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Stream: community: discussion

Topic: Extension to the Grothendieck Homotopy Hypothesis

Ben Kaminsky (Aug 07 2025 at 01:29):

Hi CT Zulip,

I'd like to start off by preemptively apologizing to the CT community for what may be difficult for some of you to read. I understand you guys do not know me, and I do not know you. But I ask that people take the time to listen to what I have to say.

I'm currently looking into how to deal with a category that doesn't conform to the Grothendieck Homotopy Hypothesis. I am afraid there is no solution I can find in the current literature. Tools in things like twisted parametric stable homotopy theory do not apply.

However, this can only be discovered if you play with transcendental space curves and execute a series of rather bizarre differential geometric/topological maneuvers followed by a group-theoretic framework that seems meaningless at first but actually evolves to Turing Completeness and beyond.

This research is related to Prof. Lurie's work on the Cobordism Hypothesis. In essence, a computational implementation based on the Baez-Dolan Conjecture (1995) and Prof. Lurie's classification of TFT's (2009). My construction is a groupoid whose morphisms are homotopy classes of paths together with explicit geometric data (e.g., winding number, chirality), and composition laws compatible with these decorations. I refer to this as a "decorated fundamental groupoid".

By doing this, one executes a limit case that falls just outside the bounds of the GHH due the particulars of this symmetric monoidal category whose structure is not compatible with a reduction to a plain homotopy type because the decorations are not homotopy invariants in the classical sense, even though the underlying space is trivially constructible. The setup is below.

Setup.
Given a smooth manifold $\mathcal{M}$ and a decoration set $\mathcal{S} = \{\tau, \chi\}$, define the decorated fundamental groupoid $\Pi^{\mathcal{S}}_1(\mathcal{M})$, with morphisms of the form $[\gamma, (\tau, \chi)]$. Composition is given by path concatenation and additive decoration.
If $\mathcal{M}$ is a 3-ball with a series of global twists, then the underlying groupoid $\Pi_1(\mathcal{M})$ is trivial. However, $\Pi^{\mathcal{S}}_1(\mathcal{M})$ is not: the decoration is inseparable from the morphism identity.

Claim.  
There exists no forgetful functor making $\Pi^{\mathcal{S}}_1$ equivalent to any classical homotopy type unless $\mathcal{S}$ is trivial.
Hence, the plain  GHH misses these decorated structures. An "Extended GHH" is needed — one that asserts:
$\text{Decorated } \infty\text{-groupoids} \simeq \text{Decorated topological spaces}$

Again, this was not intentional. I did not set out to specifically do this. I don't have anything against the CT community. Regarding technical details, a summary is below. DM me and I'll send a link to the paper for those who are curious to read the entire thing. (Warning---the entire paper is about a category-theoretic approach to the geometry of helices.) Explaining everything here would require flooding the chat with a huge amount of text that is inappropriate.

https://www.overleaf.com/read/vwytxxfdbcqn#98efca

And please, nobody panic. A simple extension to the GHH (The Extended GHH) is all that's needed. This enriches HoTT, not refutes it. No one has to change anything related to their work but eventually certain technicalities within papers should be noted.

As for why I believe this should be adopted? Here is what it boils down to:

"The Cobordism Hypothesis heavily hints that we should do this."

Do understand---it is not my intention to hide behind the work of professor Lurie or Baez, I alone will take full responsibility for my actions here, and I simply wouldn't suggest something so radical if it wasn't pressing.

For those interested, I suggest Dr. Eva Miranda's recent work on Topological Kleene Field Theory as a starting point.

https://arxiv.org/abs/2503.16100

My interpretation is just a discrete geometric version of her work, but my work began in parallel to her own, I wasn't directly inspired or even aware. If you combine my work with hers, you'll have the beginning of what amounts to a true categorical understanding of computation, both discrete and analog.

I know many people here have been seeking categories like this. If you don't wish to publicly post, my DM's are open and you have my confidence. There is a desire for a theory of computation that doesn't depend on abstractions like Turing Machines---instead just raw geometry, topology, and group theory, utilizing the true bare metal of our universe. However, I feel that if any categorical understanding is to work, it must deal with the lessons of the Blum-Shub-Smale Machine (yes that Smale), and the long shadow of Pour-El and Richard's non-computable functions. The consequences of Post-Turing complete computation are not without a heavy cost.

It is true that the GHH lies outside the ZFC. Whether or not any extension should be adopted is as much a cultural one as a mathematical one. I understand that I have no right to force my views on anyone, I only ask they be considered.

I have my reasons for why I've done things in certain ways, but for the issue of length I will leave it at this.

