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Stream: learning: questions

Topic: Extension to the Grothendieck Homotopy Hypothesis

Ben Kaminsky (Aug 08 2025 at 15:44):

Regarding a general form for decorated groupoids (no geometry) a concise definition is this:

Let $(S,*)$ be a monoid with unit $e$. Write $\mathbf{B}S$ for the one–object category whose endomorphisms are $S$ with composition given by $*$.

An S–decorated groupoid is a pair $(\mathcal{G}, d)$ where $\mathcal{G}$ is a small groupoid and $d: \mathcal{G} \to \mathbf{B}S$ is a functor.

Equivalently, for every morphism $f$ in $\mathcal{G}$, we have $d(f) \in S$ with $d(g \circ f) = d(g) * d(f)$ and $d(\mathrm{id}_x) = e$ (and $d(f^{-1}) = d(f)^{-1}$ if $S$ is a group).

A morphism of S–decorated groupoids $(\mathcal{G}, d) \to (\mathcal{H}, \delta)$ is a functor $f: \mathcal{G} \to \mathcal{H}$ with $\delta \circ F = d$.

The decorator functor $d$ does not factor through the ordinary fundamental groupoid $\Pi_1(M)$. Realizing the ordinary nerve of the groupoid forgets this extra geometric data. Hence, there is a need for an “extended” framework that lives in the setting of things like Lie groupoids.

Kevin Carlson (Aug 08 2025 at 16:29):

OK, great, so a decorated groupoid is just a groupoid over the delooping of a monoid. That's something we can work with. Now, your functor $d$ will necessarily send every morphism to an invertible element of $S,$ since functors preserve isomorphisms. So you effectively have to decorate your fundamental groupoid in a group, not a monoid. Is that OK for your examples?

Ben Kaminsky (Aug 08 2025 at 21:23):

Yes, it's necessary (we need the ability for what's called in CS terms "reversible computation") and a group object is what my main examples use.

Concretely: the abelian group $S = \mathbb{Z} \times C_2$, where:
$\tau \in \mathbb{Z}$ (twist count) has an additive inverse, and

$\chi \in C_2 = \{\pm 1\}$ (chirality) has a multiplicative inverse.

For a smooth path $\gamma$, we set $d([\gamma]) = (\tau(\gamma), \chi(\gamma))$.

Path reversal sends $(\tau, \chi) \mapsto (-\tau, \chi^{-1})$, so functoriality forces $S$ to be a group.

David Michael Roberts (Aug 09 2025 at 07:23):

Ok, so if you have a groupoid built in some funky way, the homotopy type it corresponds to may not be that of the manifold you started with. That doesn't mean the Homotopy Hypothesis doesn't work. There's no reason to assume that the Homotopy Hypothesis is broken because your data doesn't conform to what it applies to. Perhaps you are looking at parameterised spaces, not plain spaces.

Amar Hadzihasanovic (Aug 09 2025 at 07:41):

It seems to me that that is already what Ben was saying—that these decorated groupoids ought to classify some "decorated spaces"—but this doesn't seem an unusual thing nor one which would elicit any "panic". Ultimately the homotopy hypothesis is a typical statement of the form "equivalence classes of certain geometric objects are classified precisely by certain algebraic data", and it is clear that modifying either the geometric objects or the algebraic data requires adjusting the other side of the duality (unless the "new" objects are secretly an instance of the "old" objects).

Amar Hadzihasanovic (Aug 09 2025 at 07:44):

So if I understand correctly Ben is stating that the category of S-decorated groupoids is not secretly equivalent to some full subcategory of plain groupoids... but to me that seems the unsurprising option! You added some structure and ended up with something inequivalent. If it was the other way around I would have said "oh, that's interesting".

Amar Hadzihasanovic (Aug 09 2025 at 07:57):

(Also it is unclear to me whether what you are claiming would prevent, e.g., S-decorated groupoids from classifying some 2-types for which the decoration suffices to reconstruct the 2d data, so that there would be a non-trivial representation of S-decorated groupoids in homotopy types after all...)

John Baez (Aug 09 2025 at 08:18):

It's mildly interesting to use decorations on a space to turn its fundamental groupoid into a groupoid over a group. But it seems more interesting to study manifolds decorated with submanifolds to create generalizations of the fundamental groupoid where a path is not invertible if it crosses the submanifold.

Developing this idea further one is led to a generalization of the homotopy hypothesis and the more powerful [[tangle hypothesis]] which applies to stratified spaces. I called it the 'generalized tangle hypothesis', and various people have developed it. Here are a couple of papers:

• Conor Smyth and Jon Woolf, Whitney categories and the Tangle Hypothesis.

