Theories of Everything with Curt Jaimungal - The Emergence of the Super Point from Nothing

Episode Date: March 20, 2025

I'm back, baby. I've been away traveling for podcasts and am excited to bring you new ones with Michael Levin, William Hahn, Robin Hanson, and Emily Riehl, coming up shortly. They're already recorded. I've been recovering from a terrible flu but pushed through it to bring you today's episode with Urs Schreiber. This one is quite mind-blowing. It's quite hairy mathematics, something called higher category theory, and how using this math (which examines the structure of structure) allows one manner of finding "something" from "nothing." Here, "nothing" means the empty set, and "something" is defined as fermions and even 11D supergravity. It's the first time this material has been presented in this manner. Enjoy. NOTE: Link to technical details are here from Urs Schreiber: https://ncatlab.org/schreiber/show/Peri+Pantheorias As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://tinyurl.com/SpotifyTOE Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join Links Mentioned: - nLab website: https://ncatlab.org/nlab/show/HomePage - Paper on category theory: https://people.math.osu.edu/cogdell.1/6112-Eilenberg&MacLane-www.pdf - “Higher Topos Theory for Physics” (Urs’s talk): https://www.youtube.com/watch?v=GD20W6vxMI4 - “Higher Topos Theory for Physics” (Urs’s paper): https://arxiv.org/pdf/2311.11026 - Stephen Wolfram on TOE: https://www.youtube.com/watch?v=0YRlQQw0d-4 - Feynman’s thesis: https://faculty.washington.edu/seattle/physics541/2012-path-integrals/thesis.pdf - Differential cohomology in a cohesive ∞-topos (Urs’s paper): https://arxiv.org/pdf/1310.7930 - M-Theory from the Superpoint (paper): https://arxiv.org/pdf/1702.01774 - Character Map in Non-Abelian Cohomology, The: Twisted, Differential, and Generalized (textbook): https://amzn.to/4bFuz7H - TOE’s String Theory Iceberg: https://www.youtube.com/watch?v=X4PdPnQuwjY Timestamps: 00:00 Introduction 01:27 The Creation of nLab 04:36 Philosophy Meets Physics 07:55 The Role of Mathematical Language 09:32 Emergence from Nothing 16:25 Towards a Theory of Everything 22:21 The Problem with Modern Physics 25:31 Diving into Category Theory 35:30 Understanding Adjunctions 41:46 The Significance of Duality 52:54 Exploring Toposes 1:14:20 The UNEDA Lemma and Generalized Spaces 1:16:37 Charts in Physics 1:20:55 Introduction to Infinitesimal Disks 1:23:56 The Emergence of Supergeometry 1:27:33 Transitioning to Gauge Theories 1:28:11 Exploring Singularities in Physics 1:32:50 The Role of Superformal Spaces 1:36:44 Functors and Their Implications 1:40:51 From Nothing to Emergent Structures 1:43:04 Hegel's Influence on Modern Physics 1:54:07 Discovering Higher-Dimensional Structures 1:56:30 The Path to 11-Dimensional Supergravity 1:57:21 Universal Central Extensions 2:03:21 The Journey to M-Theory 2:11:19 Globalizing the Structure of Supergravity 2:15:36 Understanding Global Charges in Physics 2:23:31 Dirac's Insights into Gauge Potentials 2:30:21 The Quest for Non-Perturbative Physics 2:39:04 Conclusion Support TOE on Patreon: https://patreon.com/curtjaimungal Twitter: https://twitter.com/TOEwithCurt Discord Invite: https://discord.com/invite/kBcnfNVwqs #science #theoreticalphysics Learn more about your ad choices. Visit megaphone.fm/adchoices

Every object is in between pure nothing or pure being. And this is one of these dualities, and we'll see that it serves as the basis for a whole tower of such dualities. Today, you'll discover how category theory acts as a metaphysical microscope, revealing that contemporary physics isn't too mathematical, as critics claim, but rather lacks mathematical precision at its foundation. NYU Abu Dhabi researcher Uri Schreiber mentions that the problem isn't that physics is too mathematical, rather it's not rigorous enough. Thus, precision is needed and we have today's

specific and defined talk. Building on Hegel, Uris begins with the concept of pure emptiness and unfolds through logical necessity. You'll witness the spontaneous emergence of something called the superpoint, an 11-dimensional spacetime, and even supergravity, all without assuming their existence beforehand. What if the entire universe, spacetime, gravity, and quantum structures emerge naturally from the most fundamental logical structures possible? Uri Striber.

Ha ha ha! great to see you. Thanks for coming out. Well, thanks for inviting me. Yeah, it's a pleasure. Why don't you give an overview as to what this talk is about and what inspired it? All right. Yeah, you know, what inspired it was really your invitation. I didn't really plan to give such a talk before you invited me.

And then I thought, well, what would I say? And then I thought, well, the title of your podcast is theories of everything. So I thought, well, okay, maybe I can say something in that direction. And so then, yeah, so essentially what I ended up doing here, what I ended up compiling and what I hope I will be able to present is a little bit of an overview of various things I've done over the last, I guess I have to say decades now, less than 20 years or something. Starting with some basics in ontology, you might say, and then maybe if we have time, arrive at some actual experimental physics.

Yes, great. So the mathematicians in the audience would know you from NLab, which is a monumental website. And it's quite, it's monumental in a few respects. So one is, it's the categorification of any subject that you type in that has to do with math. It's also quick. It's also quicker than Wikipedia. I don't know why that is. Like it's extremely snappy and it's a pleasure to scroll. You mean just the pages appearing? Yes. Technically? Okay.

So tell me about the creation of NLab and what inspired that as well. All right. Yeah. So, yeah. Well, what inspired it? So underlying it is the desire, the wish to chat about math and physics while being productive, while making some net progress. Whenever you figure out something, make a note about it. That was the original driving idea behind it. There is a place where we can discuss things, but then also – well,

that was the original motivation anyway – but then also have the opportunity to record any insights, any actual advancements that have been made. So it started out as very much not an encyclopedia, which maybe is what it appears as now. But I think as the introduction, the homepage says, it's meant to be lab notes that the people may have flying around in there, on the desk or in the lab where they make notes, what they have flying around in there, you know, on their desk or in their

lab where they make notes what they have been writing on, what they have been thinking about, what they plan to do maybe, and give that all at home. And yeah, you know, it started, you know, that was actually back in the days before the modern version of social media, there was a time when people would discuss physics and math not on the commercial websites that we have these days, but on what is called the Usenet, right? Right. Maybe some of your viewers know what the Usenet, that was the free internet version before it was taken over by companies. Where we would just hang around. There was a, I guess, a group called SciPhysics Research where we would discuss.

And then later, when that kind of became problematic, some people switched to blogs to discuss on blogs, but that didn't really work out well. So on a blog, it's mostly just one person making declarations, saying something, and then other people can comment a bit, but it's not really a discussion. And yeah, anyway, so out of these inclinations, I started the N-Lab at some point. And yeah, it has been growing slowly, but surely ever since. What is it about category theory that made you take it so seriously you created an entire website about its viewpoint. So is there something different about it as a framework compared to

first-order logic or set theory or what? Does it subsume the rest? I know it's not merely about category theory, but please elaborate. Yeah, so that's a very good question. So actually, I think this is a common misunderstanding. I know many people think that kind of the N-Lab is only about category theory, and that's certainly not the case.

As anybody who has ever dived a bit deeper into it, there's lots of other subjects. So to my mind, the point of category theory is the organizing principle. The category theory is like the big index of math where things find their place, find their home. So it's the organizing principle behind stuff. And as such, it's, I think, ideal for an encyclopedic, you know, or almost encyclopedic website because it allows to relate things in the proper way. You can go and say, oh, here's a construction in that such and such field.

Oh, but look, it's just such and such limit that also appears here and there. So things get the proper context and become meaningful, I suppose. But it's also true that not so much maybe these days, but in the original years, we added just a lot of categories, just of actual category theory. I mean, what's happening with the NLIB is really mostly it's growing, you know, nobody is being paid for editing the NLIB. So what happens is people edit it when and if it's useful for them as editors.

So for me personally, I rarely go and make an edit like, you know, for somebody else's sake. I make an edit if and when I'm learning something or making notes for myself. So, back in the days when I was learning category theory, I made lots of notes in category theory and other people did. Before we get to your presentation, I want to quote something from you from the philosophy stack exchange. Like I mentioned, the mathematicians know about you from NLAB and maybe the philosophers are familiar with this because this is as you'll hear. To sum up, I think the lesson is the following. Once you have a formal system that formalizes what was previously quote-unquote just a natural philosophy, and I should give some context to this,

the questioner was asking on stack exchange, has philosophy contributed to anything outside philosophy in the past 20 years? And they were particularly thinking about the STEM fields. So your response was, to sum this up, I think one lesson is the following. Once you have a formal system that formalizes what was just natural philosophy, such as when Newton had his laws of motion nailed down, reasoning what that formal system will be far superior than what any philosophical mind unarmed with such tools may possibly achieve.

However, these formal systems, our modern theories of mathematics and physics, don't just come to us. They need to be found and finding them is generally a hard and non-trivial step. Once we have them, they appear beautiful and elegant and of an eternal nature. It makes us feel as if they've been around in our minds forever. But they have not. And this is the point where philosophical thinking may have a deep impact on the development of science.

Expand and talk about its relevance to today's talk. Right. Yeah. So actually, I was planning to end my today's presentation on such a note when we talk about maybe non-perdurbit of quantum field theory and maybe M theory and how a big problem there is that it's not just hard. I will end by saying that understanding non-perdurbit of physics like confined QC and strongly coupled solid state physics is the big or the grand

open question of theoretical physics of our time. And the question or the reason why it's so hard is that it's not just a problem that is already formulated in an existing theory where we just need to go at it and just follow the rules and it may just be tough but we essentially know what we have to do. No, it's because we're actually lacking some language. There's something actually missing. Some conceptual language level insight is lacking that will tell us what we're even looking for. And I think that is the reason why so little progress has been made on this subject, non-perturbative quantum

field theory, because there are a few proposals, of course, but by and large, there was very little to go by. And yeah, so I will maybe try to convince or I don't know, try to present some evidence today that going back to some really deep-seeming ontological foundations, maybe philosophical foundations, if you will, but all in the realm of formalized, provable math, does have something to say eventually on such open questions. And that, I think, goes very much in the direction of the quote which you just quoted, which arose actually from a, so that pose that you just quoted was made at an intermediate stage of the development that I'm going to present

today because there was at the time when I started appreciating this, some people have heard me say the Segelian aspect of parts of Topol's theory. And at that time, we were concretely interested in understanding an open question, which is what that post is referring to, namely the question of how to understand generalized differential cohomology theories, which are, as maybe I can say a bit more later, which are meant to be the actual full mathematical formalization of gauge fields, of higher gauge fields. And that wasn't quite clear actually. And that was really what I did since my PhD thesis, just thinking about how to properly

phrase the idea of generalized, meaning also higher gauge fields, like one expects to see in string theory, like RR fields, the supergravity C field and these things. What is it really? Like what is very fundamentally, what is it actually looking for when we say we want to build a model of the C field, say? And that was very much not clear. And yeah, so then we started toying with these modalities on topos, on actually topos of course, and at some point the solution appeared.

It kind of appeared in tandem with, for me at least, understanding also the role of what these modalities, the role that these modalities play in more philosophical realms. So there was a back and forth. I can't really quite tell what was first and what came later. But that was the question that this blog post, this Stack Exchange post was referring to, the question how to formulate generalized differential cohomology and hence high gauge shields in full generality and deeply. Yeah, and it turns out that it is related to this term of cohesive infinity topos, which is a term borrowed from Levere, who in turn was reading Hegel and trying very hard

to understand what is actually going on there and I would say actually succeeded, which is quite fascinating actually. Yes, you said that. Okay, so to be clear for the audience who hasn't read it, the question was about has philosophy inspired anything outside of the field of philosophy in the past 20 years? And you said yes, in differential topology, particularly with twisted cohomologies, and that's a new result within the past year or two. And then you mentioned Laverre and categories of being and nothing.

In particular, you said that there is a formal sense in which nothing and being can be combined to become becoming. And that's a formal precise mathematical sense. Is that what's going to be covered in this talk? Yeah, I have something on this. Maybe we shouldn't spend all too much time on it if I can get lost in these musings, but yes, that will appear briefly. And yeah, so I want to kind of go back to this creation story, if you wish, and show how once notions are set up that one can speak about these things,

there is a progression actually that starts literally from nothing in the technical sense of the initial object of some topos and then progresses to discover a whole lot of physics actually. It's at least fun. Also I should emphasize I'm not selling any theory or anything. I'm not going to present any hypothesis that people need to buy into. I'm just presenting some facts, some just provable facts that are just curious to look at and that everybody can make up their own mind about, but which certainly do seem suggestive

of something. Yeah. And I thought, yeah, So I thought I'll take the occasion that I'm speaking here to your audience on your podcast to go back to these old ideas and do a little bit of an exposition of them. Wonderful. All right, let's get to it. All right. Yeah, thanks.

Right. So as I just said, I'm going to try to give a bit of an exposition of work I've been doing over the last years or decades even that all revolves around a little bit of aspects of what one might call going towards a theory of everything. So these are some expositions of some theorems that you can find published in the literature. I put pointers where possible. But of course, I'm trying to give a bit of an overview and some gentle introduction first.

