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Stream: learning: reading & references

Topic: 📚The Stacks Project (Pedagogy)

José (Feb 26 2026 at 06:08):

The Stacks Project is an unparalleled reference, but its encyclopedic nature and extreme generality can make it tough to learn from directly. This thread is a collaborative space to reverse-engineer the intuition behind the text. We’re here to unpack the motivating geometry, map out learning paths, and translate the material using the categorical lenses we love. I am not associated to the official The Stacks Project, I am only a reader interested in improving my mathematical vocabulary and intuition.

How to use this channel:

• :label: Tag Translation: Drop a specific tag (e.g., "What is the actual geometric intuition behind Tag 01E4?")

• :map: Roadmaps: Ask for or share the most efficient path through a dense chapter.

• :bridge: Rosetta Stone: Compare Stacks terminology with standard texts (like Hartshorne, Vakil, or EGA/SGA).

• :counterclockwise: Categorical Perspectives: Discuss how concepts like descent, fibered categories, or topoi clarify the Stacks formalism.

• :robot: AI for Pedagogy Only: If you use AI tools to help break down dense material, please focus on sharing the pedagogical insights, analogies, or intuition it helped you uncover. Avoid pasting unverified, raw AI-generated proofs,the goal here is human understanding!

José (Feb 26 2026 at 06:38):

My motivation for studying The Stacks Project is to explore a formal framework for the Wolfram Physics Project. Stephen Wolfram has conjectured that the structures of theoretical physics (such as those covered in the Geometrical Anatomy of Theoretical Physics) emerge from the application of substitution rules to abstract symbols, often represented as hypergraphs.

In my view, this approach has succeeded to some extent by generating toy models of pseudo-Riemannian manifolds (via causal graphs) and quantum mechanics (via branchial spaces). Nevertheless, I believe that to push this further, we need a rigorous dictionary between this algebraic side (substitution rules acting on symbols/rewriting systems) and the geometric side (fiber bundles), much in the spirit of the Serre-Swan theorem. Of course, I’m not a fanatic: if it turns out to be mathematically impossible to express a fundamental theory of physics as a Wolfram model, I have no problem accepting that.

Because of this, I suspect that the generalized notions of space developed in The Stacks Project are more appropriate for the Wolfram framework than traditional spaces defined as sets of points embedded in Euclidean space. I am not interested in pursuing a career in mathematics or physics, studying The Stack Project and applying it to Wolfram's theory is just a hobby to me.

José (Feb 26 2026 at 06:43):

The main tool I use to transform the dense chapters of The Stacks Project into pedagogical audio summaries is NotebookML.

fosco (Feb 26 2026 at 09:44):

Can you be more precise about what you want to do? Also, who is we in "we are here to unpack the motivating geometry"? Also, why the emoji, that are giving away AI? Also, what is a branchial space?

David Michael Roberts (Feb 26 2026 at 09:55):

José said:

We’re here to unpack the motivating geometry, map out learning paths, and translate the material using the categorical lenses we love.

I would bet money an LLM generated this sentence

David Michael Roberts (Feb 26 2026 at 09:57):

José said:

application of substitution rules to abstract symbols, often represented as hypergraphs.

...

I believe that to push this further, we need a rigorous dictionary between this algebraic side (substitution rules acting on symbols/rewriting systems) and the geometric side (fiber bundles),

I don't think the Stacks Project will help for this at all.

José (Feb 26 2026 at 09:57):

David Michael Roberts said:

José said:

We’re here to unpack the motivating geometry, map out learning paths, and translate the material using the categorical lenses we love.

I would bet money an LLM generated this sentence

"we" are the people who are interested in contributing to this thread of developing a pedagogical approach to The Stacks Project. The people who are here to criticize in a destructive way, without making any contribution to the development of the mathematical culture and vocabulary, are not part of the "we"

José (Feb 26 2026 at 10:00):

David Michael Roberts said:

José said:

application of substitution rules to abstract symbols, often represented as hypergraphs.

