifolds formed from helices, and morphisms represent flow transformations that preserve some finite-energy invariant.
Here's the issue. What I’m running into is that the fundamental ∞-groupoid of these manifolds doesn’t seem to satisfy the actual homotopy hypothesis:
1.Not every weak homotopy equivalence between these objects corresponds to an ∞-groupoid equivalence that preserves the torsion/chirality structure.
2.In particular, the monoidal structure on morphisms is non-invertible due to flow degeneracies under chirality inversion, unless strict torsion constraints are imposed.
Is there a known way to categorify this kind of “chiral, torsion-enriched” structure? Thanks!
John Baez (Jun 23 2025 at 07:39):What do you mean, precisely, by "manifolds formed from helices" and "structured diffeomorphisms with chirality and torsion data"? I need precise definitions here.
Morgan Rogers (he/him) (Jun 23 2025 at 13:54):I agree, we need to know what the structures actually are if we are to say anything substantive about them!
Ben Kaminsky (Jun 23 2025 at 16:22):Hi! thank you for the response Dr. Baez and Rogers. I am sorry for the wall text but for additional context, the model is partly a synthesis of your work on the cobordism hypothesis and Dr. Eva Miranda’s ideas in Topological Kleene Field Theory. Initially, I was trying to model the semantics of programming languages using enriched categories — but found that directly modeling the languages themselves (especially dynamically typed ones) was too syntactically dense. Instead, I pivoted to using differential geometry to model the computational behavior through structured manifolds, formed from gluing together pairs of 3D helices. Luckily I happened to figure out an easy way to do this from some unrelated research. They act as geometric logic gates via their chirality and orientation.
Here's where things get strange. I discovered that a certain group cohomology pattern (specifically for finite abelian groups with torsion constraints) that validates a pattern in Dr. Baez’s predictions about quantum tetrahedra, mirroring the behavior he describes. But in the process, the geometric space that realizes this behavior ends up with a fundamental groupoid whose properties fall outside* (not contradict) Grothendieck’s HH.
In my category, groupoid morphisms that correspond to chirality and orientation swaps on helices that feed into a category called S-GCM (States of Geometrically Computable Manifold) The contradiction isn’t abstract! It arises naturally when trying to preserve computational invariants like chirality and torsion across flow-based morphisms. The resulting ∞-groupoid is not equivalent to the homotopy type unless you erase precisely the data (chirality,torsion) that makes the winding number computationally meaningful.
So ironically, while trying to prove Baez right in one direction, the construction breaks the categorical foundation beneath it, like stumbling into a weird Lurie-scale pathology.
Any guidance on how to formalize this or what language might exist for modeling this type of computationally enriched geometric structure would be deeply appreciated. Right now I am following Brenden Fongs work on decorated cospans. And the current paper has an extension that covers this particular limit case. Again, I don't know what the rules on posting pre-prints are but unfortunately I am afraid it's very dense. I am open to posting it though. The entire ambient category that encloses the computation model is something I call "Embedded Smooth Manifolds with Enrichments" ("ESM+"). It's defined from scratch using nothing but raw geometric calculus, topology, and group theory.
No this is not a troll post, I promise.
Ben Kaminsky (Jun 23 2025 at 20:31):Hi,
A member of the chat reached out and pointed out it's best I just post the paper. I was previously unsure about decorum. I understand it is very difficult given the entire computational model is basically defined from scratch using nothing but...helices. I use results like rule 110 to speed things up but still.
The main results are in section
5.6 "Application of Lie Groupoids to the Quasi-Fluidic Regime" relating to homotopy compuation
5.5 "The Algebra of Flow" - does the actual computational modeling using differential geometry
5.7 "Fiber Bundles and Moduli Spaces on Chiral Manifolds" - discusses how the moduli space gives rise to geometric computational processes phase space
https://drive.google.com/file/d/1Pf88th2pnK1WRzCwf5tBYZsDGRhsALVA/view?usp=sharing
Again, this is potential extension (not contradiction) to the homotopy hypothesis that aligns directly with Dr. Baez's work. It should prove valuable to the ongoing efforts in categorical application to physics. Thank you,
John Baez (Jun 23 2025 at 20:46):I asked you to define the basic terms in your first post, but instead of defining them you brought in more jargon and vague allusions to various people's work. This means you're not operating like a mathematician. To be frank, you're operating more like someone who has spent too much time talking to a large language model. I get about one email a day from people like that. It's really tragic.