Abstract. We propose a new notion of 'n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct 'fundamental Whitney n-categories' from each smooth stratified space X. These are obtained by considering the homotopy theory of smooth maps into X which are transversal to all strata. This makes concrete another idea of Baez and Dolan's which is that a suitable version of homotopy theory for stratified spaces should allow one to generalise the relationship between spaces and groupoids to one between stratified spaces and categories with duals.

• David Ayala and John Francis, The Tangle Hypothesis: dimension 1.

Abstract. We introduce an $(\infty,1)$-category ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$, the morphisms in which are framed tangles in $\mathbb{R}^n\times \mathbb{D}^1$. We prove that ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$ has the universal mapping out property of the 1-dimensional Tangle Hypothesis of Baez-Dolan and Hopkins-Lurie: it is the rigid $\mathcal{E}_n$-monoidal $(\infty,1)$-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal $(\infty,1)$-category gives link invariants, generalizing the Reshetikhin--Turaev invariants.

David Michael Roberts (Aug 09 2025 at 08:40):

By the way, I don't know if you know, but if you treat the fundamental groupoid $\Pi_1(X)$ of a manifold $X$ as a Lie groupoid with object manifold $X$, then the identity-on-objects functor $\Pi_1(X)^{discr}\to \Pi_1(X)^{LieGpd}$ from the discrete version to the Lie groupoid is a weak equivalence in the sense of being a Morita equivalence/equivalence in the bicategory of differentiable stacks/anafunctors. As a result, both of them represent the same homotopy type, in that the geometric realisations of their nerves are homotopy equivalent (and not just weakly homotopy equivalent). So discrete decorations on the Lie groupoid version of Pi_1 aren't adding more than just adding decorations to the plain topologically-discrete Pi_1.

David Michael Roberts (Aug 09 2025 at 08:42):

One might expect that for a fixed (discrete) group $G$ of decorations, the ($(\infty,1)$-)category of $G$-decorated groupoids is equivalent in an appropriate way to the ( $(\infty,1)$-)category of homotopy 1-types over $BG$.

Morgan Rogers (he/him) (Aug 09 2025 at 13:16):

Alright, now we have a definition of decorated groupoid. Amar seems to think the proposed extension of GHH is unsurprising; I at least also don't see why you expected it to elicit shock or panic either. Should we be asking what the connection with the cobordism hypothesis is next?

John Baez (Aug 09 2025 at 13:30):

David Michael Roberts said:

By the way, I don't know if you know, but if you treat the fundamental groupoid $\Pi_1(X)$ of a manifold $X$ as a Lie groupoid with object manifold $X$, then the identity-on-objects functor $\Pi_1(X)^{discr}\to \Pi_1(X)^{LieGpd}$ from the discrete version to the Lie groupoid is a weak equivalence in the sense of being a Morita equivalence/equivalence in the bicategory of differentiable stacks/anafunctors. As a result, both of them represent the same homotopy type, in that the geometric realisations of their nerves are homotopy equivalent (and not just weakly homotopy equivalent).

I hadn't noticed that throwing out the 'cohesion' on objects of the fundamental groupoid in this way doesn't amount to anything in the bicategory of differential stacks. Thanks! But now it sort of makes sense. I guess the moral is that the cohesion is in the morphisms.

David Michael Roberts (Aug 09 2025 at 13:56):

Well, it's not just about cohesion on objects. The Lie groupoid has a manifold of arrows (which is connected if X is), but each hom-set is discrete. And the discrete version is completely discrete.

Notification Bot (Aug 09 2025 at 21:17):

This topic was moved here from #community: discussion > Extension to the Grothendieck Homotopy Hypothesis by Madeleine Birchfield.

Ben Kaminsky (Aug 10 2025 at 04:17):

Dr. Carlson and co., I appreciate the help--- I will look into the literature cited on the tangle hypothesis. So at this point, I will now reveal the geometric "gotcha" to keep the discussion on track. In this category, you compute directly using group theory and cyclic groups. For example, take $C_2$ x $C_2$. The "slippage" happens because the group operations themselves are:

Chirality swap $(C_2)$
Orientation flip $(C_2)$
(And there are many more later, but I'll keep it simple.)

This is where the "rubber meets the road". You do these on the submanifolds of the structure I create. Formally, the decoration map $d:\mathrm{Mor}(\mathcal G)\to\mathcal S$ is $\mathcal S$-equivariant under composition, hence computation = applying $d$-labels via the group action. (There are, of course, specific rules for encoding the decoration set into data (this is referred to as a "geobit"), which aren't totally relevant right now.)

But as you can see, these are observer-oriented phenomena. My results are this. What we call "analog computation" is exactly like physics; you need fixed reference frames! That leaves me in a pickle since even things like Kan complexes/Segal spaces are coordinate-free by default.

