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Urs Schreiber: The Emergence of the Super Point from Nothing
March 20, 2025
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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 Unis Shriver 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 Schreiber!
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. You know, I didn't really plan to give such a talk before you invited me. And then I thought, well, what what would I say? And then I thought, well, the title of your podcast is theories of everything. So I thought, well, OK, 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. I'm 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 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 like yes, technically. OK, 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. Well, you know, whenever you figure out something, make a note about it. That was the original driving idea behind it, that there is a place where we can discuss things, but then also
What was the original motivation we but then also have the opportunity to record any insights you know any actual advancements that have been made so it started out as very much you know not an encyclopedia which maybe it is what it appears as now but i think as the introduction the home page says it's a it's meant to be left notes.
What's what they have been writing on what they have been thinking about what they plan to do maybe. And give that all a 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 sing around. There was, 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, you know, 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 to end up 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 it's, um, this is a common misunderstanding. I'm sure 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 there's lots of other subjects. So to my mind, the point of category theory is, is the organizing principle, the category theory is like the big index of math where things find its find the place, find the home.
so it's the organizing principle behind stuff and as such it's i think ideal for a for an encyclopedic you know or almost encyclopedic um website because it allows to to relate things in the proper way you can you can go and say oh here's a construction in that such and such field oh but look it's just just such and such limit that also appears here and there so things um 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 analytics really mostly, it's growing, you know, nobody is being paid for editing the analytics. So what happens is people edit it when and if it's useful for them as editors. So for me personally, I, you know,
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 like 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 n lab 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-perturbative quantum field theory and maybe m theory and how a big problem there is that it's not just um heart you know you know I will I will end by saying that understanding non-perturbative physics like confined qc and the
Strongly coupled solid-state physics is the big or the grand open question of theoretical physics of our time. 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 would tell us what we're even looking for. And I think that is that is the reason why so little progress has been made on on the subject, non perturbative quantum field theory, because there was very 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 you know going back to some really deep seeming ontological foundations maybe philosophical foundations if you will but but all in the realm of you know 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
Which arose actually from a which so that the post that you just quoted was made at an intermediate stage of the development that I'm going to present today because you know there was at the time when when I started appreciating this some people have heard me say the sagalian aspect of parts of topos theory and
And we were 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 like generalized differential cohomology theories, which are, as maybe I can say a bit more later, which are meant to be the actual, you know, full mathematical formalization of gauge fields, of higher gauge fields. And that's actually, that wasn't quite clear actually.
and i you know that was really what i did since my phd thesis just thinking about how to properly phrase the the idea of generalized meaning also higher gauge fields like like one expects to see in string theory like rf fields the super gravity c field and these things what is it really like what is very fundamentally what is it basically looking for when we say we want to build a model of the c field say and that was very much not clear and um
Yeah, so then we started toying with these modalities on topos of course, and at some point the solution appeared and it kind of appeared in tandem with me at least understanding also the role of what these modalities, the role that these modalities play in a more philosophical realm. 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 fields in full generality and deeply. Yeah, and it turns out that it is related to this term of cohesive infinity top bosses, 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 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 the veer and, 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. 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 you know 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 and it's
It's at least fun. Also, I should emphasize I'm not selling any theory or anything. I'm not going to, you know, present any hypothesis that people need to buy into. I'm just presenting some facts. I'm 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'd take the occasion that I'm speaking here in to your audience on your podcast to go back to these old ideas.
And do a little bit of an exposition of them. Yeah. Wonderful. All right. Let's get to it. All right. Yeah. Thanks. So, um, 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, um, that all revolve around a little bit of aspects of what mine might call, um, going towards a theory of everything. So these are some expositions.
Awesome theorems that have you can 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'm that i started with here shows is we want to start looking at something at the very foundations of thought.
So this is as we'll get to a little cartoon of of an adjunction category theory and by analyzing some structures that appear there which which are sort of known to some experts but actually not widely known we'll we'll see some dynamics emerge some dynamics in the in the platonic sense in the realm of ideas where concepts will emerge and eventually we'll be talking about
I'm gravity super gravity in fact the level of super gravity and then maybe then five brain and if if time and energy is sufficient little bit worried about energy but then we'll get to some actual statements about a strongly couple quantum systems with what is called topological order and ionic expectations which is what this little animated cartoon on the right is alluding to and that connects to the extra current research i'm currently doing alright and why don't you explain the title as well please
right at the title of this page right so i was thinking about how to call this and um you know i must say there's a curious um how should i say there's a curious aspect of 20th century science that some of the deepest um 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 that 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'd call this talk about theories of everything, but in a slightly more fancy
i see i see that's that's what the title is doing okay it's really what it really is it's so i'm not claiming i will of course not claim to have any theory of everything in my hand but i i don't want to claim that i have um like found some fragments of what looks like uh should belong to such a theory or at least which are noteworthy to
to take note of if one is interested in, in theories of everything, um, in an actual deep sense, um, like in the sense of starting, not just from, from the assumption of, of the notion of quantum field theory with its implied notion of space and everything, but studying deeper, like if there is a level of pure logic, that's, 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 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,
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 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 is or 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
With computing, if you wish, quality is 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,
This complaint, right? The main problem is 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
What is your favorite quantum field theory?
like me are going now and say no wait a second it's actually the opposite we're we're lacking math i think the trouble is that even the meaning of what it means to to have mathematical formulation of physics has been forgotten a little bit at times so what i'm speaking about is having um first of all precise definitions like to have definitions exact unambiguous definitions that's that's actually more important even than the the rigorous proofs that one can base on these um
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 and physics to have some precise definitions. So I think what people mean when they say
The 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 is it schematic like it's not filled with physical life but it's also not filled with mathematical precision and so. So it's maybe a mistake to to mistake that for math anyway so that's my.
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 or we're not it's not clear that
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Join with reality. I think that's what's meant.
No, 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 now is rigorous yeah it's not something that you could actually explain to a mathematician.
They will not end and that's i think one one reason why we're not seeing as much interaction between math and physics these days is maybe we did in the nineties the mathematicians pretty much i lost they do not know when this is 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 or 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-perturbative qft or m theory or something so
What we need in order to make some tangible progress for such matters, we need a language that can speak about concepts in a precise way, not just the 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, I'm going to claim without much ado here that the language for these four aspects, structure, duality, equality, and effects is categories in this order, adjunctions. And I'm going to speak a bit about those modalities and monads. So, so this technology, this, this categorical algebra is it's sometimes called is what Levier at some point identified with what Hegel in turn called the objective logic.
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making statements about what we think we know or do not know but it's a way of speaking about sort of the the world the platonic world if you will even though platonic is not quite the right version 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 of reality in a sense that's what this term means okay so here's a lightning crash course on what categories are about yeah you should stop me maybe if this goes if i'm going
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 is, you know, satisfies the expected properties. It's dissociative 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, these terms come from what you might call the archetypical examples of category. 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, um,
category shows objects are spaces of sorts physical spaces but also more abstract space like moralized spaces classifying spaces. Space quite generally and where the edges and these graphs the maps.
I call the morphisms are just maps between the spaces and this is actually going to be a big theme in just a moment that even though it's maybe not often fully recognized but in order to the physics one that just a grandian 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 your 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
of the difficulties that afflict field theory result from the fact that one is dealing with these generalized spaces and if you have 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 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
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 or before the 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 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
Add 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 is that 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 you 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 any figure in that space. Yeah okay we'll we'll get to this and then for those since um yeah just to connect maybe to some um currently fashionable verbiage um among categories is is group points 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 transformation. So when these maps here are invertible, when they have inverses, then they identify these two objects with each other in a specific way.
custom isomorphisms of course and uh 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 um the the operation of a symmetry group on that object and so um the special case of categories all morphisms are invertible are group points and they should very much be thought of as being the incarnation of the idea of of symmetry and specifically of gauge symmetry
You should think of group points. I can think of group points as being like sets equipped with gauge transformations. In fact, a very important example of a category is, for instance, the group point of say, you know, Maxwell gauge fields on a given space time. So the objects in that case are gauge field configurations and morphisms are the true 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 are non-invertible symmetries, namely a situation where there's operations between things that behave a whole lot like 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 symmetries precise, you will in the end have to be talking about
But categories and then of course so this is what is shown here then of course once you have these categories you want to you want to talk about finding images and we'll see concrete examples of this in just a moment to 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 cnd you want to say what's an image of that category in that category well it's just just a map that 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 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 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 function g, and here's its image. Then we can ask, are these two images deformable into each other? Where, if you maybe think of as you can, for the special situations where we're looking at the categories set out, 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
is respected and now we just abstract from that situation if you will we don't have actual paths here we just have these 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 well this is
This result down here counts as a deformation of this result up here by these deformations if this is compatible. So, namely, 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 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 yes all squares commute And later on of course, I will not commute that will that will there will be some home some further homotopies in there But for the moment they will just commute So that's that's the basic that's the thing you you open Wikipedia and his other book 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 it is this actually worth our time this is um this is trivial if this appears trivial and thought logical then that's good that's what it is it's not this is not doing much and i i now want to make another hot take i will actually oppose now well
And often quoted quoted saying in category theory and will want to highlight that it's not actually true so this may originate with with this reference here for a fright where.
Which is certainly historically somewhat accurate, where people said, well, so what are we trying to do here? That'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
To at least half the extent for the original article by McLean or category theory where where they actually asked what is a dual object so they they essentially observed that at that time. In algebraic topology people would would speak about certain things being naturally isomorphic to each other or 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 to make precise what it actually means for 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 me making a big deal and giving everything here a name wait I don't understand why you call it thin
Well, what do you mean? I want to I want to make a point now that there is more to category theory, a whole lot more like the whole the actual theory. Well, good. Thanks for interacting.
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, defines everything that is about categories.
But the actual theory, the category theory, the non-trivial aspects of it, also the, 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 a junctions. 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 want to 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, this is really where something not completely obvious happens and all that all the deep parts of category theory are related to a junctions. 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 a joint that are and that makes them being a joint. They're equipped with natural transformations. Remember that was these deformations of functors from so right. These two functors, they go back and forth. So as we compose them, they give an endo functor on one of these categories or 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 function, 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.
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 the
what's usually called the triangle identity suggestively yes i see yeah i draw it suggestively like this so what this is showing is i'm not 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 to this composite and the core 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
By expanding out these identity morphisms here to have some finite width and then they look like, I don't know, like a poached egg or something. But I just want to amplify it's the very same diagram that I'm not changing anything here. This is just the definition of unit or 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 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, like if you think of these as graphs, we're basically dealing with discrete spaces or something that looks like discrete spaces, at least if 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 just mentioned 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 that's a very simple example. One can make the following exercise. We can ask for that. 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, is a morphism. So where 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, the partial order on the, on the national or on the integers regarded as a category.
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. There's an endo functor on the integers that projects every number to its
It's closest smaller even number or its closest larger odd number. And these two factors I joined to each other. They do each other expressing in a nice way the, you know, what you would intuitively think should be, should be a very simple form of duality between even and odd. And actually that example, if one actually works it out, that example is kind of nice in that it, it shows, it shows both how being even is in some sense opposite to being odd. 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, the isomorphism being at one. So there's a whole lot or, how should one say, surprisingly, this baby as this example is this, 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 that 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 and 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 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 the bottom element of 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 want to actually mention here, not sure if we actually get to this, but I want to 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 reductions. 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 high dimensional space time. And you want to say that there's an equivalence that you can equivalently regard the system from the point of view from lower dimensional space time, where some of the
Degrees of freedom that you had in higher dimensions are incarnated as so-called Calusa-Klein modes. So that's a very important and 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 at least the right was known topological t-duality and in fact the duality between m and 2a so the 11 the reduction of 11 is super gravity to 10d and so i'm giving some references here there's this 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 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.
I'm giving a specific 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 he or 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.
I want to be looking at concretely as categories of categories of such generalised 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 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, you 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 physically 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. 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 bike 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
To very good avail have used this fact without thinking about what it means usually what it means actually globally for space if it's coordinates behave this way and so here category series actually comes to the rescue and in its guise as top of theory provides actually a fascinating accurate
um, 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 C's here could just be, um, like ordinary Cartesian spaces are ends.
for for any end. So just the actual original Cartesian coordinate charts of anything and maps between them say any smooth maps between our ends or they could be super Cartesian spaces Cartesian spaces with some super coordinates attached to them but we think of these C's 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 you'll see maybe may have some new potent coordinates. Well, what write some epsilons that's greater zero. What will we mean by this? Well, we will just mean the kind of define such a seed to be.
It's algebra functions, and we just declare that the algebra function is not just an actual algebra function, 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 charges that i can understand from these charged so what what is the global geometry that that is kind of modeled on these charts and the simple idea is the following it's a it's a fun bootstrap exercise so we say suppose we had boldface x that is supposed to be by 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 choice 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 it's a very physics operational kind of definition we say how do we actually understand space while we understand it by throwing stuff into it like light rays and and see what happens so we probe it.
In fact, it's very similar, also very related to the terminology of what's called probe brains in string theory. We 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, they come back to you with something. Right. Exactly. Yeah.
You can 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 that I'm exposing here is really it's going to be the notion of a sheath is so general that of course, many other concepts will fit on it that are not
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 charge 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, 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 it's kind of, that's why I'm calling it a bootstrap. It's defined here by, and we kind of need to 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, 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 sets 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. We have a map here, so this might be a smooth map between our ends or
Super geometric maps between super Cartesian space or something like this. So we and those aren't to be interpreted as coordinate transformations. Yeah, you can. The thing is, I'm not requiring them the map here to be an isomorphism. So it does. It can actually, you know, it doesn't have to be an invisible coordinate transformation. It can be, but it can just be any smooth function. Actually, we're allowing we're allowing this. 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. In effect, the whole point is that we get something that is more general than manifolds. It subsumes manifolds, but it is more general, where we just speak about not having just, you know, charts that are isomorphous onto the image, but just any maps. That's why they're called plots, any maps from the chart.
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 a with three 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 a 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 is 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 Unidad lemma, which is of course one of the is the emblematic lemma of categories here and they will make this work. Okay. So let's, let's, so this is actually fun. So let's think about this. So, so we're going to now say, well, if I have another chart, then my rule since I'm, I assume I have X. So if I have X, well, I will, I will be able to probe it also with C prime.
But then if we think of the sets of plots as being such maps, even though they are not yet, but if you 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 again, this is a commuting, this is a pullback square. So we regard it as commuting diagram. So phi prime here is the image of composing phi with F. So this is called, this is pullback, pullback of functions. So this just says that along, if you have any map of charts, you can pull back the corresponding plots and that's right. 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 this assignment is respects the composition so if i if i have two i haven't drawn this here if i if i have two maps between the charts
and I pulled 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 Cs and pulling back along an identity map should be an identity. So that's some very basic consistency conditions on our bootstrapped X to say, well, if I have such a functor, I have a good chance that I actually know no X.
So what I'm 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 taking your 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 errors going this way turn to errors going this way then it's called a pre-sheaf of sets.
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 it in words. Maybe we want to 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 in R2, 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've 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 they're
The 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
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 and 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 geometrists know this and that's why they work coordinate free.
Now you're saying okay well the differential geometers way is you first define a 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
Yes. Yes. Yes. Yes.