So very broadly as this little animation that I started with here shows is we wanna start looking at something at the very foundations of thought. So this is, as we'll get to, a little cartoon of an adjunction and category theory. By analyzing some structures that appear there, which are known to some experts,

but actually not widely known, we'll see some dynamics emerge, some dynamics in the platonic sense, in the realm of ideas where concepts will emerge, some dynamics in the platonic sense, in the realm of ideas, where concepts will emerge and eventually we'll be talking about gravity, supergravity in fact, the level of material supergravity and then maybe the M5 brain. And if time and energy is sufficient, I'm a little bit worried about energy, but then we'll get to some actual statements about a strongly coupled quantum systems with what

is called topological order and anionic excitations, which is what this little animated cartoon on the right is alluding to. And that connects to the actual current research I'm currently doing. All right. And why don't you explain the title as well, please? Right. The title of this page. Right.

So I was thinking about how to call this. And, you know, I must say there's a curious, how should I say, there's a curious aspect of 20th century science that some of the deepest thoughts have silly names, like theories of everything. It's not a very elegant name. I mean, it's not too bad. The really bad ones are like Big Bang and Black Hole, right? So these are terms I started out as jokes. So I was thinking of a more academic sounding name for theories of everything and Pantheorias is maybe a good thing. So I thought I'll call this talk about

theories of everything, but in a slightly more fancy version. I see, I see. That's what the title is doing. So I'm not claiming, I will of course not claim to have any theory of everything in my hand, but I do want to claim that I have found some fragments of what looks like should belong to such a theory, or at least which are noteworthy to take note of if one is interested in theories of everything in an actual deep sense, like in the sense of starting not just from the assumption of the notion of quantum field theory with its implied notion of spacetime and everything, but starting deeper, like at the level of pure logic.

That's maybe part of the fun aspect here that I can offer something about. Interesting. So are you going to derive space-time from logic? Or is that a hope of yours at some point? Yeah. So there's certainly a kind of emergence going on here, as I guess you can probably already see here from the title of this third item here, we will see that first the super point and then and then aspects of super space-time do emerge in some sense. I don't want to make too strong a claim here. I don't want to over claim anything, but there is certainly something like this going on where we're not just, you know, not just a space emerges in the sense that we already had a notion of space and then, oh, here's a particular one, like the bulk to a certain CFT, as people like to do it this time. The actual concept of space sort of arises, actually.

But it's maybe hard to explain without actually explaining it. So maybe we should just... Yeah, please. I'll start with something of a hot take maybe. So just to put what follows in perspective, I'm going to claim that what we want to be doing is if we're thinking about theories of everything and theories of physics in general, we want to be thinking about mathematical language as a metaphysical microscope. So I have these well-known quotes here just to appeal to authority or just to remind us all that this thought has been important to people in the past.

But I think it's being forgotten often these days where people point out a deep paradoxes or remaining paradoxes of fundamental physics by having chats about them in ordinary language. And the debates over many such topics like interpretations of quantum mechanics or the nature of the singularity and gravity and all these things, it's easy to have long debates about it that effectively lead nowhere because I think we're lacking or everyday language is lacking the terms to actually address the actual issues. And it's not meant to be surprising anymore, right, that if we speak about fundamental physics, which by the very nature is outside our realm of experience, we need a better

language than just the ordinary mesoscopic everyday language in order to speak about these things. And that language has to be mathematical of some sort. So I will try to show some aspects of how math can actually help with, you know, not just as computing quantities, but with computing, if you wish, qualities, like conceptual notions, to maybe preempt a certain debate that I know is going on in some circles. I want to maybe make the following warning.

There is in the public discussion on social networks and so forth, there's this idea, this meme that the main problem with contemporary physics, some people have this complaint, right? The main problem with contemporary physics and especially string theory is that it's too mathematical. That's what some people say, which is a curious thing because in view of what our forefathers here said and the way I perceive it, it's actually the opposite that we're actually lacking mathematical

formalizations of most of quantum field theory, certainly everything that goes beyond the standard perturbation theory and pretty much everything in string theory. And I think the resolution of this apparent paradox, how can it be that some people complain that physics has become too mathematical, like theoretical fundamental physics has become too mathematical and others like me are going now and saying, no, wait a second, it's actually the opposite, we're lacking math. I think the trouble is that even the meaning of what it means to have mathematical formulation of physics has been forgotten a little bit at times. So what I'm speaking about is having, first of all, precise definitions, like to have definitions, exact unambiguous definitions. That's actually more important even than the rigorous proofs that one can base on these.

It's a problem with lots of ongoing current contemporary physics that it's absolutely not clear at a fundamental level what people are actually talking about because things are not defined. And so that's what I want to mean by mathematical technology in physics to have some precise definitions. So I think what people mean when they say that contemporary physics, especially string theory, is too mathematical, they really mean it's too schematic. That's really what it means. Like when people talk about rains or maybe the swamp land or generalized symmetries,

all these things, it sounds very mathematical, but for the most part, it's actually absolutely not mathematical in the sense that if you point, if you show the discussions to a mathematician, you wouldn't be able to understand anything. What it really is, it's schematic. It's not filled with physical life, but it's also not filled with mathematical precision.

And so it's maybe a mistake to mistake that for math. Anyway, so that's my little hot take warning at the beginning. I think what people, what many people, not every person, but I think what many people mean when they say that modern physics is too mathematical, is particular theoretical or high energy physics, and that it becomes disjoined from experiment, and at some point you're using math that's been inspired by the math that's been inspired by, and you're exploring this mathematical space,

and it's not clear that unless you're someone like Max Tegmark who believes that the universe is mathematical and all mathematical spaces are realized somewhere, unless you believe that, then it's not clear that the realm that's being explored in much of the archive in the high energy section is of the realm of math that is joined with reality. So I think that's what's meant. As you know, on Theories of Everything, we delve into some of the most reality spiraling concepts from theoretical physics and consciousness to AI and emerging technologies.

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Thanks for tuning in and now back to our explorations of the mysteries of the universe. Joined with reality. So I think that's what's meant. Absolutely. I agree. But I was trying to add to this the following observation. You started by saying it's perceived as math because it's not related to experimental physics, right?

That's what you said at the beginning. It's no longer clear how some of these discussions actually relate to experience with physics. But at the same time, I think it's actually a mistake to say, oh, if it's not physics, then it has to be math. Because on top of that, much of this discussion is actually not math in the sense that it's not precise. It's not something that- Right.

It's not as rigorous. Yeah. It's not something that you could actually explain to a mathematician. They will not. And that's, I think, one reason why we're not seeing as much interaction between math and physics these days as maybe we did in the 90s. The mathematicians pretty much are lost. They do not know when there's this talk about all these brains and everything. They just don't know, don't have an entry point anymore. It seems like that's why I said it's schematic. It's neither physics nor math actually. Interesting.

So let's see how this works. So I want to start talking a bit about category theory. At the same time, I'm actually a bit reluctant to give anything like an introduction to category theory. So I'll drop some what I think are good ways of, you know, some buzzwords of ways of what I think is good to think about category theory. And we'll see, maybe if you have more questions, so anybody has a, if we can, of course, go deeper. But the main point I want to emphasize here in speaking about category theory now is with this

goal in mind that I mentioned at the very beginning that we maybe we want to move towards finding new theories of physics, where even the form of the theory is not currently clear, like you know non-perderbative QFT or M theory or something. To make some tangible progress for such matters, we need a language that can speak about concepts in a precise way, not just a math of quantity where we compute numbers, but some kind of formal system that allows us to move with precision and good insight in the realm of concepts and ideas and notions. And that's really what category theory is. It's like the conceptual backbone of mathematics. So as I'm saying here on this slide, beyond the mathematics of quantity, there is a mathematics of, and there's a certain order to these things, of structure, duality, quality, and effects.

It's interesting that the term duality appears here, which of course also plays a huge role in modern theoretical physics. The notion of duality that I'm going to introduce in a moment is not exactly congruent with what people elsewhere understand as duality, but there is a big overlap, actually. So it's not disjoint either. And so I'm going to claim, without much ado here, that the language for these four aspects – structure, duality, equality, and effect, is categories, in this order, adjunctions, and I'm going to speak a bit about those,

modalities and monads. So this technology, this categorical algebra, as it's sometimes called, is what Laverre at some point identified with what Hegel in turn called the objective logic. Note, I've written about Laver's theorem here on Substack. It actually went viral. It's a theorem that encompasses Girdle's incompleteness theorem, Cantor's diagonalization argument, Turing's halting problem, and even Tarski's undefineability theorem, as well as what self-reference means.

Feel free to read it and subscribe to the Substack. It's C-U-R-T-J-A-I-M-U-N-G-A-L dot org. Is what Levere at some point identified with what Hegel in turn called the objective logic by which is meant that it's a logic, it's not the subjective logic of making statements about what we think we know or do not know, but it's a way of speaking about the world, the Platonic world if you will, even though Platonic is not quite the right word here, the world of ideas that is just out there, it's objective.

It's something that is out there and we're trying to explore it. We're trying to get to the roots of reality in a sense. That's what this term means. Here's a lightning crash course on what categories are about. Yeah, you should stop me maybe if this gets too boring. No, this is great. Never feel like you're getting too technical on this podcast because this podcast is known for not skimping on the technicalities and the audience loves and they crave the technicalities.

All right. Okay. So let me give here a couple of ways of how to think of categories. So at some technical level, but that's maybe the most boring level, of course, categories are like directed graphs, but equipped with the notion of how to compose the edges of the graph, such that the composition satisfies the expected properties, it's associative and unit. And then just in order to have more jargon, one calls the vertices in these graphs objects and the edges morphisms.

And these terms come from what you might call the archetypical examples of categories. So as physicists, what you want to be thinking of, I mean, there are many uses of category theory in physics, but the primordial one I think that you should all be thinking of is categories whose objects are spaces of sorts, physical spaces, but also more abstract spaces like modularized spaces, classifying spaces, space quite generally, and where the edges on these graphs, the maps, as they call them morphisms, are just maps between these spaces. And this is actually going to be a big theme in just a moment that to, even though it's maybe not often fully recognized, but in order to do physics, one, like just Lagrangian ordinary classical field theory, one very quickly runs into the issue that one is dealing with spaces

that are actually more general than what the standard textbooks allow for. Like already, if you're just doing like scalar, bosonic, ordinary classical field theory, if your base space time is not compact, then the space of fields is just not a manifold. It's just not a smooth manifold because if the base is not compact, it's no longer fresh.

So you're outside the realm of what traditional differential geometry actually offers. And not all, but parts of the difficulties that afflict field theory result from the fact that one is dealing with these generalized spaces. And the moment you add fermions to your classical field theory, the spaces become actually super spaces, even if there's no supersymmetry. Just because there's no supersymmetry. Just because there's fermionic coordinates there, there's a Pauli exclusion principle that says that suddenly we're dealing with spaces, some of whose coordinate functions actually square to zero, which is of course nothing that ever happens on an ordinary manifold. And so again, we're out of the realm of what the traditional textbooks offer us.

And I think a very good entry point to category theories, and I'll get to that to the actual examples is think of categories as providing the context where the actual spaces that physics actually needs live and where we know how to map between them. So what it is, for instance, for the, say, what it means to have a curve in the space of fields, on the space of super fields, on the orbe folded space of super fields and all these things. And so the standard way of drawing these things is with these diagrams shown on the right here.

So the XYZ are these objects in some category or think of spaces of sorts and the arrows are the maps between them, the morphisms. And for any composable pair of such, we have the corresponding composite. So that's a cartoon picture, you have a category, where I'm omitting, of course, what most textbooks will amplify much more is the fact that this composition operation is supposed to be associative at unit.

So there is identity, morphisms, and all these objects, and so forth. Another actually very useful point or like default point of view on categories is the second or third item that I'm showing here, which is not actually as it turns out disjoint from the previous one. So do you think of the object as being data types? Think of this as a procedural declaration in the theory of computing and programming, where the objects are kind of data types consisting of all data of a certain type and the maps between them are programs that take one to

the other. So if we have enough time, we may get at the end to the aspect where we talk about maybe quantum language, quantum programming language that emerges here. And it's in that context that this perspective is the dominant one. But these perspectives are not mutually exclusive. Like a space is the data type of the data that is a point in that space or more generally any figure in that space.

Yeah, okay. We'll get to this. And then for those since, yeah, just to connect maybe to some currently fashionable verbiage, among categories is groupoids, where all the maps are invertible. So a groupoid is a generalized symmetry. It's a bunch of objects, a bunch of things that are not just sitting there as they are in a set, but that are connected by what you should think of as gauge transformations. So when these maps here are invertible, when they have inverses, then they identify these two objects in a specific way.

It calls them isomorphisms, of course. And in particular, an object can be identified with itself in several ways. Such a morphism can go from an object to itself, and that is just the operation of a symmetry group on that object. And so the special case of categories for all morphisms are invertible are groupoids and they should very much be thought of as being the incarnation of the idea of symmetry and specifically of gauge symmetry. You should think of groupoids, I can think of groupoids as being like sets equipped with

gauge transformations. In fact, a very important example of a category is for instance the groupoid groupoid of, say, Maxwell gauge fields on a given space time. So the objects in that case are gauge field configurations and morphisms are little gauge transformations between these. And so if from that perspective we lift this condition that all morphisms be invertible, then we have pretty much what these days people want to understand on our non-invertible symmetries, namely a situation where there's operations between things that behave a whole lot like what a

symmetry – you would like a symmetry to behave except that they are not actually invertible. And that's what's captured by these morphisms here. So I think if you go and make anything about non-invertible symmetry as precise, you will in the end have to be talking about categories. And then of course, so this is what is shown here. Then of course, once you have these categories, you want to talk about finding images. And we'll see concrete examples of this in just a moment to give some life to this. But let me just say it in the abstract right now.