Thank you,

Kevin Carlson (Aug 07 2025 at 04:43):

You haven’t given a complete definition of $\Pi_1^{S}(M)$ so it’s unfortunately impossible to comment, or to address any of your expository commentary.

Ben Kaminsky (Aug 07 2025 at 19:08):

Hi, $\Pi^{\mathcal S}_1(M)$ means the fundamental groupoid, but with each morphism further decorated by data from $\mathcal{S}$ (e.g., $\tau,\chi$).

The decorations being "trivial" in this case would be a space that is an achiral manifold like a torus.

Kevin Carlson (Aug 07 2025 at 21:09):

That's not a definition, though. You seem to mean something specific by $\tau$ and $\chi$, but from what you've written they're just labels for arbitrary elements of an arbitrary set. And "a groupoid with decorated morphisms" is not a concept that already exists, so you have to actually define that concept before you use it.

Ben Kaminsky (Aug 07 2025 at 23:10):

Got it, here's my full working definition.

The decorated fundamental groupoid} $\Pi_1^{\mathcal{S}}(M)$ is defined as follows:
Objects: Points $x \in M$ (as in the classical fundamental groupoid).
Morphisms: Homotopy classes of continuous paths $\gamma: [0,1] \to M$ from $x$ to $y$, together with an assignment of "decorations'' from $\mathcal{S}$ to each path, compatible with the composition law. That is, a morphism from $x$ to $y$ is an equivalence class $[\gamma, d]$, where $d$ records the geometric or combinatorial data along $\gamma$.
Composition: Given $[\gamma_1, d_1]: x \to y$ and $[\gamma_2, d_2]: y \to z$, their composite is $[\gamma_2 \ast \gamma_1, d_2 \ast d_1]$, with decorations composed via a specified rule (e.g., addition of winding numbers, concatenation of twists, etc.).

Two decorated manifolds $(M, \mathcal{S})$ and $(N, \mathcal{S}')$ have \emph{equivalent decorated groupoids} if there exists a functor between their groupoids preserving both the path composition and the decoration data.

Chi is chirality, (standard right hand rule), tau is torsion. If it helps to get an idea of the class of manifolds under group actions (chirality swaps and orientation flips) we are working with, there's a picture in the summary.

Kevin Carlson (Aug 07 2025 at 23:39):

What exactly are the decorations allowed to be, mathematically? I now see that since you write $d_2*d_1$ they're things that can be multiplied. Are they in a monoid? A category equipped with some appropriate functor? Phrases like "records geometric or combinatorial data" still aren't part of a mathematical definition, unless you've defined what you mean by "geometric or combinatorial data."

David Michael Roberts (Aug 07 2025 at 23:39):

where $d$ records the geometric or combinatorial data along $\gamma$.

what does this mean? Still not enough information to understand what you are actually defining. Moreover, you say $\mathcal{S}$ is a set, but I'm guessing it's actually meant to be a group? And, moreover, has more elements than the example...

If $\Pi_1^{\mathcal{S}}(M)$ is a groupoid, then there is a homotopy 1-type corresponding to it, namely the geometric realisation of its nerve. Even if you call it a "decorated groupoid", you haven't defined the abstract notion of what it means to be a "decorated groupoid". I can only see that it is (meant to be) a groupoid. If $\Pi_1^{(-)}$ is meant to be a functor that doesn't land in the category of groupodis, please define its codomain without reference to manifolds or geometric information (chirality etc).

Kevin Carlson (Aug 07 2025 at 23:39):

I don't want to "get an idea" of anything, yet: I want an actual definition of the mathematical structures of which you think you have some interesting geometric examples.

David Michael Roberts (Aug 07 2025 at 23:40):

Forget the geometry, define "decorated groupoid" first, and tell us why it doesn't satisfy the axioms of a groupoid.

David Michael Roberts (Aug 07 2025 at 23:41):

Because it it satisfies the axioms of a groupoid, you are definitely in a setting where the Homotopy Hypothesis is proved, and has been since the 1940s.

John Baez (Aug 08 2025 at 08:13):

Kevin Carlson said:

I don't want to "get an idea" of anything, yet: I want an actual definition of the mathematical structures of which you think you have some interesting geometric examples.

This example illustrates why it's bad to include heuristic comments aimed at helping readers understand a definition in the definition itself.

First, even experienced readers can get confused about where the definition ends and where the heuristic remarks begin. The heuristic remarks often contain vague, undefined terms that poison the precision of the definition. Second, beginners can get confused about what a definition is - how definitions actually work!

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 $BG$.