...

I believe that to push this further, we need a rigorous dictionary between this algebraic side (substitution rules acting on symbols/rewriting systems) and the geometric side (fiber bundles),

I don't think the Stacks Project will help for this at all.

Do you ever believe in the Wolfram Physics Project? Because if you don't believe that it will give rise to a theory of everything, you will think that X will not help for this at all, regardless of the nature of X.

David Michael Roberts (Feb 26 2026 at 10:03):

I'm not talking physics, just mathematics. And I think the Stacks Project's scope doesn't help with substitution rules à la Wolfram

José (Feb 26 2026 at 10:05):

fosco said:

Also, why the emoji, that are giving away AI? Also, what is a branchial space?

Is there any reason why emojies should not be used here? I recall that this is an educational thread, not a research thread. Emojies play a pedagogical role of illustrating the ideas.

Concerning Branchal space, here are several references:

https://mathworld.wolfram.com/BranchialSpace.html

https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/the-concept-of-branchial-graphs/

https://www.youtube.com/watch?v=nDJxaakH6_g

José (Feb 26 2026 at 10:06):

By the way, infrageometry is very cool.

José (Feb 26 2026 at 10:08):

David Michael Roberts said:

I'm not talking physics, just mathematics. And I think the Stacks Project's scope doesn't help with substitution rules à la Wolfram

You are saying that you are not talking about X, just Y, but that Y is not useful for X. I think that this dismissal is rather premature.

José (Feb 26 2026 at 11:23):

For now, I will focus on the algebraic structure known as a module (intuitively, a module is a generalization of a vector space and an abelian group). Later, I might move on to another algebraic structure as my favorite one. Here is an interesting problem from the chapter Shaves of Modules
Screenshot 2026-02-26 at 06-04-35 modules.pdf.png

José (Feb 26 2026 at 11:28):

A candidate for a physical interpretation (please correct it if there is something wrong, I am not a physicist, I am just an ordinary guy reading The Stacks Project and trying to understand it, intuitively).

Imagine a space made of infinitely many one-dimensional wires intersecting at a single origin point . You are studying a physical field, like temperature, across this network. To keep the math sensible and avoid infinities, the rule is that at any exact, microscopic point, only a finite number of these wire fields can be active.

The mathematician in the text engineered a pathological operator that takes a signal from one wire and scatters it across all infinitely many wires. To bypass the rules, the scattered signal is carefully sculpted to flatline to exactly zero right at the central junction.

Because it flatlines at the center, point-by-point you only ever detect a finite number of active fields. But if you look at any physical region around that junction, no matter how tiny your observation window is, it will contain segments of infinitely many wires where the signal has eventually woken up. A standard operator or matrix requires the output to have finitely many active components across a whole local region. Because this operator activates infinitely many wires inside any neighborhood of the origin, it absolutely cannot be written as a local matrix.

The unsolved problem they want emailed is the next logical step. Physicists usually build global field configurations, like gauge fields, by patching together standard local matrix descriptions. The author is asking for a rigorous proof that a fundamentally twisted field configuration exists—built using these pathological, non-matrix operators—that is mathematically impossible to describe using standard, well-behaved local algebra in any patch whatsoever.
Gemini_Generated_Image_wk0tzhwk0tzhwk0t.png

Morgan Rogers (he/him) (Feb 26 2026 at 12:14):

It is truly wild that you would open with "the algebraic structure known as a module" (phrasing which implies no awareness of how basic/fundamental this concept is) and then cite with no additional context Example 17.10.9 of the Stacks project, an example which requires at least 5 of the preceding 17 and a half chapters to understand in any depth.
Speaking as a moderator, I have no interest in moderating or even tolerating content that is significantly LLM generated. If you find an LLM helps you feel like you understand something, by all means enjoy that feeling at your leisure, but don't expect others to engage when you pull an example from the middle of a large text and then start talking about it in terms of physics while in the same breath admitting that you are not a physicist.