Yeah, no, I think that's a good way of thinking about it. I might just add that, of course, secretly, even in the standard textbooks and manifolds, secretly, of course, you need to know what RN 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
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 so that's what's going on here too we we want to say what are these global spaces both as x and we need to we need to say that it's somewhere or other they're actually determined by charge 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 the smooth function between
Manifolds more generally got it so we're going to say okay we want to say a generalized space model on these charts is such a sheath such a contra range functor with such a consistent assignment of plots and now actually of course now that we're talking categories also want to make want to make the collection of all these
Generalize 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-sheeps,
Then a map between them is not just a natural transformation between the corresponding fungus. 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
I'm called fold face F now. Well, the idea is to again read pretty much this one does for manifolds to, to reduce it again to what happens on charts. So in order to know what this map does, well, we, we remember that our plots from charts were meant to be like maps into the space X. So suppose I have a C here chart mapping into X, then if this is really a map,
of these spaces that preserves, you know, all the given smooth, whatever, super geometric structure. Well, then it should be able, it should be possible to compose these maps phi from F that will give us a chart, a generalized chart of Y, right? Because now this map from C to X has been pushed forward to a map from C to Y. And so, 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.
So this gives us a category of generalized spaces modeled on any category of charts and that such a category is called this the jargon that's called a topos or actually go topos of generalized spaces, which is the chief topos as people would say on the site of charts. So it's just jargon for exactly what I what I just said. Okay. And
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 Rodendick way back in 1965.
when you actually started saying that this is the way to do algebraic geometry. Instead of talking about locally ringed spaces, one should regard schemes and 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, of formal duals of rings. 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 locally-ringed 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
In this lower stage of categories, functions 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 the 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 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
bootstrapped, if you will, declaration of what should be the admissible maps from any chart into RC. 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 unilateral lemma, which is maybe a fun thing to notice. Everybody is sort of the unilateral lemma and how it gets mentioned all the time in category theory. And here we see that if you think of categories as being categories of generalized spaces for physics, then the unilateral lemma places 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. Cool. So this is from this encyclopedia article that I have on high topocerein 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. Yes. From a few years ago. So I love this part. Okay. Very good. Yeah. In fact, that was a, that was January of last year. I was in, I was visiting in Beijing.
At Tsinghua University, I was giving the talk there, but it was remote to the, you're right, to the Wolfram, to the Wolfram, whatever it's called, Wolfram school. Yeah. Right. So, um, 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 this firm. Yeah. Okay. I mean, let's, 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
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
And 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 shift topos over the category of Cartesian spaces, the mapping spaces from say space-time to any coefficient space
I should say that this is like the basis of all of physics, all of modern physics on a business card. Yeah. 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, 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. You should think of these things as being infinitesimal disks like
Sometimes we'll 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.
disk is so small that 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
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but actually vanishes. So the experimental physicist that was your teacher would have k equals one in his example from before? Yes. Of squaring to zero? Yes. That will be the first order infinitesimal disk. Yes. Exactly. Yeah. You can also see in the old, in the original Feynman articles, I think in his thesis where he introduced the passing that he is also playing with such epsilons. And this is a, this is a fun story because, uh,
You know, physicists must have been using this even long before Feynman, I suppose, probably for 400 years or something, this kind of trick. And it took mathematicians quite a while to fully come to grips with this, but it's exactly in this way. This now goes now by the name synthetic differential geometry, where one considers shift opposites on
on such probes or similar with infinitesimals and then notices that inside these shift top bosses one has that's why it's called synthetic inside these shift top bosses there's generalized that like the the infinitesimal disk then exists as an actual object and and all other kinds of infinitesimals so that i can for instance go and 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's it becomes literally an infinite intelligence vector becomes literally infinitesimal path in the manifold that's why it's called synthetic because the 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 applications, 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 you can speak about variation calculus without this tool, but the statement here 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 Lagrangian, if you want to look at its Lagrange equations, then your
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 um like like a dq it's like an infinitesimal disk of dimension q where 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? As you know. So this is very much like an infinitesimal disk, the super point, but with the additional property that thinks 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. And this, 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, n 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 for actually formal super geometry where formal means formal doesn't mean undefined or something it means it has these formal power series algebras here and i just want to amplify this is maybe um
Totology that might be underappreciated commonly is that even though it's called super geometry, this is a term invented really by mathematicians. This has nothing to do yet with supersymmetry. This is the super geometry 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 the corresponding fermionic fields classically are inter-commuting fields. And that is important for, for instance, the Lagrangian density for the Dirac operator, something like psi bar d psi. If psi
let's let's forget the bar for the moment psi d psi sure if psi were if cyber 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 um by the product rule
So already the Lagrangian 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-commuting. 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 of fermions, which is kind of a big deal actually, right?
If you want to be precise. Right. And then it continues. Should we keep going through the list? So these three are. 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.
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 group void themselves. And the way to do this, being quick now, is one kind of throws in the rudimentary information of what it means
of what it means that there are transformations and and a transformation like a gauge transformation of order r first or a second order and so forth it goes by this the symbol delta r so this is a bit technical this is this is a notation for the simplex that's one way of encoding transformation and higher gauge transformation so throwing in kind of directions in which things can gauge transform gives us gives us a notion of charts that models
gauge theory and higher gauge theory with ferments. 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 or default singularities. So we can ask our probes or probes to have actual singularities already in them so that we can detect singularities in the target space. Hmm.
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 propped by this to being parametric, what's called parametric stable homotopy types. And this is really this kind of fascinating issue. 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 n's that where the maps behave like maps of spheres, but the direction 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 unit lemma, think of these charts as being as being spectra as it were, like spectra of spaces, we will see the sphere spectrum in different dimensions. Okay. 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, I'll put
So here's the category. This is actually now a higher category of these negative dimensional spheres. It 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 aspect.
Alright, 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 now we want to find it. So this is the end, I think, of the little introduction to category theory and physics that I have here and 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 and
I'll be talking about mainly this this top was the shift of us to be kept by using. The first three at least and possibly the others is our probe spaces and that you know in the writings i have this goes by the name of super formulas so the generalized spaces that are pro by all this year.
so it's called super formal smooth infinity group or it's for the fact that they're smooth they're propped by the RNs formal because they are propped by infinitesimals super because they can be propped by super continuous spaces and higher group points because they they have these high gauge transformations in them all right so so that's the topos we have and now let's now let's look at one of this i have these elements this is a bit experimental of course as you know but let's see what happens so this this is like a slide that
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 going to 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 is 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. Okay. 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 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 further 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 adjoint. So this symbol here, just a moment, we'll get back there, a functor, and here is a right adjoint.
So this symbol means that this function is left and this function is right. 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 H1 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 folders so yeah very good thanks for asking actually i should have made this clearer so what is given here is
adjoined functors and other adjoined functors and I'm unwinding what extra structure that implies on our category. Yes. But in a moment actually in a moment we'll go so this is just the explanation of the method in the 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 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 we have an adjoint triple, as people say, then the corresponding endofunctors are themselves adjoined to each other. Now, them being
composites of a junctions 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 object right we can apply flat and get
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 diagram which so every x is sitting kind of in between these two extremes as for now because remember every junction 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 once we have an understanding of what this flat like, you know, him just I'm having to define things here. It's just supposed to be head. But once we know what flat actually does, we know what this extreme aspect is. And 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
Where flat and sharp are. The discrete aspect of a of a space like the set of underlying points with its discrete apology or the set of underlying points with what is called its codiscrete apology.
So like what's sometimes called discrete and codiscrete space, it will mean that every space in our category is in between being discrete and codiscrete. But this is exactly what a topology on a space means. Every topology is in between, in some sense, the discrete and the codiscrete topology. So it's a rudimentary, what we see is getting towards the 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 right adjoint if and only if, or well, if we assume that H0 is the category of sets, then this right adjoint exists, the right adjoint 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 right adjoint functor will have to choose a single object. And then one can check that the laws of adjunctions 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 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,
tautologically 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 of modalities of of ways that things that are 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, 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 Lavir, 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.
the slide isn't maybe showing everything that one could say here but this is um really what what levier came up with is what's actually going on in hegel's science of logic so hegel in the science of logic in this in this old book he kind of puts himself into the place of a really of a seer of a mystic he he just looks inside himself and says okay let me let me forget everything about and let me just kind of concentrate
So you want to make philosophy a science you cannot feel this way to the to the beginning of every thought and sorry he argues himself in a kind of poetic way into into this kind of story where is this okay so if i if i really have been all other thoughts what do i have nothing and then he over pages and pages it talks about.
This pure nothing that he sees pondering but then he gets to the point they 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 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 becomes gets into tension with with something that just exists.
with the pure being but then he said well but this this can't quite be because we we had nothing there and so he kind of feels an inner tension which eventually he says is being resolved by so out of this out of this dynamics really this logical tangent kind of feels like like a dynamics that makes something else emerge and and levieres observation was that that this is really captured by accurately
captured by what I'm drawing. This is not quite how Lévié 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 I can ask there for a further such opposition, which resolves the previous one in that so
As the composition of the functions 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 both nothing and pure being becomes what I guess Levy 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 try to do that at some point.
which i guess you can find on the n lab patreon science of logic you can you can now ask well what is this next opposition if yes if you were if you were to actually compare it to to hegel's poetry and then what can actually one can actually see that it does sort of make sense that he gets he is another opposition which again that's that's kind of what he calls the process the program the progression where where
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 Lavir'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 we can run Hegel's experiment which he had to do kind of without tools we can 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 what i'm showing
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 or it's all the time. You may just think of this as being sets, the way I finish these 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
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 subtoposis. 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 reduces 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 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 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, you know, these adjunctions, they exist. 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.
And 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 the 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 position between every object is 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 to that. 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 codiscrete 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 kind of, well,
Yes, 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 is this 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.
Monarch. So these are things they have, I gave them some sympathy, but these are things that are unknown in algebraic geometry of derived schemes or sort of just a format schemes, the reduced aspect, the co-reduced or the ramstack aspect. And it keeps climbing. And that's interesting. So we just, we're just asking in this Hegelian-Lavirian notion
So Levier never pushed this, it seems to me, beyond this first step. I think he looked at this first step and had some examples, some topos realizing that which were a bit contrived. So I don't think he looked at examples that are kind of practically important. And the insight 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 these various aspects and 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 the fermionic aspect so there's now a modality of being bosonic or being bifermionic and then it finally ends and what is interesting now and i haven't
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 okay but 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 of this real nomic aspect of a space? Well, it turns out the shape or the topological realization of an infinity stick 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. But if you forget that, you can shrink it to the point. So what if all the 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 like the technical construction 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
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 is this rheonomic as it's called. So it introduces fermions. Yeah, that is the one. So this level here, this, this line that is going to appear in just a moment, that is the one that knows about the firm ones, bosonic aspect, rheonomic aspect, bifermionic aspect. And this guy here,
is given by localization at the super point. So it's in this sense that this progression discovers first the continuum then the infinitesimals and then the super point 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. 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 so that's just a fact and so i'm saying if you know if this top of this or platonic world in which we live then we can play the hegelian game then ask okay what is 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 i'm a fear of 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 a big accomplishment of
Suggesting this formalization. He didn't really dwell on convincing examples He had some I think that I could fairly be called contrived examples that just showed the math at work but didn't really Show, you know much practical use and what I'm observing here Is that in in the actual in the actual topos that we actually want to be using in physics that we are using implicitly or not
that not only you know does this initial stage exist but the it keeps it keeps progressing and it it knows about these archetypes of our geometries the continuum the infinitesimals and the super point 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 let's go further so this is some backdrop on
on the higher topos theory that is maybe needed to 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 and 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 I
If we have the energy, let me maybe try to explain that. So, so now let's play with the super point in our in our high purpose. And the thing is, there's a pun there if we have the energy. Oh, I see. Yeah, let me 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 not a space. 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 this super point 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 super point
There is a structure of a super-Li algebra. It is as such the Abelian 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 showed the root of the following, we proved the root of the following extension. So it turns out if you start with the superpoint here and play the following game that in each step it doubles the fermionic dimension so it takes the
or one dimensional super point to the or 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 could 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 the super space times
which Green-Schwarz superstrings exist, so kind of the, I shouldn't say critical, the dimensions in which superstrings 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 this R0,2 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
This is super translation, the algebra structure emerges from R02. So how can this be? Well, let's, let's think about, so we're looking at extensions. So an extension of a Lie algebra is, is classified by a Tuco cycle, right? By a degree two element in the Chevrolet Allenberg algebra of the Lie algebra. Um, so a map from a map from the graded commutative second power of the Lie algebra with itself.
map from there to the ground field which satisfies some cosec 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. Sure. The Hermitian matrices. And there's a standard fact that the
the space times in these dimensions here, dimension three, six, and ten, 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-coast cycles on
On our zero two are exactly the information that encodes. That it costs the information of the three dimensional swimming cost space and then it continues this way. And it's quite interesting because the process down here in.
in this root of the bouquet of the tree is all in superlial to us. So these are all superlial to us and we're recovering the super translation superlial to us of these Lankovsky spacetimes. But then at some point there is no further invariant extension but there are now higher extensions that produce. So here's 11d super spacetime
So there's further extensions now here that are not extensions of super Li algebras but are what are called super L infinity algebras. So higher categorical Li 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 11D 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 spacetime like the Minkowski 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 Minkowski spacetime. Like the ordinary extensions of a super Li algebra by another super Li algebra
are encoded by a two-core cycle. This is Stunna textbook material. Now you could ask, well, what is it that a three-core cycle classifies? Well, it turns out a three-core cycle on a super-Li algebra classifies an extension by a higher form of super-Li algebra called a Li2 algebra. And so this happens with some of the brain, d-brains here, but this one, the M2 brain, is classified by a four-core cycle. And that four-core cycle is the famous, the famous bifermionic expression
on super space and that is known to be the bifurcated component of the super gravity sea field on 11 dimension super space time but it's also what is called the the west domino term of the green schwarz signal model of the kappa symmetric signal model for them to read so here we see the expression mu m2 is this forko 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 homology. So this brain bouquet kind of discovers from the super point now, first the super space times of kind of the right dimensions, then the super strings in their brain towards incarnation as sigma models.
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-core cycle happens to be either the bifurcated piece of the
dual C-field flux at level of supergravity or equivalently the west domino term for the Green-Schwarz sigma model, Green-Schwarz type signal model of the M5 band. 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 but we'll see it. So the punch line is and we'll 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 this pair of core cycles, the four core cycle and the seven core cycle, which are in a crucial relation to each other, which mimics just the equations of motion of the sea field and 11 dimensional super gravity. It just comes out. It's just kind of built in some sense, built into the super point here. And
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 to, how we actually obtain like full supergravity from this, because this is just local model space. All these are super-ly algebras. You should think of all these things appearing here as being the, as being what in an actual space-time you would see in the infinitesimal neighborhood of any point, like on a tangent space. On tangent space, it looks like a super Makovsky spacetime and there
We have these co-cycles, but now let's look at the following theorem. So we proved this just last year, actually. So it 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 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
moving frames. If you can move these co-cycles over your curved manifold, then that requirement alone is actually equivalent to the full equations of motion of 11. 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 Rory-Taschina equation. And on top of this, the
The super gravity C fields on the gauge field of the theory satisfies its equation of motion. And that is so that is repeated again on this animation set. We're just sketching it here. So. So what are we seeing here? I'm now I'm now going. I guess I've I've changed pace a bit. So we're now looking at how to see full super gravity actually appear from these ingredients that we discovered and
Okay, what are we seeing here? So we're looking at a superspace time. 11 dimensional superspace time. I didn't show the, I'm not showing the odd coordinate components. Maybe I should have added them. Sure. So this is says odd dimension 32 as befits n equals one. So this is the, this is a space time that is locally modeled on the, on the tip of this immersion diagram that we saw is locally modeled on this 11D superspace time. And we're asking,
that on this space time 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 bifurmanic expression and so we're asking well let there be
Forms that you know saying saying we have a super space-time but this I mean it is equipped with a Supermetric 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 in these field bites, which are are the things that locally identify the space-time with Mikovsky space-time
So we say, okay, let there be two flux densities, as we're going to call them, whose bifurcated component is exactly the one that it's kind of got given. And then let them have, in order to patch it up, let them have any other ordinary component, an ordinary four-form, 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, I haven't fully explained this, but it's clear that
g4 being closed it's an element of the chevrolet eilenberg algebra or l infinity algebra and it's being closed that's just a co-cycle condition and then 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 be this equation that 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 so 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 space time 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 to the square of the other. So we ask, well, what is that? What is it that classifies pairs of differential forms with that property? That's what I'm asking here.