So of course if you have two categories, C and D, you want to say what's an image of that category in that category? Well, it's just a map from C to D that maps all the objects and all the morphisms from C to D such that the structure is preserved. The only structure we have here is the composition of these morphisms. So such a functor indicated here by this assignment, such a functor f, takes every object to another object in another category, such that it does not matter whether we first send morphisms over one by one here and then compose them or send over the composite. Given somehow that we have this extra structure of the morphisms here that we're in a sense one homotopical step higher than in the realm of sets means that such functors in turn are objects of some category. They have relations between them and particularly they have symmetries between them.

There can be something called natural isomorphisms between these functors themselves, and they can be non-invertible again. And that's what's called natural transformations. So what I'm showing here is one functor F, and here's its image, another functor G, and here's its image. And then we can ask, are these two images deformable into each other? If you maybe think of, as you can, for special situations where we're looking at the categories that are the fundamental group of spaces, if you think of these objects here as points

in a space, and of these morphisms as actual paths in a space, then in topology it's well known the notion of a homotopy between two maps where we continuously deform a path here to a path here and this path here to this path here such that the composition of these paths is respected. And now we're just abstract from that situation. If you will, we don't have actual paths here. We just have these formal maps. We can still ask, well, let this object be kind of be transformed to another object,

so let there be a non-advertisable symmetry, if you will, let there be some morphism here that transforms these objects into each other. And then we want to say, well, this result down here counts as a deformation of this result up here by these deformations if this is compatible. So if these squares commute. So there's something actually I'm not showing on this slide. This is the standard, the default understanding that whenever we draw composable arrows in

different ways that go from the same object to the same object here, that we understand that we mean that we mean that the square is actually filled by inequality. So that we mean that the composite of this morphism with this morphism equals the composite of this morphism with this morphism, all this happening in the category D. So that's understood here. I will not, some people write a little symbol here. I will not do that.

I, all squares commute. And later on, of course, they will not commute. There will be some further homotopes in there. But for the moment, they will just commute. So that's the basic thing. You open Wikipedia and any of many books on categories here, they will tell you that this is what a category is. And then I think at some point, what happens is that for some, you know, there's always a little bit of debate. Do we even need to care about this?

Is this actually worth our time? This is trivial. If this appears trivial and not logical, then that's good. That's what it is. This is not doing much. And I now want to make another hot take. I will actually oppose now, well, an often quoted saying in category theory. And we'll want to highlight that it's not actually true.

So this may originate with this reference here, for I've fried, where we're just certainly historically somewhat accurate where people said, well, so what are we trying to do here? There's the saying that goes, oh, we just introduced categories here in order to speak about the functors. But why did we introduce the functors? Well, actually, we wanted to understand what are natural transformations. So the saying goes that the whole point of the setup is here to speak about natural transformations.

And that is certainly the case to at least half the extent for the original article by Eiland-Berg-McLean on category theory where they actually asked what is a dual object. So they essentially observed that at that time in algebraic topology, people would speak about certain things being naturally isomorphic to each other or naturally transformed into each other without there being an actual definition of what it actually means to be natural. Right? So this goes maybe back to our original discussion of speaking in an informal non-mathematical language for things that should have a formal definition.

And part of the motivation at least was of Ionbeck and Hulme-Laine was to make precise what it actually means for the two things to transform into each other in a natural way. And the definition that came up with is exactly what I just said, the commutativity of these things. So the transformation is the fact that this functor goes to this functor and its naturality means that we have these non-invertible symmetries coherently relating these two things. So sometimes people say, okay, so category theory is just all about these maybe categories, functors and natural transformations.

And then it does look a little bit, what's the right word? A little bit thin, right? Then we're just really looking at graphs with some composition. We can do some images. We can call certain things categories, but if we don't, we haven't really lost too much speed because after all, I mean, this is nice here, but you know, you could probably get

along without maybe making a big deal and giving everything here a name. Wait, I don't understand why you call it thin. Well, I want to make a point now that there is more to category theory, a whole lot more. Like the actual theory, well, thanks for interjecting. I want to make a point here that something deep happens now in the next step, actually. This is just the bare bones substrate on which category theory builds. The actual theory only happens in the next step. So this is this little progression that I'm indicating here. So what I've shown so far is the hierarchy of categories, functors, and natural transformations.

And that in a sense, that defines everything that is about categories. But the actual theory, the category theory, the non-trivial aspects of it, also I would say the dynamical aspect, the surprising aspect where the real meat is, where stuff is happening, where difficult proofs have to be proven, is in the next step where we introduce adjunctions and out of this growth and the universal constructions

that will help us let some physics emerge and also monadic algebra that plays another role, which maybe we should talk about another time. So I wanna emphasize that beyond categories, functors, and natural transformations, there's a next ingredient in category theory, and that is, in a sense, that is the important one. That is the notion of a junction. Of course, people will have heard this, but this is really where something not completely

obvious happens, and all the deep parts of category theory are all related to adjunctions. So what is an adjunction? An adjunction is a pair of functors L and R going between categories back and forth. So one goes from C to category D, the other one goes the other way around. Called L and R for the left and the right adjoint. And that makes them being adjoint. They're equipped with natural transformations. Remember that was these deformations of functors?

From. So, right, these two functors, they go back and forth. As we compose them, they give an endofunctor on one of these categories or on the other. And so we want a natural transformation that deforms, if you wish, our composite going back and forth to or from the identity functor, the one that just sends everything to itself. And moreover, so I'll show an example of this in a moment, like a physical example of this. Let's maybe just try to digest this.

So we have these things going back and forth and they satisfy identities. And these identities are usually shown with a diagram drawn slightly differently than what I'm doing here. So I thought it would be fun since I'm going to claim that junctions capture the intuitive notion of the philosophical notion of dualities. I draw what's usually called the triangle identity. Suggestively, yes, I see. Yeah, I draw it suggestively like this.

So what this is showing is, what I'm not doing is I'm not showing the identity morphism. So there's an identity morphism, identity factor here from C to C, which since it's the identity, I can just as well not show. And then we have L first followed by R, so this is this.

And what this is showing is that we have a natural transformation from this identity, which I'm not showing here, so this is this. And what this is showing is that we have a natural transformation from this identity, which I'm not showing here, to this composite. And the co-unit, as it's called, this other transformation goes conversely from R followed by L to this identity here on D. It goes in this direction. So usually these diagrams are shown by expanding out these identity morphisms here to have some finite widths, and then they look like, I don't know, like a poached egg or something.

But I just wanna amplify, it's the very same diagram that I'm not changing anything here. This is just the definition of unit and co-unit in an adjunction and the zigzag identities, if you draw them this way, have this fun yin-yang form where we claim that the composite of these transformations here has to be the identity on the L-functor and the converse composite of these transformations have to be the identity here.

So that is one way of abstractly defining what an adjunction is. And what I can already see maybe that the definition of adjunction is somehow on a different level than these previous definitions. I mean here, right, we had some, it looked like, if you think of these as graphs, we're basically dealing with discrete spaces or something that looks like discrete spaces, at least, and there are images in each other. It looks a bit like discrete topology, but here, this is something that you would not maybe have guessed if you're just thinking about topology. So there's something important and deep going on here. And I'll just mention some examples. This is maybe an exercise that everybody needs to do by themselves, But let me just mention some. So I want to claim that the notion of

adjunction, and that's not my original claim, this has been amplified by some people before, is that adjunction really captures a whole lot of what is intuitively the notion of duality. And as a very simple example, one can make the following exercise. We can ask for the, one can think of the natural numbers. The natural numbers as a category where each number is an object of the category and where the relation that one number is smaller than another number is a morphism. So where a morphism goes from one number to another, if and only if the previous one is smaller than the other. That's the so-called, that's the partial order on the integers regarded as a category. And it's a fun exercise to see that

if one now looks at just the even numbers and just the odd numbers, they have functors into the full set of integers, the full category of integers just by the embedding, and these functors have a joint. And so there's an endofunctor on the integers that predicts every number to its closest smaller even number or its closest larger odd number. And these two functors are joined to each other, they dual to each other expressing in a nice way the you know what you what intuitively think should be should be a very simple form of duality between even

and odd. Actually that example if one actually works it out that example is kind of nice and that it it shows it shows both how being even is in some sense opposite to being odd and and at the same time, at the same time there's a certain unification going on, because of course, just as a set, or just as a category, the even numbers are of course isomorphic to the odd numbers, with isomorphism being at one. So there's a whole lot, or how should I say, surprisingly, this baby as this example is, it has some nice philosophical insights to share.

So we could look more into this, but I just want to mention this. And then there's a primordial example that will drive our emergence story in just a moment. It's in a category where there is an initial object and a terminal object and the functors that are constant on these objects are rejoined to each other. So an initial object, and this will be our model for the emptiness or the nothingness, is an object in a category such that it has a unique morphism to any other object. To be thought of as the characteristic property that the empty set has in the category of sets. There's a unique function of sets, a

map of sets from the empty set to any other set. Maybe it's the one that takes the non-existing, that takes no element to nowhere. And similarly, there's a unique function from any set to the singleton set, the set with a single element, it takes necessarily every element to that single element. And from that you can already see that's actually a simple example of an adjunction going on here where the unit and the co-unit maps are these unique morphisms from the initial to the terminal object. So terminologically people can understand that the initial may have an analogy to something empty.

But then terminal usually sounds like something at the end, which would be the highest, like the highest form of infinity. But terminal in this analogy is actually the unit set, the set with one element. Yeah. So the terminology is, I guess, motivated from partially ordered sets. If you have a set with an order relation and you think of the relation one element being smaller, say, than the other is being amorphism, then the initial object is what some people call the bottom element of this or is it the smallest one and the terminal one is the top one, the top element where everything converges. In this sense it's charming that somehow every string of composable morphisms ends there.

I think that's why it's called terminal. But yeah, you could maybe find different words for this. And then, yeah, so just as a side remark, I wanna actually mention here, not sure if we actually get to this, but I wanna mention that some of the, since I'm claiming that adjunctions are actually dualities, that you should think of them as being

the mathematical formalization of the intuitive and or philosophical notion of duality, that at least some of the stringy dualities, dualities in string theory, which of course is a term that has a completely different history to it, and it's not used very systematically always, but some of them are actually just plain examples of junctions. And among them is, and I'm quite fond of that example, is the notion of what's called double-dimensional reduction, where you have some brains and their charges and some higher dimensional spacetime.

And you want to say that there's an equivalence, that you can equivalently regard the system from the point of view of lower dimensional spacetime, where some of the degrees of freedom that you had in higher dimensions are incarnated as so-called Kaluza-Klein modes. So that's a very important basic kind of duality in string theory. And that is actually an example of an adjunction. Well, it's an example of a higher adjunction, higher categories, but it follows exactly the same pattern.

And in particular, especially examples of this, one finds T-duality, or at least the non-topological T-duality, and in fact the duality between M and 2A, so the reduction of 11D supergravity to 10D. And so I'm giving some references here. There's a deep and detailed story to that. I don't want to get into this right now. If we have time, we can of course get into this right now, if we have time, we can of course get into this, but I just wanted to make this a side remark to indicate that besides

these baby toy examples that I'm mentioning here, there's really deep adjunctions. Of course, there's many, many more examples of adjunctions, but this is one that I think is, for physics inclined audiences, immediately recognizable as something important. Cool. Now, let's get our hands dirty, or how should we say? Let's see this in a bit more tangible form. I'm going to be talking about topos now a little bit. And so I'm giving a specific, how should one say, class of categories.

I'm highlighting a specific example of categories that is, I would argue, actually maybe the first one that any aspiring physicist who wonders if she should learn category theory should think about. It's not actually highlighted very much in the literature. You will not, besides maybe our writings, you will not find this currently being amplified too much. But I claim that this is actually the golden road to understanding, to get to the heart of the matter here. So this goes back to what I

announced at the beginning, the role of categories is helping to come to grips with the generalized spaces that play a role in physics. And I want to be looking at concretely as categories of categories of such generalized spaces. And how do we think of them? Well, being physicists, if we are physicists. So there's a surprising story in physics and math and the physics and math histories, where physicists go and write down what looks like naive computations in local coordinates without worrying or thinking about what this would actually mean globally.

A very good example is supersymmetry. We just go and say, well, what if my coordinate functions do not commute? What if they pick up a sign? And then just run with it. So you just, in physics, you just do something that can algebraically be done on a coordinate chart. Or you say something like, which is maybe even older,

you say, oh, let epsilon be a quantity that is so small that if you square it, it becomes zero. I remember actually, this is in my, when I studied physics, we had an experimental professor who actually explained to us the volume formula for the sphere in just this way. Just assume there's an epsilon that is so small that you can square to zero. And so these are the big tricks that you can do in coordinates, right? Coordinates square to zero and or they inter commute with each other or they do other funny things which you can easily do in coordinates and physicists have to very good avail have

used this fact without thinking about what it means usually what it means actually globally for space if its coordinates behave this way. And so here category series actually comes to the rescue and in its guise as topos theory provides actually a fascinating accurate characterization of what is going on. So it's a way to kind of bootstrap generalized globally defined, hence non-perturbative as people say in a moment spaces, from just declaring what happens on coordinate charts. Okay, so enough of an introduction, let's just look at it. Suppose you have a category of what I'm going to call generalized charts. So we look at some examples in a moment. So these Cs here could just be like ordinary Cartesian spaces, RNs for any N. So just the actual original Cartesian coordinate charts of anything.

And maps between them say any smooth maps between RNs. Or they could be super Cartesian spaces. Cartesian spaces with some super coordinates attached to them. But we think of these Cs as being kind of super simple, super naive things that we can handle algebraically maybe. We could even say, and we'll see this in a moment, that C may have some new potent coordinates. Well, write some epsilons that square to zero. What will we mean by this? Well, we will just mean, we'll kind of define

such a C to be its algebra of functions. And we just declare that the algebra of functions is not just an actual algebra of functions, but some other algebra that has no potent elements. And then we just declare that these maps between them are actually maps going the other way around, pullback morphisms of algebras. So this realm is the kind of naive setup of easy algebraic manipulations on simple objects that look like little contractible coordinate charts, maybe some extra bells and whistles.