José (Feb 26 2026 at 12:25):

Feel free to delete 📚The Stacks Project (Pedagogy)

I am not interested anymore in contributing to this Zulip. I made a mistake coming here and hoping people to help me to develop my mathematical vocabulary and culture. Most people only do bullying because of my use of AI for educational purposes. I am better studying alone.

Morgan Rogers (he/him) (Feb 26 2026 at 12:32):

As you wish. I did not intend to bully you. I wanted to point out that using AI in this way appears to convinced you that you understand (or are prepared to understand) things which your own writing suggests you do/are not.
There are plenty of well-tested ways to learn this material, and I can't deny that AI could be helpful for some people, but it being helpful doesn't mean it can possibly be enough on it's own. Good luck out there.

fosco (Feb 26 2026 at 14:56):

Coming here was the last in a long sequence of mistakes.

fosco (Feb 26 2026 at 14:57):

Good luck tho!

José (Feb 26 2026 at 21:46):

fosco said:

Good luck tho!

Thank you! Honestly, I don’t see any mistakes. I am interested in studying category theory and The Stacks Project. I tried to share my motivation with people in this community, but the response was more criticism of my use of AI than support.

Maybe this is one reason why many women, despite their love for mathematics, do not feel comfortable in the mathematics community: some mathematicians can be rude.

In any case, I will continue studying on my own, and eventually I will add the tools from The Stacks Project to my mathematical toolbox.

José (Feb 26 2026 at 21:51):

Morgan Rogers (he/him) said:

As you wish. I did not intend to bully you. I wanted to point out that using AI in this way appears to convinced you that you understand (or are prepared to understand) things which your own writing suggests you do/are not.
There are plenty of well-tested ways to learn this material, and I can't deny that AI could be helpful for some people, but it being helpful doesn't mean it can possibly be enough on it's own. Good luck out there.

I use AI because I don’t have the privilege of speaking daily with people who use the language of category theory. My alternative is AI or nothing. The ideal alternative would be kind people who speak the language of category theory and offer constructive rather than destructive criticism.

I know you have good intentions, but I think your view of the current state of AI, particularly NotebookML, is not very optimistic.

José (Feb 26 2026 at 22:02):

Living without AI
Here’s an example of a question I could ask without using AI: How can we translate the geometric description of what we know about physics into a purely algebraic language (for instance, by using the Serre–Swan theorem)?

Screenshot 2026-02-26 at 17-03-53 cer-000093203.pdf.png

Source: page 20 of E. Witten's Physics and Geometry.

José (Feb 26 2026 at 22:04):

For example, the algebraic analog of a trivial bundle is a free module.

José (Feb 26 2026 at 22:06):

The algebraic analog of a vector bundle over X is a finite projective C(X) module.

José (Feb 26 2026 at 22:13):

My global vision is that, when fundamental physics is formulated in purely algebraic terms, it will be more transparent from the point of view of the Wolfram Physics Project. In that project, there are many talented young people. The one who is the closest to the category-theoretic community, in my opinion, is Xerxes Arsiwalla, who is doing great contributions.

José (Feb 26 2026 at 22:25):

Why do I prefer the Wolfram Physics Project over other approaches such as String Theory or Loop Quantum Gravity? Because Stephen Wolfram is not aiming merely to unify physics, but to provide a foundational framework for all sciences.

For instance, we know that gravity can be described as the curvature of a pseudo-Riemannian manifold. Within Wolfram’s framework, however, we are encouraged to ask a broader question: could something analogous to gravity emerge in entirely different kinds of systems, say, in social networks evolving over time?

José (Feb 26 2026 at 22:32):

I think it would be fantastic to see how some of the geometric structures developed in The Stacks Project can be generated using Wolfram-style substitution rules.