John Baez (Jun 23 2025 at 20:53):I'll ask again: what precisely do you mean by "manifolds formed by helices"? If you can't explain this in a paragraph that I understand, I see no way forward here. I know plenty about manifolds and plenty about helices, so use those concepts freely.
Ben Kaminsky (Jun 23 2025 at 20:56):Two helical solids (solids formed from rotation of a 1-manifold around a helix spine) are glued together at their common boundary, forming a twin helical manifold. I then use this as a logic gate. The fundamental representation is V4. You then quantize this into quaternary. That's how you get the tetrahedra.
John Baez (Jun 23 2025 at 20:57):Okay. @Morgan Rogers (he/him) - this makes no sense:
Ben Kaminsky said:
Two helical solids (solids formed from rotation of a 1-manifold around a helix spine) are glued together at their common boundary, forming a twin helical manifold. I then use this as a logic gate. The fundamental representation is V4. You then quantize this into quaternary. That's how you get the tetrahedra.
Ben Kaminsky (Jun 23 2025 at 21:02):Dr. Baez, It's the only way to create a discrete geometry that mimics that tetrahedron state space. You use a algebraic group structure on an observer oriented phenomena. The manifold needs limited symmetry for the quantization. Computation requires a fixed frame of reference (Kleene). Same as physics.
Ben Kaminsky (Jun 23 2025 at 21:15):image.png
When the chirality and orientation of two helices is the same you interpret it as a 1, when it's different 0. This implies 00,01,10,11. It's a tetrahedral V4 state space you encountered. There's no other shape that works. You need limited symmetry for it work.
Basically using this manifold made of helices you solve geometric quantization in reverse in a very naive way. Yes, I understand this is odd. But you can do it.
Ben Kaminsky (Jun 23 2025 at 21:42):That said, this is a smooth differentiable manifold, not just a clever representation. I fully understand the idea of a Turing Complete shape is odd but people said the same about Lie groups. Helices are naturally great at encoding information. Think DNA. Hence, they can serve as computational objects. Conceptually - do I get this is very harsh. Please keep an open mind.
Again, I originally stumbled across this as a teaching tool for students to learn non-von Neumann computation. It just so happens to...solve the mystery here.
Kevin Carlson (Jun 24 2025 at 01:49):It's not a problem that any of this is conceptually harsh, surprising, etc. The problem is that it's extremely vague. If you want to get more feedback, please stop making any general claims and defenses or analogies, and try to say completely specific sentences. I can at least imagine an actual mathematical realization of the idea of gluing two copies of , say, where is some helix in 3-space, although it's far from obvious what the helix is accomplishing here that's different from using a linear spine. But you jumped in a single sentence from a simple geometric construction to "Then I use this as a logic gate." This is an absurd distance to leap so quickly if you want to have any hope of bringing an audience along.
Ben Kaminsky (Jun 24 2025 at 02:26):Thank you, Kevin. My apologies — to both you and Dr. Baez. I've just been conceptually overloaded trying to figure out how to explain. I should actually do the opposite and start directly from the beginning instead of trying to be specific. Allow me to clarify.
Let’s start with a symbolic gluing construction.
Take two helical solids of the form:
where:
is a helix-shaped curve parametrized by arclength, embedded in 3-space with a fixed handedness and curvature profile,
is the standard 2-disk fiber over each point of the helix (so this is a tubular neighborhood, giving a solid helical manifold).
Now consider forming a gluing along boundary sections of these two manifolds. The gluing is determined by:
The key idea is that these gluing rules define a symbolic flow rule — the result of a gluing operation maps to a discrete symbolic state. In particular, when two such helices are glued in a binary configuration, the resulting symbolic state space (00, 01, 10, 11) which forms a tetrahedral symmetry under , closely matching the Baez quantum tetrahedron configuration from his 2000 paper.