And it turns out that this is exactly what is the structure 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 on a multi technical conditions, finite type, rational, finite type to each logical space and l infinity algebra, whose 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 to the homotopic structure in a space and the space that appears here, the space that happens to encode
just the co-circle relation that finds these M2 and 5-band co-circles happens to be the force sphere or any other space of the rational type of the force 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
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 encoded the equations of motion of supergravity in a kind of purely cohomological or rational homotopical construction where we just say the fact that a spacetime satisfies supergravity equations of motion just means the flux synthesis of this form exactly arranged into a
4-sphere valued flat differential form where by 4-sphere valued I mean valued in the white head at infinity above the 4-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 super gravity so this is this is now getting us into this is now getting us into an actual actual new new physical realm so i 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 now almost exactly 100 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 Sov answered, at least in hindsight, in the 1930s about electromagnetism can and should be asked about all higher gauge theories, in particular about our
or 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 dimensional supergravity 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 that part of the force sphere that is detectable by differential forms, well, we can ask
We can ask for a classifying space to be called A here that classifies the actual global charges, not just approximated by the French reform data. And then we just want to ask for, so this would like in electromagnetism, this would be the, um, the monopole sector classifying space for the monopole sector. And so we ask what, where are these things comparable so that we can ask these charges source, these fluxes,
And as such, they are compatible, right? Not every set of charges can source given fluxes. So there needs to be some compatibility condition to them, right? 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 differential forms to their what's called their moduli stack. Anyway, some deformation space for this and that that receives a map in this high topos from any space, which has the property that is whitehead L infinity algebra coincides with that of the force field, which which means that it's it's our rationalization is that of the force 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 11-dimensional 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 field 11 and supergrave is then a homotopy between the charges to the fluxes in this diagram, in this diagram as it lifts in our higher topos. Of course, the canonical choice for A is the force itself, not the canonical choice, like the initial choice, the simplest choice.
and so what we've been calling hypothesis H is right so this diagram the option of this diagram is twofold it says well 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 dimensional super gravity which is an example and second the second thing is well there is actually among the infinite set of possible choices for the choice colonization
There's one choice that is kind of singled out as being the, it's the simplest one. 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 a Levenin-Rome supergravity, which has not found much attention before. There's articles by Diakonevsky, Fried and Moore,
from a few years back who talked about a model of the sea field which went in this direction but um but wasn't quite comprehensive i would say and and here we say first of all well look here's a choice to be made and second there is a there is a specific choice that kind of stands out and the you see what we're seeing here is that the usual data that goes into field theories
Yes, but the classifying space is, um,
Part of what characterizes the full field content. So hypothesis age is a hypothesis about what is the global completion of the field content of the level that the left dimensional super gravity in analogy to how I guess. Yes. So you see, I'm sorry. Yeah, I don't have a special slide for this, but let me just say it in words. So let's go back to electromagnetism. Um,
So what happened in 1850, Maxwell writes it in modern language, right? He writes a two form F two. 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. It's about it's a two form. It's close and and transfer some other conditions. It's a field strength, technically, as we would call it.
the yeah exactly the 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 yes so that's why we also call it the flux density that's maybe a more suggestive term here sure but the flux density absolutely is it's just another inclination of the field strength here's sorry and so
So then the decades pass and 1931, with a bit of paraphrasing, Dirac comes and says, no, no, this cannot be the full field content of the electromagnetic field. And 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, the gauge potentials.
What is such that our F connections the connections the local one from so that F is the differential of these A's. But but that alone is not actually the full inside because that it's quite interesting because that is something that already faraday was actually playing with faraday and makes for both the head of funny term for this I could look this up somewhere I forget now they were actually they couldn't quite nail it down but they
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. Like when you see discussions, for instance, of all these supergravities, you see people often write down potentials for these things here. They write a C3, a 3-form, and such that G4 is dC3, right?
so yes that is that is the analog but that cannot be true globally like we're interested if there's non-trivial charge in our space time then this g4 is not exact globally yes it measures it measures a certain dram class which is kind of a rational image of this full whatever aqua multi class we're seeing here and so so the whole point really
of the global charges that 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, gauge transformation, gluing data on double overlaps of these charts, which in one way for electromagnetism, this is another way to
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
only chart some global cohomological structure and that is 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 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 do i call the drug string is really is really speaking about what mathematicians will call the clutching construction for for line bundles on a sphere so it's a way of encoding the transition data that you need to have a check go cycle for your whatever data on the sphere but in speaking about this direct 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 it, understood this electromagnetic
situation in modern language is maybe Orlando Alvarez in the 80s, who has articles on Czech cohomology and Dirac charge quantization. His phrases, his 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 at a period, 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 are gluing this together. 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.
I think that's an interesting and appreciated aspect that actually to fully define just 11 dimensional 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? I don't know. Well, he hasn't written much about M-theory in the past few years.
doesn't seem like many people are working on m theory no absolutely yeah absolutely um there has not been much at all 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 that was always kind of more enigmatic in a way because 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 IKKT model is actually reduced to zero dimensions some 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 just lately and i guess there's some claims that i guess the japanese group is claiming that computer simulation shows that there are spontaneous compactifications to four dimensions more seen by numerical computation in these matrix models i don't know if that has been checked
But yeah so this is um this is what people have been playing with and now to come to your question like why why is there no further progress so i think well i suspect part of the problem is the will be alluded to at the very beginning that it's not actually clear like or it was not actually clear where even to look like
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, right? That's how people build new theories. You write a new Lagrangian, but the Lagrangian here is sort of already known.
It's 11 years supergravity 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. So 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 of the pointed 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
In physics, the problem that we don't understand this year, like analytically, apart from computer simulation, is really what haunts particle physics in that whenever QCD at non-extreme temperatures plays any role, any hydrodynamic contributions can never really be fully computed with perturbation theory. Nobody really knows what's going on there. And 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 lattice 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 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 this non-perturbative theory is not known and now it's interesting to note that
The history of this problem has it's kind of an interesting ironic turn to it. So let's look back how did so what happened in the past so way back in what is it the 50s or something or or maybe the 70s people thought about this sort of thinking hard about this problem of non-perturbable FQCD and there's very good arguments of course no real proof that
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 confinement in the non-perturbative regime, so probably the flux tubes are the actual degrees of freedom to be discussed for non-perturbative QCD. This is 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
Then comes this computation which is really still at the heart of much 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-perseverative 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 have to be understood as kind of a high dimensional unobserved bulk space time that hosts the actual observed space time, which sits in it in the form of a brain where the quarks sit on the brain and the strings.
In their need for a higher dimensional bulk may have the endpoints attached to these quarks, but my otherwise probed this bulk. And then of course, well, you know the story then, then, you know, 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 extra QCD and a whole lot of numbers can actually be crunched out there. There's actually quantitative
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 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 lot of hints or 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, then in some sense historically, that is really the answer to how do quarks behave non-perturbatively, then at least in the Schraf outline, M-theory reappears at least as the possible ultimate answer to how to actually define construct non-perturbative QFT. And so
So the question then is, how do we even go about formulating M3? Do we write on a Lagrangian or is it just another matrix model? And 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 I very much agree with this sentiment and I guess what I've been showing here is kind of 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.
Thank you so much for coming on. I appreciate you spending three hours with me.
Thank you, Kurt, for inviting me and for doing this. I appreciate it. That was a 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.
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"text": " The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science, they analyze culture, they analyze finance, economics, business, international affairs across every region."
},
{
"end_time": 53.234,
"index": 1,
"start_time": 26.203,
"text": " I'm particularly liking their new insider feature was just launched this month it gives you gives me a front row access to the economist internal editorial debates where senior editors argue through the news with world leaders and policy makers and twice weekly long format shows basically an extremely high quality podcast whether it's scientific innovation or shifting global politics the economist provides comprehensive coverage beyond headlines."
},
{
"end_time": 78.114,
"index": 2,
"start_time": 53.558,
"text": " Think Verizon, the best 5G network is expensive? Think again. Bring in your AT&T or T-Mobile bill to a Verizon store today and we'll give you a better deal. Now what to do with your unwanted bills? Ever seen an origami version of the Miami Bull?"
},
{
"end_time": 104.462,
"index": 3,
"start_time": 78.626,
"text": " 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."
},
{
"end_time": 132.5,
"index": 4,
"start_time": 105.93,
"text": " 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 Unis Shriver 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,"
},
{
"end_time": 161.544,
"index": 5,
"start_time": 132.705,
"text": " Uris Schreiber!"
},
{
"end_time": 190.913,
"index": 6,
"start_time": 162.346,
"text": " 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. You know, I didn't really plan to give such a talk before you invited me. And then I thought, well, what what would I say? And then I thought, well, the title of your podcast is theories of everything. So I thought, well, OK, maybe I can say something in that direction. And"
},
{
"end_time": 217.551,
"index": 7,
"start_time": 191.305,
"text": " 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. I'm starting with some basics in ontology, you might say, and then maybe if we have time, arrive at some actual experimental physics. Yes, great."
},
{
"end_time": 246.647,
"index": 8,
"start_time": 217.739,
"text": " 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 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 like yes, technically. OK, so tell me about the creation of NLAB and what inspired that as well."
},
{
"end_time": 274.343,
"index": 9,
"start_time": 247.807,
"text": " 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. Well, you know, whenever you figure out something, make a note about it. That was the original driving idea behind it, that there is a place where we can discuss things, but then also"
},
{
"end_time": 297.278,
"index": 10,
"start_time": 274.804,
"text": " What was the original motivation we but then also have the opportunity to record any insights you know any actual advancements that have been made so it started out as very much you know not an encyclopedia which maybe it is what it appears as now but i think as the introduction the home page says it's a it's meant to be left notes."
},
{
"end_time": 315.23,
"index": 11,
"start_time": 297.756,
"text": " What's what they have been writing on what they have been thinking about what they plan to do maybe. And give that all a home and yeah you know it started you know that was actually back in the days before."
},
{
"end_time": 344.565,
"index": 12,
"start_time": 315.623,
"text": " 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 sing around. There was, I guess, a group called SciPhysics Research where we would discuss. And then later,"
},
{
"end_time": 371.527,
"index": 13,
"start_time": 344.855,
"text": " 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, you know, 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 to end up at some point and yeah, it has been growing slowly, but surely ever since."
},
{
"end_time": 396.032,
"index": 14,
"start_time": 373.166,
"text": " 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."
},
{
"end_time": 423.609,
"index": 15,
"start_time": 397.398,
"text": " Yeah, so that's a very good question. So actually, I think it's, um, this is a common misunderstanding. I'm sure 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 there's lots of other subjects. So to my mind, the point of category theory is, is the organizing principle, the category theory is like the big index of math where things find its find the place, find the home."
},
{
"end_time": 452.637,
"index": 16,
"start_time": 424.309,
"text": " so it's the organizing principle behind stuff and as such it's i think ideal for a for an encyclopedic you know or almost encyclopedic um website because it allows to to relate things in the proper way you can you can go and say oh here's a construction in that such and such field oh but look it's just just such and such limit that also appears here and there so things um get the proper context and become meaningful i suppose"
},
{
"end_time": 476.63,
"index": 17,
"start_time": 453.626,
"text": " 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 analytics really mostly, it's growing, you know, nobody is being paid for editing the analytics. So what happens is people edit it when and if it's useful for them as editors. So for me personally, I, you know,"
},
{
"end_time": 504.991,
"index": 18,
"start_time": 477.159,
"text": " 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 like 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 n lab and maybe the philosophers are familiar with this because this is"
},
{
"end_time": 515.589,
"index": 19,
"start_time": 505.299,
"text": " 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,"
},
{
"end_time": 545.06,
"index": 20,
"start_time": 515.759,
"text": " 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."
},
{
"end_time": 575.162,
"index": 21,
"start_time": 545.572,
"text": " 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."
},
{
"end_time": 598.131,
"index": 22,
"start_time": 576.015,
"text": " Right yeah so actually I was planning to end my today's presentation on such a note when we talk about maybe non-perturbative quantum field theory and maybe m theory and how a big problem there is that it's not just um heart you know you know I will I will end by saying that understanding non-perturbative physics like confined qc and the"
},
{
"end_time": 626.971,
"index": 23,
"start_time": 598.609,
"text": " Strongly coupled solid-state physics is the big or the grand open question of theoretical physics of our time. 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"
},
{
"end_time": 654.292,
"index": 24,
"start_time": 627.398,
"text": " Language level insight is lacking that would tell us what we're even looking for. And I think that is that is the reason why so little progress has been made on on the subject, non perturbative quantum field theory, because there was very 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"
},
{
"end_time": 679.258,
"index": 25,
"start_time": 654.821,
"text": " Today that you know going back to some really deep seeming ontological foundations maybe philosophical foundations if you will but but all in the realm of you know 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"
},
{
"end_time": 703.763,
"index": 26,
"start_time": 680.247,
"text": " Which arose actually from a which so that the post that you just quoted was made at an intermediate stage of the development that I'm going to present today because you know there was at the time when when I started appreciating this some people have heard me say the sagalian aspect of parts of topos theory and"
},
{
"end_time": 729.821,
"index": 27,
"start_time": 704.684,
"text": " And we were 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 like generalized differential cohomology theories, which are, as maybe I can say a bit more later, which are meant to be the actual, you know, full mathematical formalization of gauge fields, of higher gauge fields. And that's actually, that wasn't quite clear actually."
},
{
"end_time": 758.558,
"index": 28,
"start_time": 730.282,
"text": " and i you know that was really what i did since my phd thesis just thinking about how to properly phrase the the idea of generalized meaning also higher gauge fields like like one expects to see in string theory like rf fields the super gravity c field and these things what is it really like what is very fundamentally what is it basically looking for when we say we want to build a model of the c field say and that was very much not clear and um"
},
{
"end_time": 786.049,
"index": 29,
"start_time": 759.428,
"text": " Yeah, so then we started toying with these modalities on topos of course, and at some point the solution appeared and it kind of appeared in tandem with me at least understanding also the role of what these modalities, the role that these modalities play in a more philosophical realm. So there was a back and forth. I can't really quite tell what was first and what came later."