José (Feb 26 2026 at 22:33):

But to do that, it is crucial to interpret the Wolfram model as a module.

Kevin Carlson (Feb 26 2026 at 23:18):

This seems like a wildly ambitious question on several fronts at once, uniting modern algebraic geometry, modern physics, and whatever "Wolfram-style substitution rules" are, all at once, with an eye toward sweeping applications to social science. I would suggest starting a new thread focusing on the specific question you asked, which is already quite ambitious: can we translate the concept of a pseudo-Riemannian manifold equipped with a vector bundle with a nonabelian gauge group into algebraic language, á la Serre-Swan? I have no idea, personally, and would guess that it's a wide open question, but I know very little about physics. Generally, mathematicians tend to engage better with focused, concrete questions. All the broad philosophizing following your specific question only detracts from the goal of getting productive engagement, because it's a style of discussion highly reminiscent of cranks, and people usually find time spent engaging with crankery is mostly wasted.

José (Feb 27 2026 at 00:03):

The Wolfram Physics Project is my main motivation for studying the Stacks Project, but for now I think it’s enough to focus on the latter. I completely agree that Wolfram’s ideas are extraordinarily ambitious, which is precisely why he has brought together researchers from many different fields to develop the project. There are even theological research initiatives connected to it, such as this one: “The G-d Conjecture” by Sam Senchal (livestream presentation).

José (Feb 27 2026 at 00:07):

In my case, I have the honor of helping Wolfram to develop this essay about number theory (I am mentioned in the "Thanks")

José (Feb 27 2026 at 00:27):

Kevin Carlson said:

Generally, mathematicians tend to engage better with focused, concrete questions. All the broad philosophizing following your specific question only detracts from the goal of getting productive engagement, because it's a style of discussion highly reminiscent of cranks, and people usually find time spent engaging with crankery is mostly wasted.

In Freeman Dyson’s terminology (from his essay Birds and Frogs), you’re describing mathematicians in general as “frogs.” By Dyson's classification, Stephen Wolfram would be a “bird.” That said, I agree that most cranks tend to see themselves as birds, although, of course, not all birds are cranks.

José (Feb 27 2026 at 00:33):

In my case, I'm a frog in statistics, but that's because I received academic training from professionals and I was giving consulting to Master and PhD students. Algebraic geometry is just a toolbox that I want to gradually assimilate in order to incorporate it into more sophisticated statistical models. This field is called algebraic statistics.

Joe Moeller (Feb 27 2026 at 02:45):

I don't think cranks are the same as birds, and I also think a bird should do some hopping around before it starts flapping it's wings.

Joe Moeller (Feb 27 2026 at 02:48):

Btw, I'm enjoying the conversation a lot more now. There's a difference between using LLMs to help you think about math vs using it to chat with us about it. I think there's a tension that enters my body when I get the sensation that I'm reading the thoughts of someone's LLM, rather than their thoughts perhaps aided by an LLM.

José (Feb 27 2026 at 04:08):

Joe Moeller said:

I don't think cranks are the same as birds, and I also think a bird should do some hopping around before it starts flapping it's wings.

Totally agree, this is why I wrote "I agree that most cranks tend to see themselves as birds, although, of course, not all birds are cranks."

José (Feb 27 2026 at 04:22):

Joe Moeller said:

Btw, I'm enjoying the conversation a lot more now. There's a difference between using LLMs to help you think about math vs using it to chat with us about it. I think there's a tension that enters my body when I get the sensation that I'm reading the thoughts of someone's LLM, rather than their thoughts perhaps aided by an LLM.

I am not a native English speaker. Indeed, English is my third language, after French, my second language and Spanish, my first language. Therefore, I need to use LLMs with the following prompt: "Write the following text as a native English speaker".