This is why the plain HH "runs out of steam", so to speak. The core issue is transcendentality and chirality. The "homotopical data" I need the fundamental groupoid to capture is inherently chiral; in a helix, the winding number (around the screw axis) is the geometry and vice versa. It's all one mass.

I understand that modeling analog forms of hypercomputation (again, think actual BSS, not useless oracle machines) may seem like a niche topic not worth worrying about. Still, the paper outlines that certain aspects of the Geometric Langlands (mostly related to Prof. Schozles' conjectures on twistors - again with the physics!) would eventually become easier to study using my model, and may require it. My purpose here is more to argue for the benefits of a broader view of the HH.

Morgan Rogers (he/him) said:

Alright, now we have a definition of decorated groupoid. Amar seems to think the proposed extension of GHH is unsurprising; I at least also don't see why you expected it to elicit shock or panic either. Should we be asking what the connection with the cobordism hypothesis is next?

Okay. This is where the connection to the cobordism hypothesis comes in. My assumption is this - The computation that occurs in a QFT is internal to the geometry of a Yang-Mills field, not external. The geometry itself is a form of analog computational flow. If we could model it correctly, it would open up vast new frontiers in categorical physics. I refer to these structures as "Geometrically Computable Manifolds". I define 4 "computational regimes" corresponding in parallel to a TFT in Prof. Luries' classification. I urge you to see the paper preview for the chart.

Disclaimer:
I understand anything quantum-related is dangerous territory, but this work is firmly within orthodox categorical LQG. And for the record, originally, this work wasn't even about physics; it's about computation! This is a semi-accidental kind of discovery, a cross-domain connection, not my attempt at a TOE. Personally, I do not care for unification theories at this precise moment, and I feel any attempt is fundamentally premature without a full understanding of non-abelian gauge theory.

I understand that people like the HH for its simplicity (I do as well), and this is a tradeoff. People should exercise discretion in decoration and enrichment. It's not that I don't believe in Grothendieck's vision, as one of the main conclusions of the paper (that I think Dr. Schreiber et. al. will agree with) is that his style of doing mathematics will eventually break the Great Stagnation of HEP. My message is one of irony, but more so hope. However, like I said, this will come at a cost.

Again, if anyone wants the paper, DM me, I'm all too happy to share.

Kevin Carlson (Aug 11 2025 at 08:27):

We took five or six back and forth to get you saying specific things, but you’ve slid back totally into dark generalities. Please try saying about 5% as much at a time and make it mathematical, not heuristic.

Kevin Carlson (Aug 11 2025 at 08:28):

The only precise point I see in your above message is that the map from paths to decorations is equivariant, but that’s already part of the definition. I have no idea what that has to do with coordinate systems, unless you mean that you can only actually calculate your decorations in a coordinate system, in which case you don’t have a decorated groupoid at all according to your definition.

David Michael Roberts (Aug 11 2025 at 09:58):

Ben Kaminsky said:

flooding the chat with a huge amount of text that is inappropriate.

It would appropriate to reflect on this point a bit more, and doing some self-editing to remove anything that's not precisely defined. And I don't mean give all your definitions so you can talk about stuff, just use the kind of mathematical language and ideas that are common or from the literature to explain what you think. This is how all mathematicians have to work, they generally don't try to communicate the mathematical latent space in their mind by elaborate metaphors and hand-wavy descriptions disconnected from anything anyone has seen before. And when it is done, you get things like Mochizuki's track record of increasingly incomprehensible simplistic metaphors that don't explain his baroque and quite probably incorrect claims.

John Baez (Aug 11 2025 at 11:56):

I think the situation is even worse than David and Kevin are making it. Anyone who starts by talking about "a category that doesn't conform to the Grothendieck Homotopy Hypothesis", without providing a real example, and then claims they're making progress on hypercomputation and the geometric Langlands program, and saying that they are doing "orthodox loop quantum gravity" without providing the slightest shred of evidence for any of these claims, is someone worth ignoring.

Madeleine Birchfield (Aug 11 2025 at 15:05):

Ben Kaminsky said:

I understand that modeling analog forms of hypercomputation (again, think actual BSS, not useless oracle machines) may seem like a niche topic not worth worrying about.

Perhaps then you can explain to the rest of us here

• what "analog hypercomputation" is
• what a BSS is
• what significance a BSS has in analog hypercomputation
• what relation all this has with your decorated groupoids and your extension to the homotopy hypothesis

Preferably with pointers to the existing literature on this subject.

Amar Hadzihasanovic (Aug 11 2025 at 15:13):

BSS refers to Blum-Shub-Smale machine.