And now we want to do kind of a bootstrap from that. We want to say, okay, if these are our generalized charts, what is the most general space that I can kind of build from these charts, or that I can understand from these charts? So what what is the global geometry that is kind of modeled on these charts? And the simple idea is the following. It's a fun bootstrap exercise. So we say suppose we had Boltface X that is

supposed to be by a generalized space and I don't have it actually yet. So we're bootstrapping it into existence. But we say well if we had it or once we have it, well, we will be able to ask what are the ways of plotting our charts in X. That's kind of the name of the game that we want to say X is a space that some are modeled on the geometry embodied by the seas. So there must be some way of mapping seas into the charts into X. Okay. So X is something that's unknown, but you want to know it. How do you know it?

You take something you do know and you probe it with it. Exactly, yeah. Exactly. So it's a very physics operational kind of definition. We say, how do we actually understand space? Well, we understand it by throwing stuff into it like light rays and see what happens. So we probe it. In fact, it's very similar also,

or very related to the terminology of what's called probe brains in string theory, where you say, well, here's some stringy background, and in order to understand what's actually going on there, well, let's suppose we have a little brain, so a little C, a little simple thing, that traces out some trajectory inside, and let's study these trajectories.

That's really the idea here. It's not only physics, it's just in everyday life you don't know something, you go out, you touch it or you look at it and that's how you come to know it or know more about it. Yeah, absolutely right. So it should be a very intuitively obvious thing to do and that's what we're doing here. Even us when we're speaking just with human communication, you don't know someone else, so you probe them, you ask them questions, and they come back to you with something. Right, exactly, yeah.

You probe them in other ways, that's for another podcast. There you go, yeah. Very good, exactly. That's true, and in fact, it's very good that you're saying this. In fact, the notion that I'm exposing here is really, it's going to be the notion of a sheaf. It's so general that of course many other concepts will fit on it that are not manifestly generalized spaces for physics as I'm presenting it here, which is just the angle I'm taking here. Indeed, it is being used for all kinds of things. More generally, one could say if you have any small category, then the pre-shifts on it are the things, it's the category of things that can be probed by these objects in the small category. Okay, very good. So why is it a bootstrap though? So what do the

quotation marks be? So the idea is now that we say we have this functor, I didn't actually write this, I see now, which goes from the category of charts to the category of sets. So it assigns to every chart, every generalized chart, a set. Just a plain set, which we think of as being, we think of it as being the plots. I call them plots. It's a technical term in this business. So we think of them as being the maps, the would-be maps from C into X, only that this hasn't actually been defined yet.

So that's why I'm calling it a bootstrap. It's defined hereby, and we need to make sense of it. So a priori is just any set which we're thinking of as being the set of maps from C to X of admissible maps. Like this would be the smooth maps or the super geometric maps, whatever structure is encoded in our charts. Okay, so we have these sets and that alone can't really be sufficient to understand our space. But now if we also know how these steps change as we actually move around the chart, so as we actually do more of an actual probe,

we don't just have one of our charts sitting there, but we move it a little bit. Like we have a map here, so this might be a smooth map between our ends or super geometric maps between super Cartesian spaces, something like this. And those aren't to be interpreted as coordinate transformations? Yeah, you can.

The thing is I'm not requiring the map here to be an isomorphism. So it can actually, you know, it doesn't have to be an invertible coordinate transformation. It can be, but it can just be any smooth function actually. We're allowing, we're allowing, that's why I'm calling them also generalized charts. I'm not at this point actually requiring, I'm not speaking of manifolds that are defined to be locally isomorphic to one of the charts. I'm defining something more relaxed and in fact the whole point is that we get something that is more general than manifolds. It's subsumers manifolds, but it is more general, where we just speak about not having just

is more general, where we just speak about not having just you know, charts that are isomorphism onto the image, but just any maps, that's why they're called plots, any maps from the chart into the space. If that's what is called a plot, then why don't you define, why is that an equality sign and not an equivalent sign, like with three vertical lines? Why is it here? Why do I have quotation marks here?

No, right there. So why is that not a definition? Well, the thing is, you know, that's why I'm calling it Buddha. It will actually, it will be a truth. It will be true in just a moment. But right now, you see C and X do not actually live yet in the same category because X hasn't even been constructed yet. Or, you know, X starts out kind of defining x as being this functor, this assignment. Of, it's the assignment from any c to a set.

And as such, we don't really know yet what it would actually mean to map c into x. Because c and x are currently conceptually on different footing. Right? c is something we have in our hands, some Rn, and X for a moment is just the rule to take any C to a set, right? Yes. So right now we don't know yet what it would actually mean to have this arrow here, and that's why it's quotation marks, and that's why we can't quite say it that it's actually

equal because we don't even know what this is yet. But in just a moment, in just a moment, it will all fall into place and we'll just remove the quotation marks. And that statement that we can remove the quotation marks, that incidentally is the UNEDA lemma, which is of course one of the emblematic lemma of category theory and that will make this work. Okay, so this is actually fun. So let's think about this. So we're going to now say, well, if I have another chart, then my rule, since I assume I have X, so if I have X, well, I will be able to probe it also with C prime. But then if we think of these sets of plots as being such maps, even though they are not

yet, but if we think of them, well, then given any map of charts going upwards here in my diagram, then we will get a map from plots going downwards as indicated here. Right? Given F, I can use F in order to take any plot of probe form C to a plot of probe form C prime by pre-composing it. Right? So this is a computing square, so we regard it as a computing diagram. So phi prime here is the image of composing phi with f. So this is called pullback, pullback of functions. So this just says that if you have any map of charts, you can pull back the corresponding plots.

So we want this to be true. This is the image we want to realize. And so here we're declaring it should be true. There should be a function. This was just an abstract abstract set. There should be a function of sets taking every element here to some element here to be thought of as this pulled back plot. Okay.

And so, and then we require this really to be a functor, so we just require this assignment with respect to the composition. So if I have two maps between the charts and I pull back consecutively, that should be the same thing as pulling back along their composite. So that's just the basics of composing maps. And there should be identity maps here on the seas and pulling back along an identity map should be an identity. So that's some very basic consistency conditions

on our bootstrapped X, right? To say, well, if I have such a functor, I have a good chance that I actually know X. So what I'm showing here, that is called, of course, the technical term for this is a pre-sheaf. That we say it's just technical jargon. That we say such an assignment of sets, of probes, of plots to any element, to every object in a given category, such a functorial assignment is called a pre-sheaf.

If it's contravariant, like if arrows going this way turn into arrows going this way, then it's called a pre-sheaf. If it's contravariant, like if arrows going this way, turn into arrows going this way, then it's called a pre-sheaf offset. I'm actually going to call it a sheaf. I don't know. At this point, I didn't really mean to spend too much time on it, but let me just say it in words. I don't have a graphics for that. Not right here. We could jump to it somewhere. There's one more condition that we want to impose here, it's a locality condition. Let me just say the words. Let me say, well, to probe X, it should be sufficient to use small probes. Meaning, suppose you have a C here, think of an Rn, say an R2, so you're mapping some surface into your would-be space. But now suppose you have covered the surface, you have forms an open cover, you have covered it by other copies of R2.

So we have sub, like little balls on that surface and we're covering it thereby. Well, if we know where all these little surfaces, how they go to X, and if we know that their restriction to the intersections agree, so we have a bunch of probes of our space by little disks or by little RNs such that on the intersections of these RNs, these probes coincide, well then we want to be able to say, well then that's just as well as having mapped our whole surface into X, right? We know all of these patches of it, where they go. And the requirement that this is the case, so first of all that one has a notion of what it means to cover coordinate charts by other coordinate charts. That information is called a coverage that makes this category what's called a site.

And then the requirement that our functor of plots respects this notion of gluing, that is called the sheaf condition, and a functor that satisfies this is called a sheaf. Okay, so let me see if I got this correct. So you were saying that physicists ordinarily live in charts. And the chart is the mathematician's way of speaking about coordinates. And then differential geometers know this, and that's why they work coordinate-free. Now, you're saying, okay, well, the differential geometer's way is you first define a manifold, and then you start to define charts.

So you go from the manifold and then you start to define charts. So you go from the manifold down to charts, but you're saying, well, what if you don't want your spaces to just be restricted to manifolds? What if we want to generalize even differential geometry to different geometries or different spaces? Then that is what's called a gross topos. And you do so actually in the opposite way. Instead of you first define your object and define the maps from your object to charts, you're doing it the other way. You're saying, okay, well, we have to start from what we know and then probe upward. Yes. Of course, with compatibility on the overlap and that's referred to as gluing.

Yeah, no, I think, no, that's a good way of thinking about it. I might just add that, of course, secretly, even in the Stunna textbooks and manifolds, secretly, of course, you need to know what RNn and smooth maps between Rn are before you can go further. That's something you know at the beginning. But it's of course true, as you said, that a big emphasis is put these days on the global spaces, the manifolds, in favor of the charts. But that can only be done after the definition of manifold has been made, which very much relies, of course, on the notion of charge, right?

And so that's what's going on here too. We want to say what are these global spaces, both as X, and we need to say that in some way or other, they're actually determined determined by charts that also for an ordinary definition of manifold that the actual notion of what is a smooth function between charts that actually determines what is a smooth function between manifolds more generally. Got it. So we want to say, okay, we want to say a generalized space modeled on these charts is such a sheath, such a contra-variant functor, with such a consistent assignment of plots.

And now, actually, of course, now that we're talking categories, I also want to make the collection of all these generalized spaces a category. So next I want to say, what is a map between the generalized spaces in turn, right? What if I have x and y, and let's go there, what is a map from x to y? And a quick way of saying this is that since x and y were defined as being pre-sheaves, then a map between them is not just a natural transformation between the corresponding functors. But let's look actually, let's see that this abstract idea again has a very concrete and very satisfactory incarnation or realization for our intuitive picture of spaces probed by charts. So we want to say, what is a map between these generalized spaces from boldface X to boldface

Y called boldface F now? Well the idea is to again, pretty much is what does for manifolds, to reduce it again to what happens on charts. So in order to know what this map does, well, we remember that our plots from charts were meant to be like maps into the space X. So suppose I have a C here, a chart mapping into X, then if this is really a map of these spaces

that preserves all the given smooth, whatever, super geometric structure, well then it should be possible to compose these maps, phi from the path. That will give us a chart, a generalized chart of Y, because now this map from C to X has been pushed forward to a map from c to y. And so that is what we make the definition of f. A map of, since our spaces were defined, bootstrap

by saying what their plots are, we say a map between these spaces is whatever takes plots to plots consistently by a would-be operation of post-composition. And so the requirement that you have this for all C compatible again with the maps between the charts C is what makes this a natural transformation between these functors by the rules that we introduced before. And So this gives us a category of generalized spaces modeled on any category of charts. And that such a category is called, that's the jargon, that's called a topos or actually a grottopos of generalized spaces, which is the shift topos as people would say on the site of charts. So it's just jargon for exactly what I just said. Okay.

of charts. So it's just jargon for exactly what I just said. Okay. And so my running claim here is that even if you will rarely see, I mean, I have some references here, but even if you will rarely see like spaces for physics be explained this way, it's kind of in the background. This is what one can see is happening when people actually handle generalized spaces. And I want to amplify here that this idea, this is ancient actually, this was promoted by Rodenweg way back in 1965 when he actually started saying that this is the way to do algebraic geometry.

Instead of talking about locally ringed spaces, one should regard schemes and more general algebraic spaces as being just such assignments where now C is not a category of smooth choice like it is for the applications that I have in mind here, but a category of affine schemes, right? Of formal duals of rings, algebra. So it's an old idea, but somehow, I had a quote which I didn't dig out now. There's some quote, like many years later, probably writes something where he's frustrated

that people are still talking about no clearing spaces instead of using this. Just to amplify, this is also a very good example of how the language of categories is actually useful even in its basic form. Remember, we haven't talked about adjunctions here at this point yet, so we're really just in this lower stage of categories, functors and initial transformations. We used exactly that to do something useful that is simple, but actually it already makes this terminology worthwhile, just to say what a generalized space is. So here's the statement about the unilateral lemma, which I mentioned before. One can now ask the following, right? And so this comes, this now gets us to the question

with the quotation marks here. So here's an example. I should give an example of what is a generalized space. And the default example is, well, the charts themselves are generalized spaces. Well, how so? Well, we need to say what are the plots of an actual chart. Well, we just take them to be the actual maps, right? We want the plots to be like maps, but now if the space C is a chart, well, then we can just use the maps because we had already assumed that we know them, right? That was our starting point. They had a category of charts. So we can take, we can regard any chart, generalized chart, as a generalized space by declaring its plots to be just the actual homomorphisms of plots into it, of, sorry, of charts into

it. So then there is a priori a little chance for inconsistency here because now C now suddenly exists in two guises. It exists as our original charts, which end up the definition of plot C and now it also exists as two guises. It exists as our original charts, which entered the definition of plot C and now it also exists as a generalized space. So there's now these two things that a priori could be different. There's the plots of C as just defined, the bootstrapped, if you will, declaration of what should be the admissible maps from any chart into R, C. But since we just built a category of generalized spaces and made C a generalized space, we can

also map into C as a generalized space. And a priori, at first sight, it's not actually so obvious maybe, or at least it's not completely self-evident right away that these things are actually the same. But they are the same. There's natural isomorphism between these two functors. And that is the statement of the Unital Lemma, which is maybe a fun thing to notice. Everybody is sort of the Unital Lemma and it gets mentioned all the time in category theory. And here we see that if you think of categories as being categories of general spaces for physics, then the Unital Lemma plays this very crucial pivotal role here as saying that's actually consistent.