},
{
"end_time": 813.729,
"index": 30,
"start_time": 786.954,
"text": " 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 fields in full generality and deeply. Yeah, and it turns out that it is related to this term of cohesive infinity top bosses, which is a term borrowed from Levere, who in turn was reading Hegel and trying very hard"
},
{
"end_time": 821.8,
"index": 31,
"start_time": 814.224,
"text": " To understand what is actually going on there and I would say actually succeeded, which is quite quite fascinating actually."
},
{
"end_time": 847.073,
"index": 32,
"start_time": 822.978,
"text": " 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 the veer and, and categories of being and nothing. In particular, you said that"
},
{
"end_time": 874.855,
"index": 33,
"start_time": 847.875,
"text": " 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. 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"
},
{
"end_time": 897.995,
"index": 34,
"start_time": 875.367,
"text": " creation story if you wish and show how once you know 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 and it's"
},
{
"end_time": 928.166,
"index": 35,
"start_time": 898.592,
"text": " It's at least fun. Also, I should emphasize I'm not selling any theory or anything. I'm not going to, you know, present any hypothesis that people need to buy into. I'm just presenting some facts. I'm 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'd take the occasion that I'm speaking here in to your audience on your podcast to go back to these old ideas."
},
{
"end_time": 954.872,
"index": 36,
"start_time": 928.66,
"text": " And do a little bit of an exposition of them. Yeah. Wonderful. All right. Let's get to it. All right. Yeah. Thanks. So, um, 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, um, that all revolve around a little bit of aspects of what mine might call, um, going towards a theory of everything. So these are some expositions."
},
{
"end_time": 979.65,
"index": 37,
"start_time": 955.401,
"text": " Awesome theorems that have you can 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'm that i started with here shows is we want to start looking at something at the very foundations of thought."
},
{
"end_time": 1004.923,
"index": 38,
"start_time": 980.043,
"text": " So this is as we'll get to a little cartoon of of an adjunction category theory and by analyzing some structures that appear there which which are sort of known to some experts but actually not widely known we'll we'll see some dynamics emerge some dynamics in the in the platonic sense in the realm of ideas where concepts will emerge and eventually we'll be talking about"
},
{
"end_time": 1034.923,
"index": 39,
"start_time": 1005.452,
"text": " I'm gravity super gravity in fact the level of super gravity and then maybe then five brain and if if time and energy is sufficient little bit worried about energy but then we'll get to some actual statements about a strongly couple quantum systems with what is called topological order and ionic expectations which is what this little animated cartoon on the right is alluding to and that connects to the extra current research i'm currently doing alright and why don't you explain the title as well please"
},
{
"end_time": 1057.278,
"index": 40,
"start_time": 1035.811,
"text": " right at the title of this page right so i was thinking about how to call this and um you know i must say there's a curious um how should i say there's a curious aspect of 20th century science that some of the deepest um thoughts have silly names like theories of everything it's not a very elegant name i mean it's not too bad"
},
{
"end_time": 1078.029,
"index": 41,
"start_time": 1057.807,
"text": " The really bad ones are like Big Bang and Black Hole, right? So these are terms that 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'd call this talk about theories of everything, but in a slightly more fancy"
},
{
"end_time": 1098.899,
"index": 42,
"start_time": 1078.456,
"text": " i see i see that's that's what the title is doing okay it's really what it really is it's so i'm not claiming i will of course not claim to have any theory of everything in my hand but i i don't want to claim that i have um like found some fragments of what looks like uh should belong to such a theory or at least which are noteworthy to"
},
{
"end_time": 1128.558,
"index": 43,
"start_time": 1099.548,
"text": " to take note of if one is interested in, in theories of everything, um, in an actual deep sense, um, like in the sense of starting, not just from, from the assumption of, of the notion of quantum field theory with its implied notion of space and everything, but studying deeper, like if there is a level of pure logic, that's, 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?"
},
{
"end_time": 1158.131,
"index": 44,
"start_time": 1129.514,
"text": " 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 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"
},
{
"end_time": 1186.8,
"index": 45,
"start_time": 1158.422,
"text": " 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,"
},
{
"end_time": 1217.739,
"index": 46,
"start_time": 1187.773,
"text": " 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"
},
{
"end_time": 1245.128,
"index": 47,
"start_time": 1218.66,
"text": " People point out 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."
},
{
"end_time": 1273.951,
"index": 48,
"start_time": 1245.555,
"text": " 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 is or 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"
},
{
"end_time": 1301.203,
"index": 49,
"start_time": 1274.565,
"text": " With computing, if you wish, quality is 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,"
},
{
"end_time": 1324.667,
"index": 50,
"start_time": 1301.869,
"text": " This complaint, right? The main problem is 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"
},
{
"end_time": 1343.302,
"index": 51,
"start_time": 1325.503,
"text": " What is your favorite quantum field theory?"
},
{
"end_time": 1371.817,
"index": 52,
"start_time": 1343.78,
"text": " like me are going now and say no wait a second it's actually the opposite we're we're lacking math i think the trouble is that even the meaning of what it means to to have mathematical formulation of physics has been forgotten a little bit at times so what i'm speaking about is having um first of all precise definitions like to have definitions exact unambiguous definitions that's that's actually more important even than the the rigorous proofs that one can base on these um"
},
{
"end_time": 1393.814,
"index": 53,
"start_time": 1372.09,
"text": " 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 and physics to have some precise definitions. So I think what people mean when they say"
},
{
"end_time": 1422.295,
"index": 54,
"start_time": 1394.821,
"text": " The 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."
},
{
"end_time": 1435.947,
"index": 55,
"start_time": 1422.722,
"text": " What is it schematic like it's not filled with physical life but it's also not filled with mathematical precision and so. So it's maybe a mistake to to mistake that for math anyway so that's my."
},
{
"end_time": 1461.015,
"index": 56,
"start_time": 1436.425,
"text": " 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 or we're not it's not clear that"
},
{
"end_time": 1483.319,
"index": 57,
"start_time": 1461.323,
"text": " Douglas Goldstein, CFP®, Financial Planner & Investment Advisor"
},
{
"end_time": 1507.073,
"index": 58,
"start_time": 1483.746,
"text": " 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. To stay informed, in an ever-evolving landscape, I see The Economist as a wellspring of insightful analysis and in-depth reporting on the various topics we explore here and beyond."
},
{
"end_time": 1531.749,
"index": 59,
"start_time": 1507.534,
"text": " The economist's commitment to rigorous journalism means you get a clear picture of the world's most significant developments, whether it's in scientific innovation or the shifting tectonic plates of global politics. The economist provides comprehensive coverage that goes beyond the headlines. What sets the economist apart is their ability to make complex issues accessible and engaging, much like we strive to do in this podcast."
},
{
"end_time": 1553.404,
"index": 60,
"start_time": 1531.749,
"text": " If you're passionate about expanding your knowledge and gaining a deeper understanding of the forces that shape our world, then I highly recommend subscribing to The Economist. It's an investment into intellectual growth, one that you won't regret. As a listener of Toe, you get a special 20% off discount. Now you can enjoy The Economist and all it has to offer for less."
},
{
"end_time": 1569.65,
"index": 61,
"start_time": 1553.404,
"text": " Join with reality. I think that's what's meant."
},
{
"end_time": 1586.578,
"index": 62,
"start_time": 1570.094,
"text": " No, 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."
},
{
"end_time": 1604.002,
"index": 63,
"start_time": 1586.886,
"text": " 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 now is rigorous yeah it's not something that you could actually explain to a mathematician."
},
{
"end_time": 1622.671,
"index": 64,
"start_time": 1604.565,
"text": " They will not end and that's i think one one reason why we're not seeing as much interaction between math and physics these days is maybe we did in the nineties the mathematicians pretty much i lost they do not know when this is talk about. All these brains and everything they just don't know don't have an entry point anymore."
},
{
"end_time": 1648.643,
"index": 65,
"start_time": 1623.08,
"text": " 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."
},
{
"end_time": 1674.804,
"index": 66,
"start_time": 1649.104,
"text": " And we'll see maybe if you have more questions or 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-perturbative qft or m theory or something so"
},
{
"end_time": 1703.439,
"index": 67,
"start_time": 1675.179,
"text": " What we need in order to make some tangible progress for such matters, we need a language that can speak about concepts in a precise way, not just the 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,"
},
{
"end_time": 1732.261,
"index": 68,
"start_time": 1703.848,
"text": " 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."
},
{
"end_time": 1758.131,
"index": 69,
"start_time": 1733.456,
"text": " And so I'm going to, I'm going to claim without much ado here that the language for these four aspects, structure, duality, equality, and effects is categories in this order, adjunctions. And I'm going to speak a bit about those modalities and monads. So, so this technology, this, this categorical algebra is it's sometimes called is what Levier at some point identified with what Hegel in turn called the objective logic."
},
{
"end_time": 1775.845,
"index": 70,
"start_time": 1759.292,
"text": " Note, I've written about Laver's theorem here on Substack. It actually went viral. It's a theorem that encompasses Gödel's incompleteness theorem, Cantor's diagonalization argument, Turing's halting problem, and even Tarski's undefinability theorem, as well as what self-reference means."
},
{
"end_time": 1795.708,
"index": 71,
"start_time": 1776.135,
"text": " Feel free to read it and subscribe to the Substack."
},
{
"end_time": 1825.35,
"index": 72,
"start_time": 1796.135,
"text": " making statements about what we think we know or do not know but it's a way of speaking about sort of the the world the platonic world if you will even though platonic is not quite the right version 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 of reality in a sense that's what this term means okay so here's a lightning crash course on what categories are about yeah you should stop me maybe if this goes if i'm going"
},
{
"end_time": 1852.483,
"index": 73,
"start_time": 1825.794,
"text": " 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"
},
{
"end_time": 1882.125,
"index": 74,
"start_time": 1852.858,
"text": " The edges of the graph such that the composition is, you know, satisfies the expected properties. It's dissociative 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, these terms come from what you might call the archetypical examples of category. 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, um,"
},
{
"end_time": 1896.254,
"index": 75,
"start_time": 1882.449,
"text": " category shows objects are spaces of sorts physical spaces but also more abstract space like moralized spaces classifying spaces. Space quite generally and where the edges and these graphs the maps."
},
{
"end_time": 1918.951,
"index": 76,
"start_time": 1897.159,
"text": " I call the morphisms are just maps between the spaces and this is actually going to be a big theme in just a moment that even though it's maybe not often fully recognized but in order to the physics one that just a grandian ordinary classical field theory one very quickly runs into the issue that one is dealing with spaces."
},
{
"end_time": 1948.234,
"index": 77,
"start_time": 1919.667,
"text": " 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 your 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"
},
{
"end_time": 1977.551,
"index": 78,
"start_time": 1948.695,
"text": " of the difficulties that afflict field theory result from the fact that one is dealing with these generalized spaces and if you have 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 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"
},
{
"end_time": 2004.275,
"index": 79,
"start_time": 1978.148,
"text": " 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 or before the space of super fields and all these things."
},
{
"end_time": 2033.933,
"index": 80,
"start_time": 2004.923,
"text": " And so the standard way of drawing these things is with these diagrams shown on the right here. So 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"
},
{
"end_time": 2063.268,
"index": 81,
"start_time": 2034.206,
"text": " Add 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 is that you think of the object as being data types. Think of this as a procedural declaration in"
},
{
"end_time": 2092.619,
"index": 82,
"start_time": 2063.746,
"text": " 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 you 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"
},
{
"end_time": 2116.459,
"index": 83,
"start_time": 2093.183,
"text": " of the data that is a point in that space or more generally any any figure in that space. Yeah okay we'll we'll get to this and then for those since um yeah just to connect maybe to some um currently fashionable verbiage um among categories is is group points where"
},
{
"end_time": 2143.285,
"index": 84,
"start_time": 2116.732,
"text": " 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 transformation. So when these maps here are invertible, when they have inverses, then they identify these two objects with each other in a specific way."
},
{
"end_time": 2167.995,
"index": 85,
"start_time": 2143.507,
"text": " custom isomorphisms of course and uh 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 um the the operation of a symmetry group on that object and so um the special case of categories all morphisms are invertible are group points and they should very much be thought of as being the incarnation of the idea of of symmetry and specifically of gauge symmetry"
},
{
"end_time": 2197.619,
"index": 86,
"start_time": 2168.609,
"text": " You should think of group points. I can think of group points as being like sets equipped with gauge transformations. In fact, a very important example of a category is, for instance, the group point of say, you know, Maxwell gauge fields on a given space time. So the objects in that case are gauge field configurations and morphisms are the true gauge transformations between these. And so if from that perspective we lift"
},
{
"end_time": 2226.766,
"index": 87,
"start_time": 2197.995,
"text": " this condition that all morphisms be invertible, then we have pretty much what these days people want to understand are non-invertible symmetries, namely a situation where there's operations between things that behave a whole lot like 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 symmetries precise, you will in the end have to be talking about"
},
{
"end_time": 2253.729,
"index": 88,
"start_time": 2227.312,
"text": " But categories and then of course so this is what is shown here then of course once you have these categories you want to you want to talk about finding images and we'll see concrete examples of this in just a moment to 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 cnd you want to say what's an image of that category in that category well it's just just a map that from c to d that maps all the objects"
},
{
"end_time": 2283.626,
"index": 89,
"start_time": 2254.138,
"text": " 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 in a sense"
},
{
"end_time": 2311.408,
"index": 90,
"start_time": 2284.821,
"text": " 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 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"
},
{
"end_time": 2336.903,
"index": 91,
"start_time": 2312.142,
"text": " and here's its image, another function g, and here's its image. Then we can ask, are these two images deformable into each other? Where, if you maybe think of as you can, for the special situations where we're looking at the categories set out, 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,"
},
{
"end_time": 2351.305,
"index": 92,
"start_time": 2337.363,
"text": " 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"
},
{
"end_time": 2376.391,
"index": 93,
"start_time": 2351.954,
"text": " is respected and now we just abstract from that situation if you will we don't have actual paths here we just have these 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 well this is"
},
{
"end_time": 2407.09,
"index": 94,
"start_time": 2377.517,
"text": " This result down here counts as a deformation of this result up here by these deformations if this is compatible. So, namely, 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 the square is actually filled by inequality. So that we mean that the composite of this morphism with this morphism"
},
{
"end_time": 2437.244,
"index": 95,
"start_time": 2407.398,
"text": " 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 yes all squares commute And later on of course, I will not commute that will that will there will be some home some further homotopies in there But for the moment they will just commute So that's that's the basic that's the thing you you open Wikipedia and his other book any of many books on categories here They will tell you that this is what a category is"
},
{
"end_time": 2460.947,
"index": 96,
"start_time": 2437.585,
"text": " 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 it is this actually worth our time this is um this is trivial if this appears trivial and thought logical then that's good that's what it is it's not this is not doing much and i i now want to make another hot take i will actually oppose now well"
},
{
"end_time": 2473.985,
"index": 97,
"start_time": 2461.51,
"text": " And often quoted quoted saying in category theory and will want to highlight that it's not actually true so this may originate with with this reference here for a fright where."
},
{
"end_time": 2499.206,
"index": 98,
"start_time": 2474.684,
"text": " Which is certainly historically somewhat accurate, where people said, well, so what are we trying to do here? That'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"
},
{
"end_time": 2524.957,
"index": 99,
"start_time": 2499.206,
"text": " To at least half the extent for the original article by McLean or category theory where where they actually asked what is a dual object so they they essentially observed that at that time. In algebraic topology people would would speak about certain things being naturally isomorphic to each other or or naturally transformed into each other without there being an actual definition of what it actually means to be natural."