Concerning my citation of Valerie Solanas, that was not an AI hallucination, but a fascination for problematic authors. Nevertheless, I have my limits: if an author is antisemitic, like Julius Evola, that is a big no-no for me to use it as reference, because his work has been poisoned. Nevertheless, Solanas's work is satirical, his enemy is the "patriarchy" is a concept rather than concrete people. It is true that Solanas did murder a man named Maurice Girodias, but she was diagnosed with paranoid schizophrenia. In other words, this murder was not the result of her intention, but of a mental handicap and people who are handicapped should not be discriminated against. This is the reason why I give Solanas a pass: she was mentally ill, not evil. People who were really evil and advocated for violence unapologetically: Adolf Hitler (yimakh shemo), Karl Marx, Martin Heidegger (specially in the Black Notebooks).

Now, I could ask the question: Is Kanye West evil like Hitler or just mentally ill like Solanas? I don't know Kanye, but I'm sure Nick Fuentes is pure evil and shouldn't have a platform.

José (Feb 27 2026 at 04:31):

The moderator could move these last comments to a community discussion about which authors should be cited and which authors shouldn't.

José (Feb 27 2026 at 04:39):

This is a series of lectures and debates on what to do with Heidegger, who was a great philosopher but also an advocate of genocide. Should we continue citing Heidegger? I think that is an interesting question.
Colloque «Heidegger et "les juifs"»

José (Feb 27 2026 at 04:41):

Example relevant for category theory (Heidegger in nLab): Heidegger's topology

Matteo Capucci (he/him) (Feb 27 2026 at 17:24):

Most cranks are birds who talk like they licked a frog and expect scientists to just trip-sit them as they have these wild visions.
Trip sitting is fun only with your friends, and if done rarely. It requires trust and love, which are slowly built. I agree with @Kevin Carlson and @Joe Moeller that small questions and careful steps are more digestible.

John Baez (Feb 27 2026 at 22:43):

I recommend that new mathematicians learn the art of asking well-posed yes-or-no mathematical questions that maximally further the growth of their understanding. So, not questions like

"Why do people care about Noetherian rings?"

(I'm not accusing anyone of asking this), but

"Is every finitely presented ring Noetherian?"

(followed by lots of other yes-or-no questions that pinpoint the concept of Noetherianness and its uses.)

It's very easy to answer a yes-or-no question if you know the answer, so a question or this sort is not a big burden on the answerer - unless they want to figure out answer a question that's very hard for them, which can sometimes be fun. Thus, answerers are unlikely to get annoyed by such questions.

John Baez (Feb 27 2026 at 22:46):

I am more of a bird than a frog; I flit about trying to learn a bit about many things; but I've learned that when a bird lands in a pond and asks the frogs questions, it's important to use frog manners.

David Egolf (he/him) (Feb 27 2026 at 23:35):

I would also add: I've found it very helpful to have mathematicians ask me questions, especially those that are easy for them to answer. It may be harder to find a mathematician willing to engage in this way - I think it's more work to pose a question like this, than to answer it.

So if one is struggling to think of a really good yes-or-no question, I would still encourage one to ask a question as best they can. And then one can try to find (or even be given!) a yes-or-no question in the responses one receives.

John Baez (Feb 27 2026 at 23:50):

If one isn't ready to ask a good yes-or-no question, this is a useful sign that one is at a very preliminary stage; at this stage one naturally wants to ask very big questions like "what is this about?" or "why is this interesting?", but answering these questions can be quite hard so it's also good to take on some of the hard work oneself. For example, learn a definition and some examples of things that obey that definition.

Of course some people may be eager to answer very big questions, but it's good to remember that other people, including some real experts on the subject at hand, will find them too burdensome to bother with.

As I recently wrote:

When people say "I have a quick question" they mean "I want you to write me a book".

When people say "that's a great question" they mean "here comes a bad answer".

Quick question: long answer. Great question: bad answer.

José (Feb 28 2026 at 00:59):

Here is an example of a yes-or-not question about The Stacks Project:
Is it possible to express all of our knowledge of physics within the framework of The Stacks Project?
By "all of our knowledge of physics" I meant according to Ed Witten's Physics and Geometry. Below are the explicit structures that I meant by that.