That functorial generalized geometry for physics is actually consistent. So that's a fun fact maybe to notice. Now let's get to some actual examples. So this is from this encyclopedia article that I have on high-topos theory and physics, just copied verbatim. Oh, I love this. I watched a talk of yours, I believe it was at the Wolfram Institute, or it was on higher topos theory from a few years ago.

So I love this part. Okay, very good. Yeah, in fact, that was January of last year. I was visiting in Beijing at Tsinghua University. I was giving the talk there, but it was remote to the Wolfram School. Right. So here's the, I want to say now what are the charts that we actually use in physics for the most part, like when we're talking about field theories on with Fermi.

Yeah, okay. I mean, let's go through this. So let's look at what these charts can be. I already mentioned this most emblematic version, maybe just the Cartesian space, the original notion of a chart. And the indices here, they run, right? So I'm not writing for all. So if I write Rn here, I'm thinking of this, it could be an Rn or an Rm for any natural number n here. So we're looking at any Cartesian space here. Since, right, just to repeat, since we're not currently requiring anything to be locally isomorphic to our charts, so we just probe by all possible things we have.

And so probing with Cartesian spaces, the kind of geometry that this models might be called differential topology. We're talking about spaces that are not necessarily manifolds, but that have enough smooth structure of source so we can probe them by our ends. In physics terms, maybe I should say more about this, but let me just say it the way it says here right now. In physics terms, that is the setup which allows us to speak about fields in the sense that in the schief topos over the category of Cartesian spaces, the mapping spaces from say space-time to any coefficient space, say the complex numbers or some liage, but these mapping spaces, which in general are not manifold, because

they're too highly dimensional, they just exist. I should say that this is like the basis of all of physics, all of modern physics on a business card. Now it's not the standard model, but I mean it's the language that underlies modern physics. That's right, yes. That's right, yeah. Exactly. I like this. It's quite impressive. Exactly. Yeah, that's the substrate on which things are built. Exactly, yeah. That's the substrate on which things are built. This stage is actually not as famous as this one, but it logically comes before it in some sense. I'm using the symbol little d here, which is for disk. So you should think of these things as being infinitesimal disks.

Like sometimes people say halos. These are like points in an RN with just an infinitesimal neighborhood of a point in RN around them. And the infinitesimal neighborhood is of order K. That's why this has two indices. So DN means think of a small disk in RN and the K means it's so tiny that the K plus first power of any function on that disk actually vanishes. The disk is so small, right? That its coordinate functions, if you take them to the k plus first power, don't just become smaller as small numbers, smaller than one, tend to do, but actually vanishes. Hi everyone.

Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here.

Hit subscribe and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L.org. Kurtjaimungal.org. But it actually vanishes. So the experimental physicist that was your teacher would have k equals 1 in his example from before of squaring to zero? Yes. That would be the first order infinitesimal disk.

Yes. Exactly. Yeah. You can also see in the original Feynman articles, I think in his thesis, where he introduced the passing, that he is also playing with such epsilons. This is a fun story because physicists must have been using this even long before Feynman, I suppose, probably for 400 years or something, this kind of trick. It took mathematicians quite a while to fully come to grips with this.

But it's exactly in this way. This now goes by the name synthetic differential geometry, where one considers shift oposites on such probes or similar with infinitesimals. And then notices that inside these shift oposites one has, that's why it's called synthetic. Inside these shift oposites, they have generalized that like the infinitesimal disc then exists as an actual object and all other kinds of infinitesimals. So that I can for instance, go and define

inside these top bosses, define a tangent vector to any point to literally be a map from the infinitesimal interval into the manifold such that the, you know, the foot point of the interval goes to that point. So it becomes literally an infinitesimal, an integer vector becomes literally an infinitesimal path in the manifold. That's why it's called synthetic because the usual heuristics of what differential geometry is about kind of becomes a synthetic reality. It becomes synthesized to something that actually exists.

And for physics application, this means this is really where calculus of variations takes place. In particular, so this is a bit technical, but in such a synthetic differential topos, so once you have such charts among your generalized charts, you can very naturally speak about jet bundles and differential operators and Euler-Lagrange equations. Because all of these, like all of classical variational field theory is all about, you know, kind of doing Taylor series expansion of fields in their variations and stuff. And this is very much encoded in here. So of course you can speak about variational calculus without this tool, but the same material

is that this is kind of secretly in the background. This is the archetypical geometry that models infinitesimal variations. And so this is like bare bones fields. This is like, okay, if you have a Lagrang and if you want to look at its Lagrange equations, then you're secretly using these kinds of generalized charts. But now that we're using some coordinate functions that square to zero, we can kind of keep going. And that's where the super geometry comes in, which so this r zero q is like a dq, it's like an infinitesimal disk of dimension q, and it's also first-order infinitesimal because the coordinate functions here on this thing

don't just have the property that their square vanishes, but they also have the property that any two of them actually anti-commute with each other, right? So this is very much like an infinitesimal disk, the super point, but with the additional property that things anti-commute, which implies that as infinitesimals, this has actually to be first order. That's why we don't write an index subscript one here. So that captures a common theme. People have slogans like, oh, fermions behave like they are very small or something. This has to do with the fact that these are really infinitesimals. So using these charts here, so I'm writing products here, so we can easily envision,

one can easily define, you know, it's clear what it means to take the Cartesian product here to have a space that has n ordinary dimensions, m infinitesimal dimensions of order k, and then addition q super dimensions, and all this is one, what I would call formal super Cartesian space. And the category of these surfaces aside for actually formal super geometry, where formal doesn't mean undefined or something, it means it has these formal power series, algebra here. And I just want to amplify, this is maybe a tautology that might be underappreciated commonly, is that even though it's called

supergeometry, this is a term invented really by mathematicians, this has nothing to do yet with supersymmetry. The supergeometry just refers to the fact that we have these odd coordinates. There's no assumption here that anything is supersymmetric. In fact, you need exactly this geometry. This is an important point. Whenever you consider a classical field theory with fermions, as of course you do in the Stannard model, say, or any other field theory with fermions. The fermions in any classical field theory do not actually form an ordinary geometry

by the Pauli exclusion principle. They are their corresponding fermionic fields classically are intercommuting fields. And that is important for instance, the Lagrangian density for the Dirac operator, something like psi bar d psi. If psi, let's forget the bar for the moment, psi d psi. If psi were an actual bosonic, an ordinary bosonic function, then, you know, psi d psi would be total derivative. It would be d of one half psi square, right? That's psi d psi by the product rule.

So the Stunned, already the Lagrangian density for density for a standard Dirac field, for any fermionic field, the free Lagrangian density for a fermionic field, would not actually make sense without having anti-computing. It would just disappear. It would become a total derivative and drop out. It wouldn't exist. So this is really necessary in order to speak about classical field theory with formulas, which is kind of a big deal actually, right?

If you want to be precise. Right. And then it continues. Shall we keep going through the list? So these three are- Sure. Yeah. Yeah.

But you could breeze through it. You can go through it quickly because I noticed we're only 40, 35% done. Yes. Yes. Should I speed up? Yeah. through it quickly because I noticed we're only 40-35% done. Yes, yes. Should I speed up? Yeah. So these three are kind of bread and butter. Everybody needs them.

And then we come to more, still important, but more, how should one say, exotic aspects of physics. So we might ask, well, what if we now have a gauge theory, then our spaces of plots themselves should not just be sets. So here we talked about them being just sets. But of course, for a gauge theory, for example, if X is something like the configuration space of gauge fields, then a plot into it, any two plots into it may be distinct and still related by gauge transformations. So they should actually form a groupoid themselves. And the way to do this, okay, being quick now, is one kind of throws in the rudimentary

information of what it means that there are transformations. And a transformation like a gauge transformation of order r, first order, second order and so forth, it goes by this symbol delta r. So this is a bit technical. This is a notation for the simplex. That's one way of encoding transformations and higher gauge transformations. So throwing in kind of directions in which things can gauge transform gives us a notion of charts that models gauge theory and higher gauge theory with firmness and blah blah blah. And then I can go further specifically in string theory but also in other areas. It's important that spaces like space times are not actually smooth everywhere.

They may have what's called orbefold singularities. So we can ask our probes to have actual singularities already in them so that we can detect singularities in the target space. So this is notation we invented. This is supposed to be suggestive of a little singularity, you see, like a little cone here, which has a G, which kind of comes from a G action. It's a quotient by a local G action. And finally, and this gets us really into deep waters here,

one can throw in something that behaves like spheres of negative dimension. And yeah, let's talk about this another time. If one does this right, then this actually boosts the corresponding spaces that are probed by this to being parametrized, what's called parametrized stable homotopy types. And this is really, it's kind of fascinating actually, this is really where the quantum aspect comes in, where linear, where the spaces we're talking about kind of acquire a linear halo of quantum operations around the classical geometry. Briefly talk about what it means to have a negative dimension on a sphere. Yeah, exactly.

So, I could show that. So we effectively define a category of charts of these D-minus-event ends that where the maps behave like maps of spheres, but the direction in which the arrow goes is as if they were of negative dimension. So it's just a notation for a certain category of things, where the main point is that after we apply the Unidad Lemma, think of these charts as being spectra, as it were, like spectra of spaces. We will see the sphere spectrum in different dimensions.

Yeah, let's move on. So this is from the end section of this encyclopedia article. Okay, great. So we'll leave a link to this in the description for people to learn more about. Yeah, so here's the category, this is what the category, this is actually now a higher category of these negative dimensional spheres looks like. This shows which maps it has, which transformations, and it explains how this thing. So this needs a bit of thought. This should not be done too quickly. I'm just claiming this maybe for the moment, but you can find the pointers here.

But this is of course important eventually to bring in the quantum asp. All right, yeah, so that's our examples. And now let's see. So what I'm going to do now, and now I finally come to these animations that I have. So this is the end, I think, of the little introduction to Ketogra-theory and physics that I have here. Now we want to see it at work. We want to see things happening.

We want to see this promised emergence of stuff out of the substrate that we built. I'll be talking now about mainly this topos, the shift of us that we get by using the first three at least and possibly the others as our probe spaces. And that, you know, in the writings I have, this goes by the name of superformals. So the generalized spaces that are probed by all this year, some are called superformals, smooth infinity group points for the fact that they're smooth, they're probed by the RNs, formal because they are probed by infinitesimals, super because they can be probed by supercatechism spaces, and higher group points because they have these high gauge transformations in them.

All right? So that's the topos we have. And now let's look at one of these. So I have these animators. This is a bit experimental, of course, as you know. But let's see what happens. So this is like a slide that keeps building itself. Okay. Note to the viewer, the animations in this talk loop. This means there's no need

to pause because if you wait just a few seconds, you'll see the same thing again. And what I want to do now is I want to talk about how in such a topos we now have a system, a progression of dualities of modalities. So remember, dualities means a junction, and modalities is something I'm explaining here. So I want to apply the following method, as it's called, now to this top of physics and see how things emerge in this. So we're going to be looking at a category, H1 here, which, so I'm just calling it H1

because at this point it could be anything. But you should think of it as being our category of generalized super geometric spaces of use in physics. And we're gonna see that it comes equipped with various functors to a base category, which is here called H0, which you may just as well think of for the application that comes as being the category of sets, just bare sets. And this functor, for instance here, from our very generalized spaces to sets will be

the functor that extracts the underlying set of points. So if we have a generalized space, we can just probe it by our zeros and just ask, well, what points do I see in here? And that is the set, set of points. So if we have a generalized space, we can just probe it by our zeros and just ask what points we have seen here. And then here's the set, set of points. And so if h0 is a category of sets, we get a functor here. What is being shown here now is that if we have systems of such functors that are adjoined in several ways, then we get corresponding for the structure. Let's first look at these. Suppose our base functor here, say the functor that signs points from H1 to H0, suppose it has a right joint. So this symbol here, just a moment, we'll get back there, a functor, and here is a right joint. So this symbol means that this function is left and this function is right joint. Now, when we have that, we can go back and forth.

This is what this arrow is indicating. So if we have these functions, we can go back and forth and get a function that is called sharp here that goes from edge one to itself. So you're showing that all of this that's emerging or that's being displayed with this auto playing PowerPoint or what have you is canonical. There's nothing extra that you're adding.

It's already there. Yes. So, well, okay. I do this in stages. So, on this slide, I'm saying if I have sequences of adjunctions, then, and the rest follows. So, yeah, very good. Thanks for asking, actually. I should have made this clearer. So, what is given here is adjoint functors and other adjoint functors, and I'm unwinding what extra structure that implies on our category. But in a moment, actually, in a moment we'll go, so this is just the explanation of the method.

In a moment we'll, on the next such slide, we'll actually apply this to our topos of super geometric physical spaces, and we'll see what functors actually exist there and only certain ones do. And so then we see a certain dynamics emerge. Right? Okay, so here's just an explanation. If we have an adjunction, then we get this endofunctor. If we have another adjoint on the other side, well, we get another endofunctor here and

these two functors are adjoined to each other as one can easily see by digging through the algebra by following through the definitions. If you have an adjoined triple as people say, then the corresponding endofunctors are themselves adjoined to each other. Now them being composites of adjunctions, it means that these co-unit and unit maps exist on them. So if we have this adjoined pair of monads or modalities as I'm actually going to call them on our category then it means that every object in our category H1, so every generalized space, now has kind of this flat as we're going

to pronounce it and sharp has this flat aspect to it. So for every aspect every every object, right, we can apply flat and get kind of this version or we can apply sharp and get this version. And the co-unit and the unit are maps like so that arrange this little diagram, which so every x is sitting kind of in between these two extremes as we're now because remember every adjunction is like a duality. So in some sense, this flat X is dual opposite, we should say, to the sharp X and X is in between. And in fact, this kind of means, so once we have an understanding of what this flat, like, you know, him just, I haven't really defined things yet. It's just, suppose we had, but once we know what flat actually does, we know what this extreme aspect is, you know, once we actually have sharp, we know what

this extreme aspect is. And then just by the fact that these two functions exist already makes every object kind of be in between these two extremes. Yes. Okay. And for the case that will appear in just a moment where flat and sharp are the discrete aspect of a space, like the set of underlying points with its discrete topology, or the set of underlying points with what is called its co-discrete topology, so like what's sometimes

called discrete and co-discrete space, it will mean that every space in our category is in between being discrete and co-discrete. But this is exactly what a topology on a space means. Every topology is in between, in some sense, the discrete and the co-discrete topology. So it's a rudimentary, what we see is getting towards a rudimentary form of encoding some spatial property in our objects just by the fact that there are these opposing aspects of things. So I see here pure nothing and pure being. Yeah, it starts to appear now.