},
{
"end_time": 2547.961,
"index": 100,
"start_time": 2525.64,
"text": " 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 to make precise what it actually means for 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."
},
{
"end_time": 2566.664,
"index": 101,
"start_time": 2548.541,
"text": " 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"
},
{
"end_time": 2591.391,
"index": 102,
"start_time": 2567.005,
"text": " 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 me making a big deal and giving everything here a name wait I don't understand why you call it thin"
},
{
"end_time": 2602.227,
"index": 103,
"start_time": 2591.954,
"text": " Well, what do you mean? I want to I want to make a point now that there is more to category theory, a whole lot more like the whole the actual theory. Well, good. Thanks for interacting."
},
{
"end_time": 2629.701,
"index": 104,
"start_time": 2603.439,
"text": " 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, defines everything that is about categories."
},
{
"end_time": 2659.701,
"index": 105,
"start_time": 2629.872,
"text": " But the actual theory, the category theory, the non-trivial aspects of it, also the, 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 a junctions. 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."
},
{
"end_time": 2683.968,
"index": 106,
"start_time": 2660.077,
"text": " So I want to 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, this is really where something not completely obvious happens and all that all the deep parts of category theory are related to a junctions. So what is an adjunction? An adjunction is"
},
{
"end_time": 2713.592,
"index": 107,
"start_time": 2684.514,
"text": " 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 a joint that are and that makes them being a joint. They're equipped with natural transformations. Remember that was these deformations of functors from so right. These two functors, they go back and forth. So as we compose them, they give an endo functor on one of these categories or the other. And so we want a natural transformation that"
},
{
"end_time": 2738.626,
"index": 108,
"start_time": 2714.138,
"text": " deforms, if you wish, our composite going back and forth to or from the identity function, 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."
},
{
"end_time": 2755.623,
"index": 109,
"start_time": 2738.916,
"text": " 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 the"
},
{
"end_time": 2776.237,
"index": 110,
"start_time": 2756.8,
"text": " what's usually called the triangle identity suggestively yes i see yeah i draw it suggestively like this so what this is showing is i'm not 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"
},
{
"end_time": 2795.947,
"index": 111,
"start_time": 2776.732,
"text": " 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 to this composite and the core 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"
},
{
"end_time": 2823.114,
"index": 112,
"start_time": 2795.947,
"text": " By expanding out these identity morphisms here to have some finite width and then they look like, I don't know, like a poached egg or something. But I just want to amplify it's the very same diagram that I'm not changing anything here. This is just the definition of unit or co-unit in an adjunction. And the zigzag identities, if you draw them this way, have this fun yin-yang form."
},
{
"end_time": 2848.746,
"index": 113,
"start_time": 2823.456,
"text": " 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 can already see maybe that the definition of adjunction is somehow on a different level than these previous definitions. I mean here"
},
{
"end_time": 2877.108,
"index": 114,
"start_time": 2849.497,
"text": " Right, we had some, like if you think of these as graphs, we're basically dealing with discrete spaces or something that looks like discrete spaces, at least if 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 just mentioned some examples. This is maybe an exercise that everybody needs to"
},
{
"end_time": 2903.387,
"index": 115,
"start_time": 2877.483,
"text": " 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 that's a very simple example. One can make the following exercise. We can ask for that. One can think of the natural numbers."
},
{
"end_time": 2929.957,
"index": 116,
"start_time": 2904.036,
"text": " 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, is a morphism. So where 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, the partial order on the, on the national or on the integers regarded as a category."
},
{
"end_time": 2953.626,
"index": 117,
"start_time": 2930.572,
"text": " 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. There's an endo functor on the integers that projects every number to its"
},
{
"end_time": 2983.217,
"index": 118,
"start_time": 2954.053,
"text": " It's closest smaller even number or its closest larger odd number. And these two factors I joined to each other. They do each other expressing in a nice way the, you know, what you would intuitively think should be, should be a very simple form of duality between even and odd. And actually that example, if one actually works it out, that example is kind of nice in that it, it shows, it shows both how being even is in some sense opposite to being odd. And at the same time,"
},
{
"end_time": 3012.602,
"index": 119,
"start_time": 2983.814,
"text": " 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, the isomorphism being at one. So there's a whole lot or, how should one say, surprisingly, this baby as this example is this, it has some nice philosophical insights to share. So we could look more into this, but I just want to mention this."
},
{
"end_time": 3038.473,
"index": 120,
"start_time": 3013.285,
"text": " 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."
},
{
"end_time": 3065.657,
"index": 121,
"start_time": 3038.814,
"text": " 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 that takes necessarily every element to that single element."
},
{
"end_time": 3094.48,
"index": 122,
"start_time": 3066.374,
"text": " 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 and 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 in this analogy is actually the unit set the set with one element"
},
{
"end_time": 3116.544,
"index": 123,
"start_time": 3096.527,
"text": " 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"
},
{
"end_time": 3139.104,
"index": 124,
"start_time": 3117.551,
"text": " The initial object is the bottom element of 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."
},
{
"end_time": 3167.602,
"index": 125,
"start_time": 3140.623,
"text": " And then, yeah, so just as a side remark, I want to actually mention here, not sure if we actually get to this, but I want to 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,"
},
{
"end_time": 3198.012,
"index": 126,
"start_time": 3168.507,
"text": " And it's not used very systematically always, but some of them are actually just plain examples of reductions. 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 high dimensional space time. And you want to say that there's an equivalence that you can equivalently regard the system from the point of view from lower dimensional space time, where some of the"
},
{
"end_time": 3217.398,
"index": 127,
"start_time": 3199.019,
"text": " Degrees of freedom that you had in higher dimensions are incarnated as so-called Calusa-Klein modes. So that's a very important and 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."
},
{
"end_time": 3246.032,
"index": 128,
"start_time": 3217.773,
"text": " and in particular especially examples of this one finds t-duality or at least at least the right was known topological t-duality and in fact the duality between m and 2a so the 11 the reduction of 11 is super gravity to 10d and so i'm giving some references here there's this 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 but i just wanted to make this a side remark to indicate"
},
{
"end_time": 3275.572,
"index": 129,
"start_time": 3246.305,
"text": " 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."
},
{
"end_time": 3305.794,
"index": 130,
"start_time": 3276.186,
"text": " I'm giving a specific 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 he or 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."
},
{
"end_time": 3325.503,
"index": 131,
"start_time": 3306.476,
"text": " 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."
},
{
"end_time": 3352.193,
"index": 132,
"start_time": 3326.067,
"text": " I want to be looking at concretely as categories of categories of such generalised 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 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."
},
{
"end_time": 3378.08,
"index": 133,
"start_time": 3352.705,
"text": " A very good example is supersymmetry. We just go, you 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 physically 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. Remember, actually, this is"
},
{
"end_time": 3405.657,
"index": 134,
"start_time": 3378.37,
"text": " 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 bike 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"
},
{
"end_time": 3427.927,
"index": 135,
"start_time": 3406.596,
"text": " To very good avail have used this fact without thinking about what it means usually what it means actually globally for space if it's coordinates behave this way and so here category series actually comes to the rescue and in its guise as top of theory provides actually a fascinating accurate"
},
{
"end_time": 3457.329,
"index": 136,
"start_time": 3428.49,
"text": " um, 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 C's here could just be, um, like ordinary Cartesian spaces are ends."
},
{
"end_time": 3481.92,
"index": 137,
"start_time": 3457.824,
"text": " for for any end. So just the actual original Cartesian coordinate charts of anything and maps between them say any smooth maps between our ends or they could be super Cartesian spaces Cartesian spaces with some super coordinates attached to them but we think of these C's as being kind of super simple super naive things that we can handle algebraically maybe"
},
{
"end_time": 3494.957,
"index": 138,
"start_time": 3482.551,
"text": " We could even say and we'll see this in a moment that you'll see maybe may have some new potent coordinates. Well, what write some epsilons that's greater zero. What will we mean by this? Well, we will just mean the kind of define such a seed to be."
},
{
"end_time": 3524.804,
"index": 139,
"start_time": 3495.282,
"text": " It's algebra functions, and we just declare that the algebra function is not just an actual algebra function, 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."
},
{
"end_time": 3555.196,
"index": 140,
"start_time": 3525.265,
"text": " 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 charges that i can understand from these charged so what what is the global geometry that that is kind of modeled on these charts and the simple idea is the following it's a it's a fun bootstrap exercise so we say suppose we had boldface x that is supposed to be by generalized space"
},
{
"end_time": 3577.278,
"index": 141,
"start_time": 3555.828,
"text": " 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"
},
{
"end_time": 3599.002,
"index": 142,
"start_time": 3577.756,
"text": " The choice 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 it's a very physics operational kind of definition we say how do we actually understand space while we understand it by throwing stuff into it like light rays and and see what happens so we probe it."
},
{
"end_time": 3621.817,
"index": 143,
"start_time": 3599.445,
"text": " In fact, it's very similar, also very related to the terminology of what's called probe brains in string theory. We 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."
},
{
"end_time": 3649.872,
"index": 144,
"start_time": 3622.108,
"text": " 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, they come back to you with something. Right. Exactly. Yeah."
},
{
"end_time": 3674.497,
"index": 145,
"start_time": 3650.282,
"text": " You can 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 that I'm exposing here is really it's going to be the notion of a sheath is so general that of course, many other concepts will fit on it that are not"
},
{
"end_time": 3704.684,
"index": 146,
"start_time": 3675.128,
"text": " . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."
},
{
"end_time": 3720.708,
"index": 147,
"start_time": 3705.316,
"text": " 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 charge a set just a plain set which we think of as being we think of it as being the plots"
},
{
"end_time": 3750.811,
"index": 148,
"start_time": 3721.186,
"text": " I call them plots. It's a technical term, 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 it's kind of, that's why I'm calling it a bootstrap. It's defined here by, and we kind of need to 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, whatever structure is encoded in our charts."
},
{
"end_time": 3777.261,
"index": 149,
"start_time": 3750.998,
"text": " 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 sets 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. We have a map here, so this might be a smooth map between our ends or"
},
{
"end_time": 3804.053,
"index": 150,
"start_time": 3777.261,
"text": " Super geometric maps between super Cartesian space or something like this. So we and those aren't to be interpreted as coordinate transformations. Yeah, you can. The thing is, I'm not requiring them the map here to be an isomorphism. So it does. It can actually, you know, it doesn't have to be an invisible coordinate transformation. It can be, but it can just be any smooth function. Actually, we're allowing we're allowing this. I'm calling them also generalized charts. I'm not at this point actually requiring. I'm not speaking of manifolds,"
},
{
"end_time": 3829.462,
"index": 151,
"start_time": 3804.462,
"text": " That are defined to be locally isomorphic to one of the charts. I'm defining something more relaxed. In effect, the whole point is that we get something that is more general than manifolds. It subsumes manifolds, but it is more general, where we just speak about not having just, you know, charts that are isomorphous onto the image, but just any maps. That's why they're called plots, any maps from the chart."
},
{
"end_time": 3844.889,
"index": 152,
"start_time": 3830.094,
"text": " 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 a with three three vertical lines why is it here why do i have quotation marks here no right there so why is that not a definition"
},
{
"end_time": 3875.486,
"index": 153,
"start_time": 3845.862,
"text": " 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"
},
{
"end_time": 3903.251,
"index": 154,
"start_time": 3875.828,
"text": " Currently conceptually on different footing, right? C is something we have in our hands some RN and X for a moment is just a 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 is 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."
},
{
"end_time": 3929.07,
"index": 155,
"start_time": 3903.404,
"text": " And that statement that we can remove the quotation marks that incidentally is the Unidad lemma, which is of course one of the is the emblematic lemma of categories here and they will make this work. Okay. So let's, let's, so this is actually fun. So let's think about this. So, so we're going to now say, well, if I have another chart, then my rule since I'm, I assume I have X. So if I have X, well, I will, I will be able to probe it also with C prime."
},
{
"end_time": 3954.94,
"index": 156,
"start_time": 3930.401,
"text": " But then if we think of the sets of plots as being such maps, even though they are not yet, but if you 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"
},
{
"end_time": 3985.128,
"index": 157,
"start_time": 3955.469,
"text": " probe form C to a plot of probe form C prime by pre-composing it, right? So again, this is a commuting, this is a pullback square. So we regard it as commuting diagram. So phi prime here is the image of composing phi with F. So this is called, this is pullback, pullback of functions. So this just says that along, if you have any map of charts, you can pull back the corresponding plots and that's right. So we want this to be true. This is the image we want to realize. And so here we're declaring"
},
{
"end_time": 4009.974,
"index": 158,
"start_time": 3985.725,
"text": " 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 this assignment is respects the composition so if i if i have two i haven't drawn this here if i if i have two maps between the charts"
},
{
"end_time": 4039.497,
"index": 159,
"start_time": 4010.316,
"text": " and I pulled 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 Cs and pulling back along an identity map should be an identity. So that's some very basic consistency conditions on our bootstrapped X to say, well, if I have such a functor, I have a good chance that I actually know no X."
},
{
"end_time": 4064.019,
"index": 160,
"start_time": 4039.753,
"text": " So what I'm 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 taking your 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 errors going this way turn to errors going this way then it's called a pre-sheaf of sets."
},
{
"end_time": 4094.889,
"index": 161,
"start_time": 4065.572,
"text": " 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 it in words. Maybe we want to say, well, to probe X, it should be sufficient to use small probes. Meaning, suppose you have a C here. Think of an RN."
},
{
"end_time": 4124.565,
"index": 162,
"start_time": 4095.418,
"text": " Say in R2, 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've 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 they're"
},
{
"end_time": 4150.981,
"index": 163,
"start_time": 4125.282,
"text": " The 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"
},
{
"end_time": 4179.838,
"index": 164,
"start_time": 4151.374,
"text": " 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 and called a sheaf. Okay. So let me see if I got this correct."
},
{
"end_time": 4192.585,
"index": 165,
"start_time": 4180.64,
"text": " 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 geometrists know this and that's why they work coordinate free."
},
{
"end_time": 4210.811,
"index": 166,
"start_time": 4193.148,
"text": " Now you're saying okay well the differential geometers way is you first define a 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"
},
{
"end_time": 4230.879,
"index": 167,
"start_time": 4210.811,
"text": " Yes. Yes. Yes. Yes."