I’m working toward formalizing how these gluing spaces induce a state space with topological logic gates, but I want to proceed one step at a time.
My question is this:
Is there a known formalism for tracking symbolic state changes across gluing operations of solid manifolds, especially when parametrized by chirality and join angle?
Ben Kaminsky (Jun 24 2025 at 02:47):Also, I should note that yes, helices are essential. The point is to start small but be able to encode large amounts of wild algebraic behavior with as many variables as possible. Transcendental curves naturally posses this, so that way you increase the total chaos of system as you ramp up behavior.
Morgan Rogers (he/him) (Jun 24 2025 at 08:18):You keep saying logic gate. I think I understand what the inputs of this "gate" are, namely the orientations of the helices being glued (I'm with Kevin on still being unclear why you need these helices rather than just the orientation data), but what is the output supposed to be? A logic gate takes in bits and spits out bits.
John Baez (Jun 24 2025 at 08:27):I'm going to mute this conversation. There are just too many vague allusions and not enough concrete details - no results commensurate with the amount of text.
John Baez (Jun 24 2025 at 08:37):It actually reminds me of when I'm dreaming about math.
Ben Kaminsky (Jun 24 2025 at 16:27):Morgan Rogers (he/him) said:
You keep saying logic gate. I think I understand what the inputs of this "gate" are, namely the orientations of the helices being glued (I'm with Kevin on still being unclear why you need these helices rather than just the orientation data), but what is the output supposed to be? A logic gate takes in bits and spits out bits.
Thank you, Dr. Rogers — that is a completely fair question.
You're right that I’ve been using the term “logic gate” in a metaphorical sense, but let me clarify:
This is a non–von Neumann model of computation. There is no central processor, no global memory tape, and no separation between control flow and spatial configuration. The “bit flips” here are not symbolic transitions — they are topologically instantiated as changes in chirality, torsion, and curvature at the junctions of helical manifolds.
So when I say “logic gate,” I’m referring to a geometric interaction rule: when two helices are joined with certain orientations and flow directions, the resulting composite manifold deterministically encodes an output torsion structure — effectively implementing a function (but realized geometrically, not symbolically).
This is closer in spirit to:
The Fredkin/Toffoli billiard ball model,
Or to a reversible cellular automaton,
But in my case, each “wire” (i.e., manifold segment) carries its own local symbolic algebra, defined by flow constraints and join angle.
The full computation emerges when you compose many of these “gates” (manifold interactions) into a symbolic flow lattice. This happens later on after the basic definitions. Unlike cellular automata, this lattice is not uniformly typed — the underlying symbolic structure is geometry-dependent.
Hope that helps — and happy to provide a minimal formal example if it clarifies things further.
Again, I pursued this model partially for students. Nondeterministic quantum computing is difficult for students to grasp, but cellular automata are too simple. I thought geometry would be a good compromise.
Ben Kaminsky (Jun 24 2025 at 16:42):John Baez said:
I'm going to mute this conversation. There are just too many vague allusions and not enough concrete details - no results commensurate with the amount of text.
I understand the frustration. But if I may invoke Freeman Dyson: this is necessarily a bird paper, not a frog paper. There is no other option. It surveys structure across computation, geo-topology, and symbolic dynamics — and that means it doesn't always arrive in the form one domain expects.
If the presentation feels vague, I take responsibility, but I hope you'll come back eventually.
Kevin Carlson (Jun 24 2025 at 16:55):Your description of what the gluing of two manifolds along their boundary is is exceedingly strange. The normal way of gluing two manifolds together is simply via a diffeomorphism between their boundaries; it's a pushout, a simple and familiar categorical construction. I know what chirality is, but can't see how it could affect the result of your gluing; I have almost no idea what your discussion of join angle means, or how the orientation of the gluing planes could possibly affect your result; and it's entirely beyond me what a symbolic labeling of each boundary "sector" (what's a sector?) is or what it has to do with gluing two manifolds together.
Kevin Carlson (Jun 24 2025 at 16:57):I also don't know what it means to glue two helices in a "binary configuration", why this apparently squares the resulting symbolic state space, what manifold, if any, is supposed to be acting on (the glued helical guy is the only plausible example, but I don't see any such action), or how any of this has accomplished anything different than just working with classical Boolean logic gates. You've probably read math papers before: generally, words are fully defined in an unambiguous way (for instance, in a way that can be reduced to a well-defined set-theoretic construction) before they're used in a load-bearing way in a discussion.