Geometry of Spacetime and General Relativity = { Riemannian geometry, pseudo-Riemannian manifold, Ricci scalar, Ricci tensor, Riemann tensor, Riemannian metric, flat space Lorentz metric, Minkowski space, compact Ricci-flat six manifold, Levi-Civita connection, covariant derivative, tangent bundle, Lichnerowicz Laplacian }

Gauge Theory and Group Representations = { principal bundle, vector bundles, dual bundle, structure group, gauge group, simple group, exceptional group, Lorentz group, group of automorphisms, Lie algebra, Cartan-Killing form, complex conjugate representation, spinor representation, antisymmetric part of the tensor product, Yang-Mills connection }

Clifford Algebras, Spinors, and Fermions = { Clifford algebra, Clifford module, spin bundle, spin structure, Dirac operator, families of elliptic operators }

Topology and Global Invariants = { topological invariants, Euler characteristic, Hirzebruch signature, de Rham complex }

String Theory and Conformal Field Theory = { Riemann surfaces, Riemann sphere, affine space, complex structure, moduli space of Riemann surfaces, infinite dimensional Lie algebras, affine Lie algebras, Virasoro algebra, Lie algebra of diffeomorphisms, integrable highest weight modules, highest weight cohomology theory of the Virasoro algebra, modular forms, holomorphic line bundle, determinant line bundle, meromorphic function }

Functional Analysis and Quantum Field Theory = { Hilbert space, vector space, linear operator, Laplacian, quadratic functional, quadratic form, bilinear functional, inner product, measure, semigroup, symmetric tensor, Euclidean space }

Kevin Carlson (Feb 28 2026 at 01:01):

No.

José (Feb 28 2026 at 01:01):

What would be an example of a structure from physics that fails to be expressed within the framework of The Stacks Project?

Kevin Carlson (Feb 28 2026 at 01:03):

Hilbert space.

David Michael Roberts (Feb 28 2026 at 01:14):

The path integral formalism

David Michael Roberts (Feb 28 2026 at 01:21):

Or to be extremely pedestrian: solving any real complicated PDE that arises in practice.

José (Feb 28 2026 at 05:23):

Thanks. I’m not looking to contradict you; rather, I want to emphasize the crucial distinction between emergent physics and fundamental physics.

From the perspective of the Wolfram Physics Project, the continuous structures that define modern physics, such as Hilbert spaces and partial differential equations, are essentially "low-resolution" models. While they are useful for calculation, they are not fundamental. Instead, the "high-resolution" reality is a finite, discrete structure governed by integral bilinear forms on Z-modules (a structure mentioned in The Stacks Project). Of course, this is not a theorem, but a consequence of an assumption about nature called the physical Church-Turing thesis, formulated by David Deutsch [1] and Stephen Wolfram [2] the same year (1985).

[1] Deutsch, David. "Quantum theory, the Church–Turing principle and the universal quantum computer." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 400.1818 (1985): 97-117.

[2] Wolfram, Stephen. "Undecidability and intractability in theoretical physics." Physical Review Letters 54.8 (1985): 735.

José (Feb 28 2026 at 05:23):

Comparison of the 1985 "Twin" Foundations: Wolfram vs. Deutsch

José (Feb 28 2026 at 05:28):

If the word Ruliad is new to you... well, this is the Ruliad

José (Feb 28 2026 at 05:41):

What I’m trying to understand is how the mathematical structures discussed in The Stack Project might be generated, at least approximately, by the Ruliad (the entangled limit of all possible computations). See “What Is The Ruliad?” by Stephen Wolfram for further context.

José (Feb 28 2026 at 05:43):

Why do I want to do this? Out of simple curiosity. There might be applications to the theory of complex systems (it is too early to think of applications, because even the theory of the Ruliad is still in its infancy).