Amar Hadzihasanovic (Aug 11 2025 at 15:16):

@John Baez You can add "Schozle" to the list containing "Einstien" and "Feynmann" (if I remember them correctly) ;)

Madeleine Birchfield (Aug 11 2025 at 15:23):

Amar Hadzihasanovic said:

BSS refers to Blum-Shub-Smale machine.

So then the "analog forms of hypercomputation" being referred to here is real computation.

Madeleine Birchfield (Aug 11 2025 at 16:08):

Ben Kaminsky said:

My purpose here is more to argue for the benefits of a broader view of the HH.

This would be significantly easier if there was articles in the established literature with such a broader view of the homotopy hypothesis. But since the only thing we currently have is an unverified (and thus possibly mathematically incorrect) and possibly incomplete paper, there really isn't anything we can really do yet that is actual mathematics instead of (possibly baseless) philosophical speculation.

First demonstrate, by publishing an article in a reputable journal that passes peer review etc, that e.g. a broader view of the homotopy hypothesis is useful in real computation (independent of any references to Langlands or physics), and then we can get back to this discussion.

Ben Kaminsky (Aug 12 2025 at 03:53):

Madeleine Birchfield said:

Ben Kaminsky said:

My purpose here is more to argue for the benefits of a broader view of the HH.

This would be significantly easier if there was articles in the established literature with such a broader view of the homotopy hypothesis. But since the only thing we currently have is an unverified (and thus possibly mathematically incorrect) and possibly incomplete paper, there really isn't anything we can really do yet that is actual mathematics instead of (possibly baseless) philosophical speculation.

First demonstrate, by publishing an article in a reputable journal that passes peer review etc, that e.g. a broader view of the homotopy hypothesis is useful in real computation (independent of any references to Langlands or physics), and then we can get back to this discussion.

Understood. However, it's also challenging to test an idea first without discussing it with others. I wanted a sharp critique here first, I'm perfectly alright with it! I know that being the first is never easy.

It really boils down to just doing raw differential topology/geometry and group theory with the right algebraic assumptions. If anyone reading this thread later wishes to check the math, just ask me. My understanding is that publicly posting a full preprint is socially frowned upon, so I chose a summary.

Regarding reference frames, Dr. Carlson, the ambient category naturally admits them, so I assumed it was included. If they need to be qualified to the groupoid itself, well...yes, that's an oversight on my part. I'll be sure to fix it. Thank you very much for the catch, Dr. Carlson.

And Prof. Baez, your Rosetta Stone paper was invaluable to me. But analog computation was the one thing you could not address in it. I'm well and truly sorry, Prof. Baez, but my results indicate that a full categorical understanding of computation will prove to be a difficult subject. It will require geometry, in addition to topology.

Still, I think the chat has spoken; they don't want to discuss this further, and I won't push it.

Madeleine Birchfield (Aug 12 2025 at 03:55):

Ben Kaminsky said:

My understanding is that publicly posting a full preprint is socially frowned upon, so I chose a summary.

It is not socially frowned upon. You can submit your preprint to the publicly available arXiv and many mathematicians do submit preprints to the arXiv. Some mathematicians also will put preprints on their personal or institute website.

Madeleine Birchfield (Aug 12 2025 at 04:06):

Ben Kaminsky said:

Understood. However, it's also challenging to test an idea first without discussing it with others. I wanted a sharp critique here first, I'm perfectly alright with it! I know that being the first is never easy.

You are currently dealing with multiple new ideas and just throwing all of them at us right now. There are

• models of real computation
• your decorated groupoids and extensions to the homotopy hypothesis
• relations of your model to the cobordism hypothesis
• relations of your model to various aspects of the geometric Langlands program
• relations of your model to Yang-Mills quantum field theory
• relations of your model to topological quantum field theory
• relations of your model to loop quantum gravity

One can write a half dozen papers for each of these topics alone. Trying to tackle all of these is just way too much for a single paper.

So if I have a suggestion, it's to put aside these other connections to cobordism hypothesis / geometric Langlands / physics for the time being and just focus on your model of real computation and your extension of the homotopy hypothesis and how they relate to each other. You can work on the others once you've finished with this topic first and send it through peer review to publication.

And if you do want to talk more in depth about e. g. the cobordism hypothesis with other people, you can create a separate thread for that; let's stay on topic about the homotopy hypothesis.

Kevin Carlson (Aug 12 2025 at 19:44):

Ben Kaminsky said:

Still, I think the chat has spoken; they don't want to discuss this further, and I won't push it.

I think this chat is happy to discuss lots of things at great length and I see no evidence anyone's not interested in discussing your topic, except perhaps from John. All your respondents are probably pretty skeptical for a number of reasons but we're encouraging you to state whatever you have clearly and precisely, one thought at a time. I doubt anyone here is going to attempt to read your paper, given how little you've been able to communicate clearly even in dialogue; it's only harder in monologue.