Ben Kaminsky (Jun 24 2025 at 17:34):Hi Dr. Carlson,
Thank you again. Let me try to clarify the formal structure a bit more.
First, regarding the gluing operation: originally, I did use a standard diffeomorphism between boundaries. However, it was pointed out by another reader that this model lacked flexibility in the topology. So the gluing now proceeds via a flexible spline-based map that still respects smoothness but allows additional symbolic parameters to persist — notably chirality and join angle.
Here are the core symbolic definitions:
The reason helices are essential is that each one possesses an internal axis of symmetry, allowing a canonical reference frame for measuring join angle. Without that symmetry, join angle becomes frame-dependent and ambiguous. The helix gives you natural local data that makes symbolic flow configuration tractable and well-defined.
Now, for the algebraic structure:
Each helical segment is assigned a symbolic boundary type:
where is chirality (left/right) and is flow orientation (in/out). This gives four symbolic boundary classes:
The Klein four-group acts naturally on this space:
The action of tracks the symbolic transformation of boundaries under gluing. That is, two helical segments with different symbolic types are glued via a spline map, and the result belongs to an equivalence class determined by the orbit of the action.
So the symbolic computation is not just about the topology of the glued manifold — it's about how symbolic flow data transforms through structure-preserving gluing.
Ben Kaminsky (Jun 24 2025 at 17:43):
By doing this the state space generated by this abelian action on a smooth glued manifold generates a tetrahedron, that's why Dr. Baez's work on the tetrahedron connection was brought up.
Morgan Rogers (he/him) (Jun 24 2025 at 17:47):Ben Kaminsky said:
The “bit flips” here are not symbolic transitions — they are topologically instantiated as changes in chirality, torsion, and curvature at the junctions of helical manifolds.
So when I say “logic gate,” I’m referring to a geometric interaction rule: when two helices are joined with certain orientations and flow directions, the resulting composite manifold deterministically encodes an output torsion structure — effectively implementing a function (but realized geometrically, not symbolically).
I think I follow this, but there is a clear problem: the resulting manifold is not a helix. Moreover, based on the earlier diagram illustrating the four possibilities, the resulting manifold doesn't have a boundary to glue along. This seems easy enough to fix (and maybe it's what you have in mind): take a piece-wise helix instead?
Even if I've understood something, I don't know what the "symbolic structure" or "symbolic flow lattice" are.
Ben Kaminsky (Jun 24 2025 at 18:19):Yes, you are correct, it's not a helix. It's a new manifold I call a "twin helical manifold". Despite extensive literature search no papers exists on it. It doesn't naturally exist in nature. In fact, according to Dr. Baez's research it probably only exists inside the pre-quantum geometry of quantum tetrahedra. So I...had to make up a name for it.
With regards to the gluing, you only glue two helices together to form one smooth manifold and that's it. To increase the "geometric capacity" of the system, then you compose these twin helical manifolds in a lattice. That's why I use the term "GCM lattice" in the paper. A Geometrically computatble manifold "GCM" is a smooth manifold with:
The symbolic structure is a labeling of local chart sectors on the manifold boundary with discrete symbolic types (as we discussed: chirality × orientation), allowing symbolic computation via gluing morphisms.
The symbolic flow lattice is the graph (or higher category) formed by arranging these GCMs together according to symbolic boundary compatibility — like a cellular automaton, but where each edge encodes an enriched morphism with flow constraints and torsion-preserving join data. I technically use the term "Fluidic State System" in the paper since it's also written for a comp sci audience. So at it's core it's analogous to a "Finite State Machine", but more flexible due to ability of the architecture of computing model to become analog (again..be aware this only happens way later once I seriously start to mess with the algebra to achieve "maximal power").
So the goal is to define a compositional, non–von Neumann computational framework where symbolic information is embedded geometrically.
Morgan Rogers (he/him) (Jun 25 2025 at 07:58):[I spoke to Ben in DMs.]