},
{
"end_time": 4259.36,
"index": 168,
"start_time": 4230.879,
"text": " Yeah, no, I think that's a good way of thinking about it. I might just add that, of course, secretly, even in the standard textbooks and manifolds, secretly, of course, you need to know what RN 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"
},
{
"end_time": 4288.746,
"index": 169,
"start_time": 4260.145,
"text": " 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 so that's what's going on here too we we want to say what are these global spaces both as x and we need to we need to say that it's somewhere or other they're actually determined by charge 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 the smooth function between"
},
{
"end_time": 4308.319,
"index": 170,
"start_time": 4289.275,
"text": " Manifolds more generally got it so we're going to say okay we want to say a generalized space model on these charts is such a sheath such a contra range functor with such a consistent assignment of plots and now actually of course now that we're talking categories also want to make want to make the collection of all these"
},
{
"end_time": 4328.063,
"index": 171,
"start_time": 4308.865,
"text": " Generalize 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-sheeps,"
},
{
"end_time": 4355.196,
"index": 172,
"start_time": 4328.37,
"text": " Then a map between them is not just a natural transformation between the corresponding fungus. 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"
},
{
"end_time": 4381.544,
"index": 173,
"start_time": 4355.708,
"text": " I'm called fold face F now. Well, the idea is to again read pretty much this one does for manifolds to, to reduce it again to what happens on charts. So in order to know what this map does, well, we, we remember that our plots from charts were meant to be like maps into the space X. So suppose I have a C here chart mapping into X, then if this is really a map,"
},
{
"end_time": 4410.828,
"index": 174,
"start_time": 4381.715,
"text": " of these spaces that preserves, you know, all the given smooth, whatever, super geometric structure. Well, then it should be able, it should be possible to compose these maps phi from F that will give us a chart, a generalized chart of Y, right? Because now this map from C to X has been pushed forward to a map from C to Y. And so, so that is what we make the definition of F, a map of since our spaces were defined,"
},
{
"end_time": 4440.128,
"index": 175,
"start_time": 4411.22,
"text": " 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."
},
{
"end_time": 4462.534,
"index": 176,
"start_time": 4441.067,
"text": " So this gives us a category of generalized spaces modeled on any category of charts and that such a category is called this the jargon that's called a topos or actually go topos of generalized spaces, which is the chief topos as people would say on the site of charts. So it's just jargon for exactly what I what I just said. Okay. And"
},
{
"end_time": 4492.568,
"index": 177,
"start_time": 4463.029,
"text": " 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 Rodendick way back in 1965."
},
{
"end_time": 4520.589,
"index": 178,
"start_time": 4493.08,
"text": " when you actually started saying that this is the way to do algebraic geometry. Instead of talking about locally ringed spaces, one should regard schemes and 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, of formal duals of rings. So it's an old idea, but somehow"
},
{
"end_time": 4543.046,
"index": 179,
"start_time": 4521.476,
"text": " 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 locally-ringed 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"
},
{
"end_time": 4566.084,
"index": 180,
"start_time": 4543.541,
"text": " In this lower stage of categories, functions 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."
},
{
"end_time": 4579.787,
"index": 181,
"start_time": 4566.493,
"text": " So here's an example. I should give an example of what is the 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"
},
{
"end_time": 4606.578,
"index": 182,
"start_time": 4580.077,
"text": " 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"
},
{
"end_time": 4634.462,
"index": 183,
"start_time": 4606.92,
"text": " 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 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"
},
{
"end_time": 4660.725,
"index": 184,
"start_time": 4634.855,
"text": " bootstrapped, if you will, declaration of what should be the admissible maps from any chart into RC. 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"
},
{
"end_time": 4689.411,
"index": 185,
"start_time": 4660.896,
"text": " 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 unilateral lemma, which is maybe a fun thing to notice. Everybody is sort of the unilateral lemma and how it gets mentioned all the time in category theory. And here we see that if you think of categories as being categories of generalized spaces for physics, then the unilateral lemma places this very crucial pivotal role here as saying that's actually consistent."
},
{
"end_time": 4720.111,
"index": 186,
"start_time": 4690.469,
"text": " 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. Cool. So this is from this encyclopedia article that I have on high topocerein 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. Yes. From a few years ago. So I love this part. Okay. Very good. Yeah. In fact, that was a, that was January of last year. I was in, I was visiting in Beijing."
},
{
"end_time": 4747.381,
"index": 187,
"start_time": 4720.538,
"text": " At Tsinghua University, I was giving the talk there, but it was remote to the, you're right, to the Wolfram, to the Wolfram, whatever it's called, Wolfram school. Yeah. Right. So, um, 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 this firm. Yeah. Okay. I mean, let's, let's go through this. So let's look at,"
},
{
"end_time": 4771.578,
"index": 188,
"start_time": 4748.012,
"text": " 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"
},
{
"end_time": 4801.476,
"index": 189,
"start_time": 4771.8,
"text": " 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"
},
{
"end_time": 4828.131,
"index": 190,
"start_time": 4801.92,
"text": " And 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 shift topos over the category of Cartesian spaces, the mapping spaces from say space-time to any coefficient space"
},
{
"end_time": 4855.452,
"index": 191,
"start_time": 4828.524,
"text": " I should say that this is like the basis of all of physics, all of modern physics on a business card. Yeah. 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."
},
{
"end_time": 4876.664,
"index": 192,
"start_time": 4856.049,
"text": " Exactly, 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. You should think of these things as being infinitesimal disks like"
},
{
"end_time": 4905.026,
"index": 193,
"start_time": 4877.073,
"text": " Sometimes we'll 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."
},
{
"end_time": 4918.797,
"index": 194,
"start_time": 4905.35,
"text": " disk is so small that 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"
},
{
"end_time": 4940.247,
"index": 195,
"start_time": 4919.36,
"text": " 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."
},
{
"end_time": 4960.64,
"index": 196,
"start_time": 4940.247,
"text": " 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."
},
{
"end_time": 4987.978,
"index": 197,
"start_time": 4961.937,
"text": " but actually vanishes. So the experimental physicist that was your teacher would have k equals one in his example from before? Yes. Of squaring to zero? Yes. That will be the first order infinitesimal disk. Yes. Exactly. Yeah. You can also see in the old, in the original Feynman articles, I think in his thesis where he introduced the passing that he is also playing with such epsilons. And this is a, this is a fun story because, uh,"
},
{
"end_time": 5009.514,
"index": 198,
"start_time": 4988.848,
"text": " You know, physicists must have been using this even long before Feynman, I suppose, probably for 400 years or something, this kind of trick. And it took mathematicians quite a while to fully come to grips with this, but it's exactly in this way. This now goes now by the name synthetic differential geometry, where one considers shift opposites on"
},
{
"end_time": 5038.302,
"index": 199,
"start_time": 5009.906,
"text": " on such probes or similar with infinitesimals and then notices that inside these shift top bosses one has that's why it's called synthetic inside these shift top bosses there's generalized that like the the infinitesimal disk then exists as an actual object and and all other kinds of infinitesimals so that i can for instance go and and define inside these top bosses define a tangent vector to any point to literally be a map from the"
},
{
"end_time": 5064.394,
"index": 200,
"start_time": 5038.882,
"text": " infinitesimal interval into the manifold such that the you know the foot point of the interval goes to that point so it's it becomes literally an infinite intelligence vector becomes literally infinitesimal path in the manifold that's why it's called synthetic because the the usual heuristics of what differential geometry is about kind of becomes a synthetic reality it becomes synthesized to something that actually exists"
},
{
"end_time": 5093.336,
"index": 201,
"start_time": 5065.776,
"text": " And for physics applications, 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"
},
{
"end_time": 5123.524,
"index": 202,
"start_time": 5093.951,
"text": " 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 you can speak about variation calculus without this tool, but the statement here 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 Lagrangian, if you want to look at its Lagrange equations, then your"
},
{
"end_time": 5154.053,
"index": 203,
"start_time": 5124.377,
"text": " 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 um like like a dq it's like an infinitesimal disk of dimension q where and it's also first-order infinitesimal because the coordinate functions here on this thing"
},
{
"end_time": 5181.391,
"index": 204,
"start_time": 5155.06,
"text": " 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? As you know. So this is very much like an infinitesimal disk, the super point, but with the additional property that thinks 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."
},
{
"end_time": 5212.329,
"index": 205,
"start_time": 5183.609,
"text": " 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. And this, 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, n infinitesimal dimensions of"
},
{
"end_time": 5236.22,
"index": 206,
"start_time": 5212.705,
"text": " 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 for actually formal super geometry where formal means formal doesn't mean undefined or something it means it has these formal power series algebras here and i just want to amplify this is maybe um"
},
{
"end_time": 5265.981,
"index": 207,
"start_time": 5236.681,
"text": " Totology that might be underappreciated commonly is that even though it's called super geometry, this is a term invented really by mathematicians. This has nothing to do yet with supersymmetry. This is the super geometry 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."
},
{
"end_time": 5295.145,
"index": 208,
"start_time": 5266.323,
"text": " The fermions in any classical field theory do not actually form an ordinary geometry by the Pauli exclusion principle. They are the corresponding fermionic fields classically are inter-commuting fields. And that is important for, for instance, the Lagrangian density for the Dirac operator, something like psi bar d psi. If psi"
},
{
"end_time": 5316.715,
"index": 209,
"start_time": 5295.725,
"text": " let's let's forget the bar for the moment psi d psi sure if psi were if cyber 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 um by the product rule"
},
{
"end_time": 5346.254,
"index": 210,
"start_time": 5317.278,
"text": " So already the Lagrangian 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-commuting. 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 of fermions, which is kind of a big deal actually, right?"
},
{
"end_time": 5365.247,
"index": 211,
"start_time": 5347.159,
"text": " If you want to be precise. Right. And then it continues. Should we keep going through the list? So these three are. 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."
},
{
"end_time": 5387.312,
"index": 212,
"start_time": 5365.589,
"text": " 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."
},
{
"end_time": 5414.497,
"index": 213,
"start_time": 5387.892,
"text": " 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 group void themselves. And the way to do this, being quick now, is one kind of throws in the rudimentary information of what it means"
},
{
"end_time": 5444.377,
"index": 214,
"start_time": 5415.503,
"text": " of what it means that there are transformations and and a transformation like a gauge transformation of order r first or a second order and so forth it goes by this the symbol delta r so this is a bit technical this is this is a notation for the simplex that's one way of encoding transformation and higher gauge transformation so throwing in kind of directions in which things can gauge transform gives us gives us a notion of charts that models"
},
{
"end_time": 5470.674,
"index": 215,
"start_time": 5445.128,
"text": " gauge theory and higher gauge theory with ferments. 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 or default singularities. So we can ask our probes or probes to have actual singularities already in them so that we can detect singularities in the target space. Hmm."
},
{
"end_time": 5495.06,
"index": 216,
"start_time": 5471.032,
"text": " 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,"
},
{
"end_time": 5521.886,
"index": 217,
"start_time": 5496.135,
"text": " And, yeah, let's talk about this another time. If one does this right, then this actually boosts the corresponding spaces that are propped by this to being parametric, what's called parametric stable homotopy types. And this is really this kind of fascinating issue. 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."
},
{
"end_time": 5547.619,
"index": 218,
"start_time": 5522.381,
"text": " 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 n's that where the maps behave like maps of spheres, but the direction which the arrow goes is as if they were of negative dimension. So, it's just a notation for a certain category of things."
},
{
"end_time": 5577.159,
"index": 219,
"start_time": 5548.319,
"text": " where the main point is that after we apply the unit lemma, think of these charts as being as being spectra as it were, like spectra of spaces, we will see the sphere spectrum in different dimensions. Okay. 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, I'll put"
},
{
"end_time": 5604.48,
"index": 220,
"start_time": 5577.449,
"text": " So here's the category. This is actually now a higher category of these negative dimensional spheres. It 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 aspect."
},
{
"end_time": 5632.398,
"index": 221,
"start_time": 5605.469,
"text": " Alright, 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 now we want to find it. So this is the end, I think, of the little introduction to category theory and physics that I have here and 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 and"
},
{
"end_time": 5650.469,
"index": 222,
"start_time": 5632.705,
"text": " I'll be talking about mainly this this top was the shift of us to be kept by using. The first three at least and possibly the others is our probe spaces and that you know in the writings i have this goes by the name of super formulas so the generalized spaces that are pro by all this year."
},
{
"end_time": 5678.985,
"index": 223,
"start_time": 5651.186,
"text": " so it's called super formal smooth infinity group or it's for the fact that they're smooth they're propped by the RNs formal because they are propped by infinitesimals super because they can be propped by super continuous spaces and higher group points because they they have these high gauge transformations in them all right so so that's the topos we have and now let's now let's look at one of this i have these elements this is a bit experimental of course as you know but let's see what happens so this this is like a slide that"
},
{
"end_time": 5705.674,
"index": 224,
"start_time": 5679.377,
"text": " 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."
},
{
"end_time": 5735.691,
"index": 225,
"start_time": 5706.613,
"text": " 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"
},
{
"end_time": 5744.684,
"index": 226,
"start_time": 5735.93,
"text": " and we're going to 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"
},
{
"end_time": 5773.268,
"index": 227,
"start_time": 5745.026,
"text": " for the application that comes is 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. Okay. 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 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,"
},
{
"end_time": 5794.241,
"index": 228,
"start_time": 5773.746,
"text": " that are adjoined in several ways, then we get corresponding further 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 adjoint. So this symbol here, just a moment, we'll get back there, a functor, and here is a right adjoint."
},
{
"end_time": 5821.903,
"index": 229,
"start_time": 5794.548,
"text": " So this symbol means that this function is left and this function is right. 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 H1 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."
},
{
"end_time": 5838.114,
"index": 230,
"start_time": 5823.814,
"text": " 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 folders so yeah very good thanks for asking actually i should have made this clearer so what is given here is"
},
{
"end_time": 5867.79,
"index": 231,
"start_time": 5838.626,
"text": " adjoined functors and other adjoined functors and I'm unwinding what extra structure that implies on our category. Yes. But in a moment actually in a moment we'll go so this is just the explanation of the method in the 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 a certain dynamics emerge"
},
{
"end_time": 5896.015,
"index": 232,
"start_time": 5868.268,
"text": " 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 we have an adjoint triple, as people say, then the corresponding endofunctors are themselves adjoined to each other. Now, them being"
},
{
"end_time": 5924.292,
"index": 233,
"start_time": 5896.681,
"text": " composites of a junctions 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 object right we can apply flat and get"
},
{
"end_time": 5949.991,
"index": 234,
"start_time": 5924.735,
"text": " 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 diagram which so every x is sitting kind of in between these two extremes as for now because remember every junction 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"
},
{
"end_time": 5979.821,
"index": 235,
"start_time": 5951.425,
"text": " In fact, this kind of means so once we once we have an understanding of what this flat like, you know, him just I'm having to define things here. It's just supposed to be head. But once we know what flat actually does, we know what this extreme aspect is. And 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"
},
{
"end_time": 5995.725,
"index": 236,
"start_time": 5979.821,
"text": " Where flat and sharp are. The discrete aspect of a of a space like the set of underlying points with its discrete apology or the set of underlying points with what is called its codiscrete apology."
},
{
"end_time": 6026.22,
"index": 237,
"start_time": 5996.647,
"text": " So like what's sometimes called discrete and codiscrete space, it will mean that every space in our category is in between being discrete and codiscrete. But this is exactly what a topology on a space means. Every topology is in between, in some sense, the discrete and the codiscrete topology. So it's a rudimentary, what we see is getting towards the rudimentary form of encoding some spatial property in our objects, just by the fact that there are these opposing aspects"
},
{
"end_time": 6052.312,
"index": 238,
"start_time": 6026.8,
"text": " 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."
},
{
"end_time": 6081.305,
"index": 239,
"start_time": 6053.046,
"text": " And one can check, one can see that this functor has a right adjoint if and only if, or well, if we assume that H0 is the category of sets, then this right adjoint exists, the right adjoint 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 right adjoint functor will have to choose a single object. And then one can check that the laws of adjunctions say that single object has to be the terminal object. The object, every other one has a unique map to it. Right."