So this is what we'll see more in the next diagram, but I'm just amplifying here that our base suppose of sets here has a canonical functor down to the point where the point now means the category which is the single category with the single object and only an identity morph is mounted. And one can check. One can see that this functor has a writer joint even only if our, well, if we assume that H0 is the category of sets, then this writer joint exists, the writer joint to the functor to the point exists.

So it will choose a single object in H0 since the point has a single object. So our writer joint functor will have to choose a single object. And then one can check that the laws of adjunction say that single object has to be the terminal object. The object, every other one has a unique map to it. Right. And similarly, one will see that the, now I'm almost catching up here, with the left rejoin exists, it will have to pick the initial object. So that's why these functions are the functions constant on the initial and the terminal object. So that means that every object is in between pure nothing, like the initial object, or pure being, like the

thing that is, but in its most trivial form. And this is of course tautological in itself, but it is an opposition and it is one of these dualities and we'll see that it serves as the basis for a whole tower of such dualities that we can see. For instance, we can next ask for a yet further left adjoint here, it will produce yet the further endofunctor and they will arrange in such a tower of in such a progression as I say of modalities of ways that things in our topos can be. So that is the method that I'm going to show in a moment. This comes maybe finally to our first little punchline

here. I'm going to now look at this progression in the next animated slide. We'll see this progression in following these rules at work in our top boss of super geometric spaces. And we'll see what is it that emerges out of this initial opposition between nothing and pure being. And it turns out there emerges a very interesting... And it turns out there emerges a very interesting. A very interesting. So I noticed it says that this is part of some work that was just a decade old at this point, and it's inspired by Laverre, who was inspired by Hegel. Yeah. Yeah, this is right. Yeah. So, okay, right. I should have said this.

So this is really in sketch form at this point here. I mean, the slide isn't maybe showing everything that one could say here, but this is really what Levere came up with is what's actually going on in Hegel's science of logic. So Hegel in the science of logic, in this old book, he kind of puts himself into the place of a seer, of a mystic. He just looks inside himself and says, okay, let me forget everything about and let me just kind of concentrate. So he wants to make philosophy a science.

He kind of fields his way to the beginning of every thought. And so he argues himself in a kind of poetic way into this kind of story where he says, okay, so if I really abandon all other thoughts, what do I have? Well, I have nothing. And then he, over pages and pages, he talks about this pure nothing that he's pondering. But then he gets to the point that he says, well, I've been pondering pure nothing now for quite a while, so apparently it is something, right?

After all, it exists. So and then he feels that there's a certain internal opposition happening where the pure nothingness that he assumed to have started with kind of gets into tension with something that just exists with the pure being. But then he says, well, but this can't quite be because we had nothing there. And so he kind of feels an inner tension, which eventually he says is being resolved by. which eventually he says is being resolved by, so out of this dynamics really, this logical tangent, kind of he feels it like a dynamics that makes something else emerge. And Lavir's observation was that this is really captured by, accurately captured by what I'm

drawing, this is not quite how Lavir drew it, that's what I'm drawing as this progression of these modalities. So Lévié observed that from adjoint triples of functors, you get these adjoint monads or adjoint modalities that express opposing dual opposite aspects of one thing. And then one can ask therefore for a further such opposition which resolves the previous one in that so as the composition of the functors up here implies the things that are purely sharp for the sharp modality actually include both nothing and pure being. So in this sense, the opposition between these two aspects gets resolved or unified in this opposition because they both nothing and pure being becomes what I guess Léviol likes to call becoming because that's kind of what matches here with

the terminology which then has this other duality to it and so forth. And then you have another opposition here and one can, and I tried to do this at some point which I guess you can find on the NLab Patreon, Science of Logic. You can now ask, well, what is this next opposition if you were to actually compare it to Hegel's poetry? And then one can actually see that it does sort of make sense that Hegel sees another opposition which again, that's kind of what he calls the process, the progression, where there's intrinsic oppositions that get resolved only to find new oppositions to get resolved

and so forth. So this is this, I guess what's sometimes called the dialectic method or something. And Levere's insight was that this Hegelian poetry, as I would call it, actually is nicely matched to exactly this math of adjoined monads, of, I guess he calls it adjoined cylinders. Right. And so what we do now is we kind of, we say, okay, now we've actually formalized the science of logic. So now we can run Hegel's experiment, which he had to do kind of without tools. We can now ask, so, okay, what is it that emerges out of the primordial nothingness? And so what is it that emerges if we just keep playing this, if we keep growing this

tower of modalities here. And so that is what I'm showing in this next animation. So I should say I've typed this a few years back here, so I didn't retype it. So it speaks about infinity group words all the time. You may just think of this as being sets, the way I've introduced things. So not to get hung up on this. So the basic, what we're seeing here is this. The full topos of generalized super geometric spaces on the far left, just disappeared. And we're looking at this kind of situation that we just explained in the abstract.

A functor all the way down to the point, which then successively factors through subtoposes. First of sets, the bare geometrically discrete things that have no geometry on them, just points. And then we'll see that there's further factorizations to a category of the reduced spaces, where reduced is in the sense of reduced schemes as it's used in algebraic geometry, meaning those that have no infinitesimal extension, that are just ordinary non-infinitesimally thickened spaces.

The technical term for that is reduced. And then as a further subcategory, we'll find in a moment here appear the bosonic parts. Just a moment, here it comes. So this is the subcategory of all those spaces that even though they're potentially super geometric they just happen to actually have no fermionic aspects to them. They're just ordinary bosonic spaces. And so the punch, the point here is that these adjunctions that on the previous slide I just assumed, I said

well suppose we have them, then what follows? They just appear now here. There's no way these adjunctions, if they exist, they are unique. So you either have them or you don't have them. So we just feel which we have and this development down here shows how successively these adjunctions factorize in such a way. Then for each such pair, so all these little symbols are missing, so every arrow on top of another one is left adjoined to the one at the bottom.

So whenever we have such a pair going back and forth between any of these topos, this means there's a corresponding modality induced, a certain operation on H, I should say here, that is given by going all the way down here and coming back. So that will extract some extreme aspect, physical aspect of our spaces. And the corresponding progression of these extreme aspects is shown here. So we start with an opposition between pure nothingness and pure being as we did. So that's at the very beginning of this animation.

That's the the function that goes all the way down to the point and just comes back. So nothing really happens, but still there's this, there's an initial opposition between every object is sits between the initial and the terminal object. And then we observe that this gets resolved by this next operation, which is just the one we've shown before, sharp, which in fact plays this role of forming the codiscrete objects on the underlying points of a given space. So we see that things become spatial in some rudimentary form. There's a discrete aspect and a codiscrete aspect there. Yes.

But then further adjoints appear. The next one that just disappeared here is the so-called shape modality, which says, well, that apart from just having discrete and co-discrete aspects based on the underlying points, there's actually also a shape to our spaces that is not just embodied by the points. So the shape operation in the actual model corresponds to forming what's sometimes called the topological realization of an infinity stack. So it's the thing that sees the actual homotopical shape of something.

That's why it's called the shape. So more and more of such aspects appear. Then after the shape operation, there's an infinitesimal version of the flat and the shape. There's something that sees an infinitesimal shape of things, which in algebraic geometry is known as the Dram stack monad. So these are things that are known in algebraic geometry of derived schemes or sorry of just of formal schemes, the reduced aspect, the co-reduced or the Rahn mistake aspect. And it keeps climbing and that's interesting. So we're just asking in this Hegelian-Lavirian notion.

So Lavir never pushed this, it seems to me, beyond this first step. I think he looked at this first step and had some examples, some toposes realizing that, which were a bit contrived. So I don't think he looked at examples that are kind of practically important. And the inside is that actually applying this to our generalized categories of physical spaces, not only does this have some importance in itself, but this progress actually continues and kind of discovers all or rediscovers one could say these various aspects in our spaces have like the pure being aspect, the differential topological aspect, the homotopical aspect, the infinitesimal aspect.

And then next, and that's interesting, it rediscovers the fermionic aspect. So there's now a modality of being bosonic or being bifermionic, and then it finally ends. What is interesting now, and I haven't previously talked about this, maybe I should have made a dedicated slide for that too, is that the modalities that appear in the middle here, and now it would be good of course to be able to stop this animation, I can't, so we have to just live with it. But the three that appear here in the middle, they are special in that these three are given

what is called bilocalization. So you can ask, well, how do we compute the shape or the infinitesimal shape or this real nomic aspect of a space? Well, it turns out the shape or the topological realization of the linfegistic is the answer to asking, well, what if I had this space and I want to just look at its homotopical meaning, meaning that the real line being as it is contractible, having no homotopical content, it just has smooth content, kind of. But if you forget that, you can shrink it to the point. line plots and all the RN plots into my space, if they're actually not equal, but gauge equivalent to the point, what is it that remains? What can I still see in my space?

And that is the technical instructions that I'm alluding to here is saying that the shape monad is actually what's called the localization at the RNs. So it shrinks them away and asks what is that remains and that's why it's called the shape. You no longer see local differential structure, but you still see some global shape. For instance, the circle, if you look at the shape of the circle, then you end up with what's sometimes called the categorical circle or the simplicial circle. You not only have a smooth manifold, but you still have kind of the rudimentary information that there is a point and there is a way of going once around and come back to that point.

So just this global gauge transformation, that there is a large global gauge transformation is retained. Right. And so I said this, so the three modalities in the middle, they have this property that they're all localizations and as such they pick out a distinguished object. So the shape here picks out the continuum line, the Rn, the R1 actually. So it knows there is the continuum, whereas the infinitesimal shape modality picks out the one dimensional discs, the infinitesimal intervals.

Yes, yes. And then finally, and that's kind of a punch line here. Now it just says disappeared again. And then finally, there is one more. It's this rheonomic as it's called. So it introduces fermions? Yeah, that is the one. So this level here, this line that is going to appear in just a moment, that is the one

that knows about the Fermions, bosonic aspect, trinomic aspect, bifermionic aspect. And this guy here is given by localization at the superpoint. So it's in this sense that this progression discovers first the continuum, then the infinitesimals, and then the superpoint is being the objects that characterize this whole progression. And so it's in this sense that, as this diagram keeps showing, that we kind of went just by applying this, what Hegel called the objective logic, we kind of went from nothing to finding geometric structure culminating in the super point.

So in this diagrammatic playing of nothing to a super point, a so-called super point, is there more than just analogies here? Is there a rigorous proof behind this? No, there's no analogies here. It's just, I mean, all this is a proof. Like what I'm showing here is just a fact. So I'm kind of illustrating a theorem here. It's a fact that this progression of modalities and adjunctions exists in this topos, right? So that's just a fact. And so I'm saying if this topos is our Platonic world in which we live, then we can play the Hegelian game there and ask, okay, what is the structure that we see emerge out of the initial opposition of nothingness and pure being?

And then it's just a fact now. This is a theorem. It's not an analogy or something. Is this theorem going back to 1965 from Grothendieck or to 2013 with you? No, that's my observation. So this is quite recent. Yeah. So this is what I wrote in this habilitation thesis, differential cohomology in a cohesive

topos. Yeah. Right. So the thing is that, as I just mentioned, so even though Levere, he made this, I think, or should call it a big accomplishment of suggesting this formalization, he didn't really dwell on convincing examples. He had some, I think that could fairly be called contrived examples that just showed the math at work but didn't really show much practical use.

And what I'm observing here is that in the actual topos that we actually want to be using in physics, that we are using, implicitly or not, that not only does this initial stage exist, but it keeps progressing and it knows about these archetypes of our geometries, the continuum, the infinitesimals and the superpoint. Beautiful. Okay, so what else is there? If you've shown almost everything from nothing, it's not just something, it's quite a plethora from nothing.

Exactly. So now let's... What else is there? So now let's go further. So this is some backdrop on the higher-topos theory that is maybe needed to formalize or just to phrase basic physics, field theory and stuff. So let's see what more fun we can have now that we kind of got the super point experience. Here's another incarnation of this, another rendering simply of this progression that

we've just seen. But let's maybe keep going. So now let's see. So I'm now making this fun sounding claim that actually from the super point emerges 11 dimensional super gravity. And if we have the energy, let me maybe try to explain that. So now let's play with the super point in our hypopos. And the thing is, there's a pun there if we have the energy.

Oh, I see. Yeah, let's see if we can go through this. So now I'm going to look. So we have this kind of atom, as one might call it, the atom of space or super space kind of came to us in this emerging form. It's not sitting anywhere yet, right? It's a zero-dimensional space, but it's not anywhere. It's just kind of the concept of this space that came to us.

