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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 Ă 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.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 Ă 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.