},
{
"end_time": 6102.534,
"index": 240,
"start_time": 6082.039,
"text": " And similarly, one will see that 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,"
},
{
"end_time": 6128.626,
"index": 241,
"start_time": 6102.824,
"text": " tautologically 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 of modalities of of ways that things that are topos can be."
},
{
"end_time": 6156.408,
"index": 242,
"start_time": 6129.667,
"text": " 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."
},
{
"end_time": 6180.674,
"index": 243,
"start_time": 6156.766,
"text": " And it turns out there emerges a very interesting, 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 Lavir, 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."
},
{
"end_time": 6207.602,
"index": 244,
"start_time": 6181.374,
"text": " the slide isn't maybe showing everything that one could say here but this is um really what what levier came up with is what's actually going on in hegel's science of logic so hegel in the science of logic in this in this old book he kind of puts himself into the place of a really of a seer of a mystic he he just looks inside himself and says okay let me let me forget everything about and let me just kind of concentrate"
},
{
"end_time": 6227.21,
"index": 245,
"start_time": 6208.302,
"text": " So you want to make philosophy a science you cannot feel this way to the to the beginning of every thought and sorry he argues himself in a kind of poetic way into into this kind of story where is this okay so if i if i really have been all other thoughts what do i have nothing and then he over pages and pages it talks about."
},
{
"end_time": 6248.473,
"index": 246,
"start_time": 6227.21,
"text": " This pure nothing that he sees pondering but then he gets to the point they 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 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 becomes gets into tension with with something that just exists."
},
{
"end_time": 6276.288,
"index": 247,
"start_time": 6248.746,
"text": " with the pure being but then he said well but this this can't quite be because we we had nothing there and so he kind of feels an inner tension which eventually he says is being resolved by so out of this out of this dynamics really this logical tangent kind of feels like like a dynamics that makes something else emerge and and levieres observation was that that this is really captured by accurately"
},
{
"end_time": 6305.845,
"index": 248,
"start_time": 6276.596,
"text": " captured by what I'm drawing. This is not quite how Lévié 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 I can ask there for a further such opposition, which resolves the previous one in that so"
},
{
"end_time": 6332.927,
"index": 249,
"start_time": 6306.408,
"text": " As the composition of the functions 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 both nothing and pure being becomes what I guess Levy likes to call becoming."
},
{
"end_time": 6343.507,
"index": 250,
"start_time": 6333.439,
"text": " 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 try to do that at some point."
},
{
"end_time": 6368.097,
"index": 251,
"start_time": 6344.002,
"text": " which i guess you can find on the n lab patreon science of logic you can you can now ask well what is this next opposition if yes if you were if you were to actually compare it to to hegel's poetry and then what can actually one can actually see that it does sort of make sense that he gets he is another opposition which again that's that's kind of what he calls the process the program the progression where where"
},
{
"end_time": 6394.804,
"index": 252,
"start_time": 6368.422,
"text": " 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 Lavir'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."
},
{
"end_time": 6425.623,
"index": 253,
"start_time": 6395.776,
"text": " 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 we can run Hegel's experiment which he had to do kind of without tools we can 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 what i'm showing"
},
{
"end_time": 6453.302,
"index": 254,
"start_time": 6426.203,
"text": " 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 or it's all the time. You may just think of this as being sets, the way I finish these 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"
},
{
"end_time": 6476.578,
"index": 255,
"start_time": 6453.712,
"text": " 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 subtoposis. 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"
},
{
"end_time": 6501.8,
"index": 256,
"start_time": 6477.073,
"text": " spaces where reduces 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 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"
},
{
"end_time": 6513.848,
"index": 257,
"start_time": 6502.346,
"text": " 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."
},
{
"end_time": 6541.135,
"index": 258,
"start_time": 6514.514,
"text": " And so the punch, the point 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, you know, these adjunctions, they exist. 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."
},
{
"end_time": 6568.951,
"index": 259,
"start_time": 6541.459,
"text": " And 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."
},
{
"end_time": 6595.299,
"index": 260,
"start_time": 6569.701,
"text": " And the corresponding progression of these extreme aspects is shown here. So we start with the 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 position between every object is between the initial and the terminal object."
},
{
"end_time": 6624.258,
"index": 261,
"start_time": 6595.64,
"text": " 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 to that. Yes. But then further adjoints appear. The next one that just disappeared here is the so-called shape modality."
},
{
"end_time": 6653.302,
"index": 262,
"start_time": 6624.787,
"text": " which says, well, that apart from just having discrete and codiscrete 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 kind of, well,"
},
{
"end_time": 6675.162,
"index": 263,
"start_time": 6653.865,
"text": " Yes, 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 is this 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."
},
{
"end_time": 6700.213,
"index": 264,
"start_time": 6676.049,
"text": " Monarch. So these are things they have, I gave them some sympathy, but these are things that are unknown in algebraic geometry of derived schemes or sort of just a format schemes, the reduced aspect, the co-reduced or the ramstack aspect. And it keeps climbing. And that's interesting. So we just, we're just asking in this Hegelian-Lavirian notion"
},
{
"end_time": 6727.654,
"index": 265,
"start_time": 6700.708,
"text": " So Levier never pushed this, it seems to me, beyond this first step. I think he looked at this first step and had some examples, some topos realizing that which were a bit contrived. So I don't think he looked at examples that are kind of practically important. And the insight 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"
},
{
"end_time": 6756.561,
"index": 266,
"start_time": 6728.507,
"text": " and kind of discovers all or rediscovers one could say these these various aspects and 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 the fermionic aspect so there's now a modality of being bosonic or being bifermionic and then it finally ends and what is interesting now and i haven't"
},
{
"end_time": 6772.91,
"index": 267,
"start_time": 6756.954,
"text": " 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 okay but but the three that appear here in the middle"
},
{
"end_time": 6796.442,
"index": 268,
"start_time": 6773.234,
"text": " 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 of this real nomic aspect of a space? Well, it turns out the shape or the topological realization of an infinity stick is the answer to asking, well, what if I had this space"
},
{
"end_time": 6826.049,
"index": 269,
"start_time": 6796.852,
"text": " 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. But if you forget that, you can shrink it to the point. So what if all the line plots and all the RN plots into my space, if they're actually not equal, but gauge equivalent to the point,"
},
{
"end_time": 6851.715,
"index": 270,
"start_time": 6826.613,
"text": " What is it that remains? What can I still see in my space? And that is like the technical construction 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"
},
{
"end_time": 6882.005,
"index": 271,
"start_time": 6852.602,
"text": " 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"
},
{
"end_time": 6900.879,
"index": 272,
"start_time": 6882.654,
"text": " 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"
},
{
"end_time": 6929.599,
"index": 273,
"start_time": 6901.715,
"text": " 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 is this rheonomic as it's called. So it introduces fermions. Yeah, that is the one. So this level here, this, this line that is going to appear in just a moment, that is the one that knows about the firm ones, bosonic aspect, rheonomic aspect, bifermionic aspect. And this guy here,"
},
{
"end_time": 6955.247,
"index": 274,
"start_time": 6930.623,
"text": " is given by localization at the super point. So it's in this sense that this progression discovers first the continuum then the infinitesimals and then the super point 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"
},
{
"end_time": 6985.435,
"index": 275,
"start_time": 6955.811,
"text": " 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. 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"
},
{
"end_time": 7008.097,
"index": 276,
"start_time": 6985.776,
"text": " Topos right so so that's just a fact and so i'm saying if you know if this top of this or platonic world in which we live then we can play the hegelian game then ask okay what is 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 i'm a fear of something."
},
{
"end_time": 7038.336,
"index": 277,
"start_time": 7009.121,
"text": " 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 a big accomplishment of"
},
{
"end_time": 7066.766,
"index": 278,
"start_time": 7039.087,
"text": " Suggesting this formalization. He didn't really dwell on convincing examples He had some I think that I could fairly be called contrived examples that just showed the math at work but didn't really Show, you know much practical use and what I'm observing here Is that in in the actual in the actual topos that we actually want to be using in physics that we are using implicitly or not"
},
{
"end_time": 7093.831,
"index": 279,
"start_time": 7067.363,
"text": " that not only you know does this initial stage exist but the it keeps it keeps progressing and it it knows about these archetypes of our geometries the continuum the infinitesimals and the super point 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 let's go further so this is some backdrop on"
},
{
"end_time": 7122.244,
"index": 280,
"start_time": 7094.224,
"text": " on the higher topos theory that is maybe needed to 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 and 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 I"
},
{
"end_time": 7148.507,
"index": 281,
"start_time": 7123.012,
"text": " If we have the energy, let me maybe try to explain that. So, so now let's play with the super point in our in our high purpose. And the thing is, there's a pun there if we have the energy. Oh, I see. Yeah, let me 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"
},
{
"end_time": 7177.637,
"index": 282,
"start_time": 7148.916,
"text": " in this emerging form. It's not sitting anywhere yet, right? It's not a space. 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 this super point 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 super point"
},
{
"end_time": 7208.2,
"index": 283,
"start_time": 7178.37,
"text": " There is a structure of a super-Li algebra. It is as such the Abelian 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?"
},
{
"end_time": 7235.759,
"index": 284,
"start_time": 7208.746,
"text": " And it's a fun exercise which originally started with John Huerta in this article here called M Theory from the Superpoint where we showed the root of the following, we proved the root of the following extension. So it turns out if you start with the superpoint here and play the following game that in each step it doubles the fermionic dimension so it takes the"
},
{
"end_time": 7264.531,
"index": 285,
"start_time": 7236.408,
"text": " or one dimensional super point to the or 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 could 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 the super space times"
},
{
"end_time": 7294.565,
"index": 286,
"start_time": 7264.804,
"text": " which Green-Schwarz superstrings exist, so kind of the, I shouldn't say critical, the dimensions in which superstrings 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 this R0,2 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"
},
{
"end_time": 7321.459,
"index": 287,
"start_time": 7295.213,
"text": " This is super translation, the algebra structure emerges from R02. So how can this be? Well, let's, let's think about, so we're looking at extensions. So an extension of a Lie algebra is, is classified by a Tuco cycle, right? By a degree two element in the Chevrolet Allenberg algebra of the Lie algebra. Um, so a map from a map from the graded commutative second power of the Lie algebra with itself."
},
{
"end_time": 7346.51,
"index": 288,
"start_time": 7321.954,
"text": " map from there to the ground field which satisfies some cosec 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."
},
{
"end_time": 7376.954,
"index": 289,
"start_time": 7347.432,
"text": " 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. Sure. The Hermitian matrices. And there's a standard fact that the"
},
{
"end_time": 7406.493,
"index": 290,
"start_time": 7377.312,
"text": " the space times in these dimensions here, dimension three, six, and ten, 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-coast cycles on"
},
{
"end_time": 7423.78,
"index": 291,
"start_time": 7407.022,
"text": " On our zero two are exactly the information that encodes. That it costs the information of the three dimensional swimming cost space and then it continues this way. And it's quite interesting because the process down here in."
},
{
"end_time": 7444.292,
"index": 292,
"start_time": 7424.48,
"text": " in this root of the bouquet of the tree is all in superlial to us. So these are all superlial to us and we're recovering the super translation superlial to us of these Lankovsky spacetimes. But then at some point there is no further invariant extension but there are now higher extensions that produce. So here's 11d super spacetime"
},
{
"end_time": 7473.933,
"index": 293,
"start_time": 7444.855,
"text": " So there's further extensions now here that are not extensions of super Li algebras but are what are called super L infinity algebras. So higher categorical Li 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 11D 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."
},
{
"end_time": 7503.148,
"index": 294,
"start_time": 7474.616,
"text": " that on 11-dimensional super spacetime like the Minkowski 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 Minkowski spacetime. Like the ordinary extensions of a super Li algebra by another super Li algebra"
},
{
"end_time": 7531.067,
"index": 295,
"start_time": 7504.002,
"text": " are encoded by a two-core cycle. This is Stunna textbook material. Now you could ask, well, what is it that a three-core cycle classifies? Well, it turns out a three-core cycle on a super-Li algebra classifies an extension by a higher form of super-Li algebra called a Li2 algebra. And so this happens with some of the brain, d-brains here, but this one, the M2 brain, is classified by a four-core cycle. And that four-core cycle is the famous, the famous bifermionic expression"
},
{
"end_time": 7552.466,
"index": 296,
"start_time": 7531.596,
"text": " on super space and that is known to be the bifurcated component of the super gravity sea field on 11 dimension super space time but it's also what is called the the west domino term of the green schwarz signal model of the kappa symmetric signal model for them to read so here we see the expression mu m2 is this forko cycle oops that was too quick we'll see it again"
},
{
"end_time": 7580.691,
"index": 297,
"start_time": 7552.773,
"text": " 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 homology. So this brain bouquet kind of discovers from the super point now, first the super space times of kind of the right dimensions, then the super strings in their brain towards incarnation as sigma models."
},
{
"end_time": 7608.609,
"index": 298,
"start_time": 7581.852,
"text": " 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-core cycle happens to be either the bifurcated piece of the"
},
{
"end_time": 7626.578,
"index": 299,
"start_time": 7609.377,
"text": " dual C-field flux at level of supergravity or equivalently the west domino term for the Green-Schwarz sigma model, Green-Schwarz type signal model of the M5 band. And it's these two co-cycles that we want to be focusing on, which are shown here. G4, oops, it's gone again."
},
{
"end_time": 7655.384,
"index": 300,
"start_time": 7627.5,
"text": " But but we'll see it. So the punch line is and we'll 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 this pair of core cycles, the four core cycle and the seven core cycle, which are in a crucial relation to each other, which mimics just the equations of motion of the sea field and 11 dimensional super gravity. It just comes out. It's just kind of built in some sense, built into the super point here. And"
},
{
"end_time": 7686.015,
"index": 301,
"start_time": 7656.476,
"text": " 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 to, how we actually obtain like full supergravity from this, because this is just local model space. All these are super-ly algebras. You should think of all these things appearing here as being the, as being what in an actual space-time you would see in the infinitesimal neighborhood of any point, like on a tangent space. On tangent space, it looks like a super Makovsky spacetime and there"
},
{
"end_time": 7711.664,
"index": 302,
"start_time": 7686.323,
"text": " We have these co-cycles, but now let's look at the following theorem. So we proved this just last year, actually. So it 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,"
},
{
"end_time": 7739.974,
"index": 303,
"start_time": 7712.483,
"text": " and we ask that they globalize now over a non 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"
},
{
"end_time": 7767.159,
"index": 304,
"start_time": 7740.538,
"text": " moving frames. If you can move these co-cycles over your curved manifold, then that requirement alone is actually equivalent to the full equations of motion of 11. 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 Rory-Taschina equation. And on top of this, the"
},
{
"end_time": 7796.834,
"index": 305,
"start_time": 7767.602,
"text": " The super gravity C fields on the gauge field of the theory satisfies its equation of motion. And that is so that is repeated again on this animation set. We're just sketching it here. So. So what are we seeing here? I'm now I'm now going. I guess I've I've changed pace a bit. So we're now looking at how to see full super gravity actually appear from these ingredients that we discovered and"
},
{
"end_time": 7826.869,
"index": 306,
"start_time": 7797.381,
"text": " Okay, what are we seeing here? So we're looking at a superspace time. 11 dimensional superspace time. I didn't show the, I'm not showing the odd coordinate components. Maybe I should have added them. Sure. So this is says odd dimension 32 as befits n equals one. So this is the, this is a space time that is locally modeled on the, on the tip of this immersion diagram that we saw is locally modeled on this 11D superspace time. And we're asking,"
},
{
"end_time": 7856.681,
"index": 307,
"start_time": 7827.551,
"text": " that on this space time 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 bifurmanic expression and so we're asking well let there be"
},
{
"end_time": 7880.879,
"index": 308,
"start_time": 7857.278,
"text": " Forms that you know saying saying we have a super space-time but this I mean it is equipped with a Supermetric 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 in these field bites, which are are the things that locally identify the space-time with Mikovsky space-time"
},
{
"end_time": 7899.684,
"index": 309,
"start_time": 7881.766,
"text": " So we say, okay, let there be two flux densities, as we're going to call them, whose bifurcated component is exactly the one that it's kind of got given. And then let them have, in order to patch it up, let them have any other ordinary component, an ordinary four-form, ordinary seven-form component."