But now let's kind of hold the microscope over the superpoint that just emerged and see what extra structure is in there. And of course, there's other universal constructions we can do with the junctions. We can look at what's called universal center. So the superpoint carries the structure of a super-Li algebra. It is such that a billion super-Li algebra on a single odd generator. And with that structure, we want to ask, well, what kind of emerges out of that object in the sense of now in a more, how should I say, more classical sense that we ask, what are

the universal extensions in the realm not just of super-Li algebras but of higher super-Li algebras that emerge out of it? And it's a fun exercise which originally started with John Huerta in this article here called M Theory from the Superpoint where we proved the root of the following, we proved the root of the following extension. So it turns out if you start with the super point here and play the following game, then in each step it doubles the fermionic dimension. So it takes the odd one dimensional super point to the odd two dimensional one and then

forms what's called the universal central extension, equivalent under, so there's a technical thing equivalent under the external automorphism group, so there's some intrinsic equivalence condition one can compose. Then again, the double form the extension, double form the extension, double form the extension. Then this process discovers it runs exactly through first the super space times in which green-short super strings exist. So kind of the, I shouldn't say critical, the dimensions in which super strings can exist and then it keeps going. So let's talk about the first step here.

So this extension here, so what is going on here? So let's talk about the first step here. This extension here, so what is going on here? So let's talk about this. R02, in which sense, so this is still just a super point. So the first interesting step is happening here where I suddenly claim that three-dimensional n equals one super Minkowski spacetime with its super translation, Lie algebra structure, emerges from R02. So how can this be? Well, let's think about it.

Looking at extensions. So an extension of a Lie algebra is classified by a two-co cycle, by a degree two element in the Chevrolet-Alenberg algebra of the Lie algebra. So a map from the graded commutative second power of the Lie algebra with itself, a map from the graded commutative second power of the Li-Ajba with itself, a map from there to the ground field, which satisfies some Cossacker condition. And now, since our superpoint has a single odd direction, such a two-form on it, which would classify an extension, is actually a symmetric form, since it's an odd two form on an odd space. So what we're really asking and asking for this first step here, that just reappears again, this first step here, we're asking kind of for the maximal

invariant number of symmetric forms on just R2, on these two odd coordinates. But since we're working over the real numbers, these are just the symmetric real matrices, the Hermitian matrices. And there's a standard fact that the space times in these dimensions here, dimension 3, 6 and 10, can equivalently be understood as being the Hermitian matrices with coefficients in the real numbers, the complex numbers and the quaternions. And this kind of explains, so I was trying to kind of indicate how the proof works here. This explains how this first step can happen here.

The two independent non-trivial two-course cycles on R02 are exactly the information that encodes the information of the three-dimensional Suwimikowski space. And then it continues this way. And it's quite interesting because the process down here in this root of the bouquet of the tree is all in superlialtvas. So these are all superlialtvas and we're recovering the supertranslation superlialtvas of these Minkowski spacetimes. But then at some point, there is no further invariant extension, but there are now higher extensions that produce.

So here's 11D superspace time. So there's further extensions now here that are not extensions of super-L algebras, but are what are called super-L infinity algebras. So higher categorical Lie algebra extensions. They turn out to correspond first to the strings, so the F1 brains in these dimensions, then to the corresponding D brains, then from 11 to the M2 brain and the M5 brain. In which sense? In which sense?

Let me say in which sense this is true. Let's focus maybe on this top guy here. So here's the statement that on 11-dimensional super space time, like the Mnicovsky form, the flat or maybe infinitesimal locus of any space. So there's an extension now by what's called the M2-Li3 algebra. So there's a four-core cycle actually on super Mnikovsky space time. Like the ordinary extensions of a super Li algebra by another super Li algebra are encoded by a two-co cycle. This is Stunner textbook material. Now you could ask, well what is it that a three-co cycle classifies? Well, it turns out a three-co cycle on a super Li

algebra classifies an extension by a higher form of super Li algebra called a Li-2 algebra. And so this happens with some of the brain, the brains here. But this one, the M2 brain, is just by a four-core cycle. And that four-core cycle is the famous, the famous bifermionic expression on super-space time that is known to be the bifermionic component of the supergravity C field on 11-dimensional super-space time. But it's also what is called the West-Sumino term of the Green-Schwarz signal model, of the kappa-symmetric signal model for the M2 brain.

So here we see the expression. Mu M2 is this Foucault cycle. Oops, that was too quick. We'll see it again. So what this diagram, what we call the brain bouquet as a variation of the old name of the brain scan, which just tabulated the sum of these brains by computing super-lead algebraic cohomology. So this brain bouquet kind of discovers from the super point now first the super spacetimes of kind of the right dimensions, then the super strings in their brain towards incarnation as sigma models, the D brains, and then in

its kind of tip here, the M brains. And the key point now that is interesting is that, so this is a four-core cycle that classifies the M2 brain extension, which again in turn carries the seven-core cycle that classifies one further extension corresponding to the M5 brain. So again, the seven-coast cycle happens to be either the bifurcated piece of the dual C-field flux 11 R supergravity or equivalently the West-domino term for the Green-Schwarz-type signal model of the M5 brain. And it's these two co-cycles that we want to be focusing on, which are shown here. G4, oops, it's gone again.

But we'll see it. So the punchline is, and maybe let's look at it one more time and then we'll over. So from the super point, we discover that up in 11 dimensional super spacetime, there is this pair of core cycles, a four core cycleicle and a seven-cosicle, which are in a crucial relation to each other, which mimics just the equations of motion of the sea field and 11-dimensional supergravity. It just comes out. It's just kind of built, in some sense built into the super point here.

And it kind of grows into this, it blossoms, if you will, into the structure up here. And then in the next step, I want to show how we actually obtain like full supergravity from this, because this is just local model space. All these are super-Liagvas. You should think of all these things appearing here as being what in an actual space-time you would see in the infinitesimal neighborhood of any point, like on a tangent space. On a tangent space, it looks like a superman Kovsky spacetime and there we have these cos cycles. But now let's look at the following theorem. So we proved this just last year actually.

So turns out if you ask, I'll show it maybe, I'll show a better version of this in a moment, but if you take these two co-cycles that ended up being the tip of this diagram here that classified the M2 and M5 brain extension, and we ask that they globalize now over a non-flat curved super-space-time. flat, curved super space time. So that roughly, if you have some super space time, which exists in our topos by the previous construction, so that every tangent space carries in a consistent fashion this core cycle. How so? So if it globalizes, if you can kind of move it in the sense of Cartesian geometry of moving

frames, if you can move these corecycles over your curved manifold, then that requirement alone is actually equivalent to the full equations of motion of 11 or more supergravity. So it means that your super spacetime is actually a solution to the Einstein equation in 11D, that it carries the super Einstein equations where the the Gravitino field satisfies its Rorita-Schwinger equation and on top of this the the super gravity C field so the gauge field of the theory satisfies its equation of motion. And that is so let us repeat it again on this animation set. We're just sketching it here. So what are we seeing here?

I'm now going. I guess I've changed pace a bit. So we're now looking at how to see full supergravity actually appear from these ingredients that we discovered. And, okay, what are we seeing here? So we're looking at a super space time. 11 dimensional super space time. I didn't show the, I'm not showing the odd coordinate components. Remember, I should have added them.

Sure. So this is says odd dimension 32 as befits n equals 1. So this is the space time that is locally modeled on the tip of this emergent diagram that we saw. It's locally modeled on this 11-E super spacetime. And we're asking that on this spacetime we have two forms, super forms as it were, of this form, where G4 is of this form. And this is just, even though it was too quick maybe to see it on the previous slide, but it's just exactly the four-coast cycle that classifies the M2 brain extension.

So this is kind of the intrinsic super component that knows about the existence of the M2 brain. And similarly for G7, we have this bifurcating expression. And so we're asking, well, let there be forms that, you know, saying we have a super space time, but this I mean it is equipped with a super metric structure being a space time and the metric structure is since we have fermions in first order form. So it's encoded by these by these field bite forms. So everything is expressed in these field bites, which are the things that locally identify

the space time with Movsky space time. So we say, okay, let there be two flux densities, as we're going to call them, whose bifurminant component is exactly the one that is kind of God given. And then let them have, in order to patch it up, let them have any other ordinary component, an ordinary four-formant, ordinary seven-form component. And then this co-cycle condition that makes these things be co-cycles that appeared in the previous diagram, exactly these equations here, that g4 is closed and g7, that the differential of g7 is the square of g4 up to some factor.

That is the co-cycle condition that, you know, I'm just saying this now, I haven't fully explained this, but it's clear that g4 being closed, it's an element of the Chevrolet-Ilenberg algebra or L infinity algebra and it's being closed, that's just the co-cycle condition. And then one has to just see what it means for the M5 plane extension to just a co-cycle condition. And then one has to just see what it means for the M5 plane extension to be a co-cycle, which is not defined down here, it's defined up here. So kind of pushing it down gives, makes a co-cycle condition be this equation

that says that d of g7 is g4 square. Okay, I guess, I don't know, it's maybe getting a bit technical here. But the thing is that this bottom piece here, this information we're requesting on each tangent space is the piece that came to us from this progression. It emerged to us kind of from nothing. And we're now asking just for that structure to be globalized over a spacetime. And so the first thing we want to do is we want to ask, well, in order to put this into

a proper categorical formulation, we need something like a classifying space for the solutions to these equations here. So the key co-cycle equation of our fluxes were that one of them was closed and the other one had the differential equal or proportional them was closed and the other one had the differential equal or proportional to the square of the other. So we ask, well, what is that? What is it that classifies pairs of differential forms with coefficients not in an ordinary Lie algebra, but in a higher Lie algebra that is called the whitehead L infinity algebra of the four sphere. So there is a way of assigning to each, under multi-technical conditions, finite type, rational, finite type, to each

topological space, an L infinity algebra whose ordinary bracket is what's called the whitehead bracket of that space. So it's some kind of infinitesimal approximation in a way, or rational approximation actually, to the homotopic structure in a space. And the space that appears here, the space that happens to encode just the code cycle relation that finds these M2 and M5 brain code cycles happens to be the four sphere or any other space of the rational type of the four sphere. So that we can recognize the information we need here, the local form data on our manifold,

as being exactly flat or closed, one should say, S4-valued differential form, which has these two components and the flatness encodes exactly the scope-cycle condition, which is actually equivalent to the supergravity equations of motion. So you see what we have achieved here now is that we have fact that the space-time satisfies supergravity equations of motion just means that flux synthesis of this form exactly arranged into a four-sphere valued flat differential form, whereby four-sphere valued I mean valued in the whitehead L infinity above the four-sphere. And that is of enormous use now, because as you keep seeing in the diagram that builds here, we can now ask what is the global field content of supergravity. So this is now getting us into an actual new physical realm.

So I make the bold claim that this has been to a large part, not completely, but to a large part actually be ignored. It's known for a long time, like for almost exactly a hundred years that in electromagnetism, that of course the electric and magnetic field strengths that Faraday and Maxwell talked about in the 1850s are not the full field content of the field of electromagnetism. There's a global aspect of the field of electromagnetism, which sometimes goes by the name of Dirac charge quantization or of existence of magnetic monopoles or something. So there's some global structure which is not entirely seen by just the differential form data that is the flux densities of the fields. And what we're seeing here now is that the same question that Dirac sort of answered

at least in hindsight in the 1930s about electromagnetism can and should be asked about all higher gauge theories, in particular about our supergravity theory where this four-form and the seven-form play the role of higher analogs of the Faraday tensor two form of electromagneticism. And so we want to ask now what could be a global completion of the field content of 11-interdimensional supergravity that that in addition to this local flux data knows about global topological structure encoded in the field, knows about global charges that could source these fluxes.

And the diagram that keeps building up here and disappearing again, that shows how these things can be done. So now that we know that the field content of the fluxes is encoded in something with various in that part of the force sphere that is detectable by differential forms, well we can ask for a classifying space to be called A here that classifies the actual global charges not just approximated by differential form data. And then we just want to ask, so this would like in electromagnetism, this would be the

monopole sector, classifying space for the monopole sector. And so we ask, where are these things comparable so that we can ask these charges towards these fluxes and as such they are compatible. Not every set of charges can source given fluxes so there needs to be some compatibility condition to them. If you have a magnetic monopole somewhere it sources very particular fluxes and not any fluxes. And so that question of where these things can talk to each other

so that they can be compared is answered by the top part of this diagram, which appears now in just a moment. So it turns out that we can push forward, we can send these close-to-vertex forms to what's called their moduli stack. Anyway, some deformation space for this. And that receives a map in this higher topos from any space which has the property that its whitehead L infinity algebra coincides with that of the fourth sphere, which means that it's our rationalization is that of the fourth sphere.

Any such space can be put here. And any such space has such a map. It's the character map, the generalization of the Chan character in K-theory. And so this kind of answers the question of what is it that serves as consistent flux quantization laws for left fundamental supergravity. It's any space, classifying space for the charges as any space, subject to the condition that it's kind of the part of it that can be seen by differential form data is the same as that of the force sphere. And then if you choose any such space, then the full field content of fielded level of supergravity is then a homotopy between the charges to the fluxes in this

diagram as it lives in our higher topos. Of course, the canonical choice for A is the force field itself, not the canonical choice, like the initial choice, the simplest choice. And so what we've been calling hypothesis age is, right? So this diagram, the option of this diagram is twofold. It says both, first of all, here's the actual rule, how to globally complete the field content of, well, any higher gauge theory of Maxwell type, first of all, but specifically here now 11-meter supergravity, which is an example.

And second, the second thing is, well, there is actually among the infinite set of possible choices for the charge colonization. There's one choice that is kind of singled out as being the, it's the simple one. It's for instance in terms of CW complexes, it's the one with a single cell, the simplest. It's just as the force cell. And but anyway, whatever choice you make, it gives a definition of global field content of 11-gram supergravity, which has not found much attention before. There's articles by Diakonis Kofrid and Moore from a few years back who talked about a model

of the sea field, which went in this direction, but wasn't quite comprehensive, I would say. And here we say, first of all, well, look, here's a choice to be made. And second, there is a specific choice that kind of stands out. And you see, what we're seeing here is that the usual data that goes into field theories, as they usually formulate in arangian form, this differential form data is not in general, and we know this from electromagneticism, it's not in general the full field content actually, it's just kind of chart-wise field content. It's lacking all the global structure.

So first of all, this is saying, what is it that actually completes, in this case, the left knee supergravity globally? Yes. So hypothesis H is a claim about the classifying space? Yeah, it's a right, but the classifying space is part of what characterizes the full field content. So hypothesis H is a hypothesis about what is the global completion of the field content of the 11- dimensional supergravity, an analogy to how...

Yeah, yes, yes. So you see... I see. Yeah, I don't have a special slide for this, but let me just say a few words. So let's go back to electromagnetism. So what happened? In 1850, Maxwell writes in modern language,

he writes a two-form, F2, so such a G with two indices, and says that is the electromagnetic field. So it's built from its components are the electric and the magnetic field. But it's a two-form, it's closed and it's satisfied some other conditions. It's a field strength, technically, as we would call it. Yeah, exactly. The field strength or, you know, if you integrate it over a sphere, for instance, around a monopole, it measures the flux through that sphere that emanates from the charge that's carried by the monopole. So that's why we also call it the flux density.

That's maybe a more suggestive term here. But the flux density absolutely is just another incarnation of the field strength here. So then the decades pass and 1931 with a bit of paraphrasing Dirac comes and says, no, this cannot be the full field content of the electromagnetic field. He really says two things at this point. He says, well, first of all, there must be a gauge potential. We must locally be looking for one forms A, right, the so-called vector, yeah, the gauge potentials, such that our F-

Or the connections. Yeah, the connections, yeah, the local one-forms. So, F is the differential of these As. But that alone is not actually the full insight because that is quite interesting because that is something that already Faraday was actually playing with. Faraday and Maxwell both, they had a funny term for this. I could look this up somewhere. I forget now. They couldn't quite nail it down, but they recognizably realized that something like what we now call A is actually a good way to encode the field equations.

But this is not the full insight. When you see discussions, for instance, of all these super gravities, people often write down potentials for these things here. They write a C3, a 3-form, such that G4 is dC3, right? So that is the analog. But that cannot be true globally. We're interested if there's non-trivial charge in our space-time, then this G4 is not exact globally. It measures a certain DRAM class, which is kind of a rational image of this full whatever ACOMOLG class we're seeing here.

And so the whole point really of the global charges, the G4, is not generally globally exact. So there is not actually generally a global gauge potential. So the gauge potentials are more subtle than just potential forms. They actually forms like A or C3 here on each chart together with further information, gate transformation, gluing data on double overlaps of these charts, which in one way, like for electromagnetism, this is another way to speak of all this local transition data is to say, well, the vector potential forms have to be the local

incarnations of a connection on a circle, principal bundle, right? So this is really what this diagram, what this innocent looking homotopy here encodes. It's something, as the notation next to the arrow meant to indicate, it's something that is locally of the form of a gauge potential as people usually write down for G4 and G7. So it's locally a C3 form and a C6 form whose differential is locally equal to the restrictions of these things. But there's more data.

There's some very subtle transition data that gives these gauge potentials defined on each open or on each chart some global cohomological structure. And that is what can be encoded very neatly in this high topostereotic construction shown here. And so, right, so I wanted to say, so this was really the second main insight of Dirac. So first of all, in 1930, first of all, there is a gauge potential and it's a physically observable thing. But

second, and this is kind of in his original language, encoded in this notion of Dirac strings, which maybe in modern mathematical language, one would say differently. But anyway, this is what Dirac called the Dirac string. He's really speaking about what mathematicians would call the clutching construction for line bundles on a sphere. So it's a way of encoding the transition data that you need to have a check co-sitel for your whatever data on the sphere. But in speaking about this Dirac string, at least in hindsight, Dirac is observing that there is global topological structure in how these gauge potentials glue on charts. And that is where the degree two chromology charge lives. That is sort of the actual monopole charge, if you will.

And it's that old step from the 1930s that was understood back then, at least in hindsight by Dirac. I think the first one that actually understood this electromagnetic situation in modern language is maybe Orlando Alvarez in the 80s, who has articles on Czech cosmology and Dirac charge quantization. His phrases, these things in the modern language that I'm kind of alluding to here. Anyway, so that insight from back then has never really been much adopted to all these fancy higher gauge theories that appear, particularly in supergravity, where one must ask exactly the same question,

what is it that really defines the full global field content? It's the gauge potentials, yes, but furthermore, some topological data gluing this together. Yes. And so with this kind of theoretical language, it's kind of natural. So this is what I'm showing as the content of this book we wrote, the character map in non-abelian cohomology. It's clear which problem one has to solve when one needs to fill these diagrams. And then one can see what are the choices that one has to make to define a global completion.

And I think that's an interesting and I appreciate underappreciated aspect that actually to fully define just the level of the measure of supergraphy, not just even talking about any other fancy stuff like quantization or other lifts, it actually requires more data than is usually considered. So what does Ed Whitten think about this? Well, I don't know.

I don't know. Well he hasn't written much about M-theory in the past few years. It doesn't seem like many people are working on M-theory. No, absolutely. Yeah, absolutely. There has not been much at all. No, that's right. Why do you think that is?

Yeah, okay. Let me come to that. No, that's right. Yeah. Why do you think that is? Yeah. Okay. Let me come to that. Let me say one thing. There has been a surprising and pleasant revival though of interest in the matrix models just in the last months actually. Yes.

Right. Remember one of the candidates for defining at least some corners of M-theory was the BFSS matrix model and already had this type to be cousin, the IKKT matrix model. And that was always kind of more enigmatic in a way because where the BFSS matrix model reduced all of M-theory to kind of the quantum mechanics, right, in zero space and one time dimensions of these matrices, in the IKKKT model is actually reduced to zero dimensions. Somewhat in some sense, IKT matrix model is matrix mechanics on a point.

And that has received, so in some sense, it looks a bit deeper or more interesting. And that has received a surprising amount of attention just lately. And I guess there are some claims that, I guess the Japanese group is claiming that computer simulation shows that there are spontaneous compactifications to four dimensions, some are seen by numerical computation in these matrix models. I don't know if that has been checked.

But yeah, so this is what people have been playing with. And now to come to your question, like why is there no further progress? So I think, well, I suspect part of the problem is the, what we alluded to at the very beginning, that it's not actually clear, like, or it was not actually clear, where even to look. How do you go about, as a traditional physicist, to find M theory? Well, the only thing you can really do is you can write down a Lagrangian. That's how people build new theories. You write a new Lagrangian.

But the Lagrangian here is sort of already known. It's 11-D supergraphy with higher curvature correction. So that alone cannot be it. There's something else that is actually needed. The question is actually somewhat deeper. What else, how else besides writing down a Lagrangian can we actually do to construct a non-perturbed theory? And observe I never wrote down a Lagrangian here. The equations of motion of supergravity just came to us actually differently. But since we're talking about these things now, how about, and since energy becomes an issue, how about I jump just now to the outlook, because that maybe addresses what I designated

as the outlook here. So let's see, what am I saying here? So I would say that there's a grand open problem in contemporary theoretical physics, which is really the non-perturbative aspects of quantum systems. And in this diagram on the right, I'm showing just how enormous actually the uncharted territory is. If we remember that doing perturbation theory really means when in infinitesimal neighborhoods, if we're just looking at formal power series, in off the point of field space that is kind

of around the free and classical fields. Perturbation theory means to behave just infinitesimally in both the coupling constant and the Planck constant. So this huge realm of non-perturbative physics that has remained pretty much uncharted theoretically, but which is known, deeply known to be crucially relevant in physics. The problem that we don't understand this year, analytically, apart from computer simulation, is really what haunts particle physics in that whenever QCD at non-extreme temperatures plays any role, any hydronic contributions can never really be fully computed with perturbation theory.

Nobody really knows what's going on there. It leads to huge issues in all these discussions of whether or not people see a new effect beyond the standard model or not. These anomalies, they keep usually growing until somebody finally cranks up their latest gauge theory computer and does the computation which cannot otherwise be done. So this is a big issue. And then currently also in condensed matter theory, where all the interest is now on strongly coupled and correlated systems, again the same problem arises that there's no actual

fundamental theory for this. For instance, effects like the fractional quantum Hall effect or similar topologically ordered systems are seen in the lab, but there's no actual derivation of their description from first principles. The existing theories for say the fractional quantum Hall effect are ad hoc effective theories that are made up on the spot. Of course, in a very clever fashion and there's nothing wrong with it, but it's not derived from fundamental physics since so that's non-perturbed theory is

not known and now it's interesting to note that, um, the history of this problem has, uh, it's kind of an interesting, ironic turn to it. Um, so, so let's look back. How, how did, so what happened in the past? ironic turn to it. So let's look back. So what happened in the past? So way back in what is it, the 50s or something, or maybe the 70s, people thought about this, started thinking hard about this problem of non-perturbative QCD. And there's very good arguments, of course no real proof, that in the confined regime

QCD must somehow be controlled by the flux tubes that are thought to emerge between oppositely charged quarks, confining them, holding them together. So the idea back then was that, well, since that is the most important phenomenological effect in the confines in the non-perturbative regime, so let's probably the flux tubes are the actual degrees of freedom to be discussed for non-perturbative QCD. This really where the idea of strings originates from, that you say, well, okay, let's write down action functionals, dynamics for these strings that are the flux tubes. And then comes this computation, which is really still at the heart of much of the,

how should one say, of the derangement of the field. So you write down the quantum dynamics of these flux tubes and find that they only make sense in more than four dimensions. Five maybe if you use Liouville theory or if you use critical strings then even 10 or 26. So this is where strings seem to get really weird back then where people didn't know what to make of it. The origin of the hypothesis of strings was very much rooted exactly in experiment. Understand the last, if you will, the big but last remaining gap of the standard model, the actual non-preservative sector.

So introduce the strings and now suddenly you're faced with this new situation that you have this high dimensional space, which eventually, in fact, Poliakov said this right away at the beginning, but it was eventually realized to be, have to be understood as kind of a higher dimensional unobserved bulk space time that, that hosts the actual observed space time, which sits in it in the form of a brain where the quarks kind of sit on the brain and the strings in their, in their need for a higher dimensional bulk may have the endpoints attached to these quarks, but my otherwise probed this bulk.

Then of course, well, you know the story, then holography develops and there is a form of it called holographic QCD now where you ask, well, can we build a string model that holographically describes actual QCD? A whole lot of numbers can actually be crunched out there. There's actually quantitative experimental comparisons there. But kind of the irony of this whole project is that, of course, the strings were introduced

in order to capture the non-perturbative aspect of QCD, but the strings now were themselves only understood perturbatively in perturbative aspect of QCD with the strings now were themselves only understood perturbatively in perturbative string theory. The big difference is that as opposed to the original field theory, its reformulation in terms of these strings now comes with a whole network of hints of what the non-perturbative completion would be. This is really what the second superstring revolution is about. The observation that if you have strings, then there's brains which have non-perturbative

couplings and so forth. And so this working title M-theory really refers to the question of how to make sense of strings non-perturbatively. But if you remember at this point, and in some sense historically that is really the answer to how to works. Behave not perturbed if you then then at least in the rough outline and theory reappears at least as the. Is the possible ultimate answer to how to actually define construct non-perturbative QFT. The question then is, how do we even go about formulating entry?

Do we write down a Lagrangian or is it just another matrix model? I would argue that, and now I'm coming to an end, I would argue that language has been missing here. We're looking for a new kind of theory. And so some of the old starting points maybe need to be not revised but replaced by something that goes deeper. And I guess one can see a hint of this in this famous quote by Witten, which he himself attributes to Amartya about how string theory is so perplexing

because it seems to need math that would only be developed in the following centuries to find its true formulation. And maybe, you know, I very much agree with this sentiment. And I guess what I've been showing here is suggestive of possibly being parts of an answer in this direction. Well, that's a beautiful talk and I appreciate you premiering it here on this channel. I'd like to also wrap it up with a quote from you from that same philosophical stack exchange. In fundamental physics, it is, or at least was in the 1990s, common to declare that with a certain awe and pride that quantum gravity, non-perturbative string theory, and the like will force us to do things like quote-unquote radically rethink the foundations of reality or something similar.

Unfortunately, that rethinking has mostly been what I think is a fair bit naive. One can't just talk about it. It needs to have both a technical understanding of the core mathematics up to that very edge, where we do understand the formal laws of nature, and a trained, profound, philosophical mind who can stand at this cliff, stare into the misty clouds beyond, and suggest directions along which further solid ground of formalism may be found. Once it is found, true, then the philosophers should probably step back and watch those mathematicians and physicists build a tar road over it and then run heavy trucks back and forth into uncharted territories.

But before that is possible, the new stable ground has to be found first. Urs, thank you so much for coming on. The new stable ground has to be found first. Urs, thank you so much for coming on. I appreciate you spending three, three hours with me. Thank you Kurt for inviting me and for doing this. I appreciate it. That was fun experience. I've received several messages, emails, and comments from professors saying that they recommend theories of everything to their students and that's fantastic. If you're a professor or a lecturer and there's a particular standout episode that your students can benefit from, please do share and as always feel free to contact me. New update! Started a sub stack.

Writings on there are currently about language and ill-defined concepts as well as some other mathematical details. Much more being written there. This is content that isn't anywhere else. It's not on theories of everything. It's not on Patreon. Also full transcripts will be placed there at some point in the future. Several people ask me, hey Kurt, you've spoken to so many people in the fields of theoretical

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