},
{
"end_time": 7930.657,
"index": 310,
"start_time": 7901.681,
"text": " 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, I haven't fully explained this, but it's clear that"
},
{
"end_time": 7961.203,
"index": 311,
"start_time": 7931.459,
"text": " g4 being closed it's an element of the chevrolet eilenberg algebra or l infinity algebra and it's being closed that's just a co-cycle condition and then 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 be this equation that that says that d of g7 is g4 square okay i guess i don't know it's maybe getting a bit"
},
{
"end_time": 7985.367,
"index": 312,
"start_time": 7961.561,
"text": " technical here but the thing is that this bottom piece here so 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 space time and so the first thing we want to do is we want to ask well"
},
{
"end_time": 8015.35,
"index": 313,
"start_time": 7986.323,
"text": " 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 to the square of the other. So we ask, well, what is that? What is it that classifies pairs of differential forms with that property? That's what I'm asking here."
},
{
"end_time": 8037.756,
"index": 314,
"start_time": 8016.681,
"text": " And it turns out that this is exactly what is the structure 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"
},
{
"end_time": 8067.756,
"index": 315,
"start_time": 8038.285,
"text": " To each on a multi technical conditions, finite type, rational, finite type to each logical space and l infinity algebra, whose 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 to the homotopic structure in a space and the space that appears here, the space that happens to encode"
},
{
"end_time": 8097.21,
"index": 316,
"start_time": 8068.131,
"text": " just the co-circle relation that finds these M2 and 5-band co-circles happens to be the force sphere or any other space of the rational type of the force 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"
},
{
"end_time": 8127.585,
"index": 317,
"start_time": 8098.968,
"text": " 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 encoded the equations of motion of supergravity in a kind of purely cohomological or rational homotopical construction where we just say the fact that a spacetime satisfies supergravity equations of motion just means the flux synthesis of this form exactly arranged into a"
},
{
"end_time": 8142.619,
"index": 318,
"start_time": 8128.882,
"text": " 4-sphere valued flat differential form where by 4-sphere valued I mean valued in the white head at infinity above the 4-sphere and that is of enormous"
},
{
"end_time": 8170.879,
"index": 319,
"start_time": 8143.507,
"text": " use now because as you keep seeing in the diagram that builds here we can now ask what is the global field content of super gravity so this is this is now getting us into this is now getting us into an actual actual new new physical realm so i 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 now almost exactly 100 years that"
},
{
"end_time": 8200.811,
"index": 320,
"start_time": 8172.892,
"text": " 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"
},
{
"end_time": 8219.531,
"index": 321,
"start_time": 8201.22,
"text": " 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 Sov answered, at least in hindsight, in the 1930s about electromagnetism can and should be asked about all higher gauge theories, in particular about our"
},
{
"end_time": 8249.258,
"index": 322,
"start_time": 8220.009,
"text": " or 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 dimensional supergravity 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."
},
{
"end_time": 8272.312,
"index": 323,
"start_time": 8250.043,
"text": " 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 that part of the force sphere that is detectable by differential forms, well, we can ask"
},
{
"end_time": 8302.09,
"index": 324,
"start_time": 8273.473,
"text": " We can ask for a classifying space to be called A here that classifies the actual global charges, not just approximated by the French reform data. And then we just want to ask for, so this would like in electromagnetism, this would be the, um, the monopole sector classifying space for the monopole sector. And so we ask what, where are these things comparable so that we can ask these charges source, these fluxes,"
},
{
"end_time": 8333.097,
"index": 325,
"start_time": 8303.473,
"text": " And as such, they are compatible, right? Not every set of charges can source given fluxes. So there needs to be some compatibility condition to them, right? 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"
},
{
"end_time": 8359.889,
"index": 326,
"start_time": 8333.814,
"text": " that we can push forward, we can send these close differential forms to their what's called their moduli stack. Anyway, some deformation space for this and that that receives a map in this high topos from any space, which has the property that is whitehead L infinity algebra coincides with that of the force field, which which means that it's it's our rationalization is that of the force any such space can be put here."
},
{
"end_time": 8388.217,
"index": 327,
"start_time": 8360.265,
"text": " 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 11-dimensional 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."
},
{
"end_time": 8417.773,
"index": 328,
"start_time": 8388.712,
"text": " is the same as that of the force sphere. And then if you choose any such space, then the full field content of field 11 and supergrave is then a homotopy between the charges to the fluxes in this diagram, in this diagram as it lifts in our higher topos. Of course, the canonical choice for A is the force itself, not the canonical choice, like the initial choice, the simplest choice."
},
{
"end_time": 8445.435,
"index": 329,
"start_time": 8418.473,
"text": " and so what we've been calling hypothesis H is right so this diagram the option of this diagram is twofold it says well 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 dimensional super gravity which is an example and second the second thing is well there is actually among the infinite set of possible choices for the choice colonization"
},
{
"end_time": 8470.845,
"index": 330,
"start_time": 8445.674,
"text": " There's one choice that is kind of singled out as being the, it's the simplest one. 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 a Levenin-Rome supergravity, which has not found much attention before. There's articles by Diakonevsky, Fried and Moore,"
},
{
"end_time": 8498.677,
"index": 331,
"start_time": 8471.152,
"text": " from a few years back who talked about a model of the sea field which went in this direction but um but wasn't quite comprehensive i would say and and here we say first of all well look here's a choice to be made and second there is a there is a specific choice that kind of stands out and the you see what we're seeing here is that the usual data that goes into field theories"
},
{
"end_time": 8527.841,
"index": 332,
"start_time": 8499.155,
"text": " Yes, but the classifying space is, um,"
},
{
"end_time": 8551.664,
"index": 333,
"start_time": 8528.336,
"text": " Part of what characterizes the full field content. So hypothesis age is a hypothesis about what is the global completion of the field content of the level that the left dimensional super gravity in analogy to how I guess. Yes. So you see, I'm sorry. Yeah, I don't have a special slide for this, but let me just say it in words. So let's go back to electromagnetism. Um,"
},
{
"end_time": 8573.797,
"index": 334,
"start_time": 8552.073,
"text": " So what happened in 1850, Maxwell writes it in modern language, right? He writes a two form F two. 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. It's about it's a two form. It's close and and transfer some other conditions. It's a field strength, technically, as we would call it."
},
{
"end_time": 8600.333,
"index": 335,
"start_time": 8574.104,
"text": " the yeah exactly the 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 yes so that's why we also call it the flux density that's maybe a more suggestive term here sure but the flux density absolutely is it's just another inclination of the field strength here's sorry and so"
},
{
"end_time": 8631.067,
"index": 336,
"start_time": 8601.254,
"text": " So then the decades pass and 1931, with a bit of paraphrasing, Dirac comes and says, no, no, this cannot be the full field content of the electromagnetic field. And 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, the gauge potentials."
},
{
"end_time": 8655.606,
"index": 337,
"start_time": 8631.544,
"text": " What is such that our F connections the connections the local one from so that F is the differential of these A's. But but that alone is not actually the full inside because that it's quite interesting because that is something that already faraday was actually playing with faraday and makes for both the head of funny term for this I could look this up somewhere I forget now they were actually they couldn't quite nail it down but they"
},
{
"end_time": 8681.254,
"index": 338,
"start_time": 8656.493,
"text": " 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. Like when you see discussions, for instance, of all these supergravities, you see people often write down potentials for these things here. They write a C3, a 3-form, and such that G4 is dC3, right?"
},
{
"end_time": 8704.275,
"index": 339,
"start_time": 8681.578,
"text": " so yes that is that is the analog but that cannot be true globally like we're interested if there's non-trivial charge in our space time then this g4 is not exact globally yes it measures it measures a certain dram class which is kind of a rational image of this full whatever aqua multi class we're seeing here and so so the whole point really"
},
{
"end_time": 8733.712,
"index": 340,
"start_time": 8705.128,
"text": " of the global charges that 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, gauge transformation, gluing data on double overlaps of these charts, which in one way for electromagnetism, this is another way to"
},
{
"end_time": 8757.363,
"index": 341,
"start_time": 8734.838,
"text": " 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"
},
{
"end_time": 8780.742,
"index": 342,
"start_time": 8757.637,
"text": " 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"
},
{
"end_time": 8809.718,
"index": 343,
"start_time": 8781.22,
"text": " only chart some global cohomological structure and that is 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 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"
},
{
"end_time": 8830.316,
"index": 344,
"start_time": 8810.316,
"text": " What do i call the drug string is really is really speaking about what mathematicians will call the clutching construction for for line bundles on a sphere so it's a way of encoding the transition data that you need to have a check go cycle for your whatever data on the sphere but in speaking about this direct string."
},
{
"end_time": 8860.469,
"index": 345,
"start_time": 8830.759,
"text": " 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 it, understood this electromagnetic"
},
{
"end_time": 8889.172,
"index": 346,
"start_time": 8861.305,
"text": " situation in modern language is maybe Orlando Alvarez in the 80s, who has articles on Czech cohomology and Dirac charge quantization. His phrases, his 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 at a period, particularly in supergravity, where one must ask exactly the same question. What is it"
},
{
"end_time": 8918.78,
"index": 347,
"start_time": 8889.872,
"text": " that really defines the full global field content. It's the gauge potentials, yes, but furthermore, some topological data are gluing this together. 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."
},
{
"end_time": 8949.309,
"index": 348,
"start_time": 8919.684,
"text": " I think that's an interesting and appreciated aspect that actually to fully define just 11 dimensional 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? I don't know. Well, he hasn't written much about M-theory in the past few years."
},
{
"end_time": 8961.032,
"index": 349,
"start_time": 8949.582,
"text": " doesn't seem like many people are working on m theory no absolutely yeah absolutely um there has not been much at all no that's right yeah why do you think that is"
},
{
"end_time": 8985.333,
"index": 350,
"start_time": 8962.227,
"text": " 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."
},
{
"end_time": 9010.691,
"index": 351,
"start_time": 8985.725,
"text": " and that was that was always kind of more enigmatic in a way because 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 IKKT model is actually reduced to zero dimensions some in some sense IKT matrix model is matrix mechanics on a point"
},
{
"end_time": 9038.831,
"index": 352,
"start_time": 9011.596,
"text": " 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 just lately and i guess there's some claims that i guess the japanese group is claiming that computer simulation shows that there are spontaneous compactifications to four dimensions more seen by numerical computation in these matrix models i don't know if that has been checked"
},
{
"end_time": 9062.005,
"index": 353,
"start_time": 9039.258,
"text": " But yeah so this is um this is what people have been playing with and now to come to your question like why why is there no further progress so i think well i suspect part of the problem is the will be alluded to at the very beginning that it's not actually clear like or it was not actually clear where even to look like"
},
{
"end_time": 9077.756,
"index": 354,
"start_time": 9063.302,
"text": " 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, right? That's how people build new theories. You write a new Lagrangian, but the Lagrangian here is sort of already known."
},
{
"end_time": 9107.227,
"index": 355,
"start_time": 9078.217,
"text": " It's 11 years supergravity 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"
},
{
"end_time": 9137.022,
"index": 356,
"start_time": 9107.466,
"text": " Energy becomes an issue. How about I jump just now to the outlook because that maybe addresses. So 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"
},
{
"end_time": 9166.698,
"index": 357,
"start_time": 9137.415,
"text": " Doing perturbation theory really means when in infinitesimal neighborhoods, if we're just looking at formal power series of the pointed 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"
},
{
"end_time": 9196.425,
"index": 358,
"start_time": 9167.875,
"text": " In physics, the problem that we don't understand this year, like analytically, apart from computer simulation, is really what haunts particle physics in that whenever QCD at non-extreme temperatures plays any role, any hydrodynamic contributions can never really be fully computed with perturbation theory. Nobody really knows what's going on there. And it leads to huge issues in all these discussions of whether or not people see a new effect beyond the standard model or not."
},
{
"end_time": 9221.391,
"index": 359,
"start_time": 9197.005,
"text": " These anomalies, they keep usually growing until somebody finally cranks up their lattice 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,"
},
{
"end_time": 9252.039,
"index": 360,
"start_time": 9222.261,
"text": " effects like the fractional quantum hall effect or 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 this non-perturbative theory is not known and now it's interesting to note that"
},
{
"end_time": 9280.947,
"index": 361,
"start_time": 9252.432,
"text": " The history of this problem has it's kind of an interesting ironic turn to it. So let's look back how did so what happened in the past so way back in what is it the 50s or something or or maybe the 70s people thought about this sort of thinking hard about this problem of non-perturbable FQCD and there's very good arguments of course no real proof that"
},
{
"end_time": 9299.906,
"index": 362,
"start_time": 9283.37,
"text": " 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,"
},
{
"end_time": 9329.633,
"index": 363,
"start_time": 9300.589,
"text": " Since that is the most important phenomenological effect in the confinement in the non-perturbative regime, so probably the flux tubes are the actual degrees of freedom to be discussed for non-perturbative QCD. This is 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"
},
{
"end_time": 9357.517,
"index": 364,
"start_time": 9330.23,
"text": " Then comes this computation which is really still at the heart of much 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."
},
{
"end_time": 9372.398,
"index": 365,
"start_time": 9357.875,
"text": " 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-perseverative sector. So introduce the strings and now suddenly"
},
{
"end_time": 9397.381,
"index": 366,
"start_time": 9372.892,
"text": " 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 have to be understood as kind of a high dimensional unobserved bulk space time that hosts the actual observed space time, which sits in it in the form of a brain where the quarks sit on the brain and the strings."
},
{
"end_time": 9425.623,
"index": 367,
"start_time": 9398.166,
"text": " In their need for a higher dimensional bulk may have the endpoints attached to these quarks, but my otherwise probed this bulk. And then of course, well, you know the story then, then, you know, 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 extra QCD and a whole lot of numbers can actually be crunched out there. There's actually quantitative"
},
{
"end_time": 9454.855,
"index": 368,
"start_time": 9426.596,
"text": " 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 string theory. The big difference is that as opposed to the original field theory, its reformulation in terms of these strings now comes with"
},
{
"end_time": 9481.084,
"index": 369,
"start_time": 9455.708,
"text": " a lot of hints or 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"
},
{
"end_time": 9505.725,
"index": 370,
"start_time": 9481.374,
"text": " If you remember at this point, then in some sense historically, that is really the answer to how do quarks behave non-perturbatively, then at least in the Schraf outline, M-theory reappears at least as the possible ultimate answer to how to actually define construct non-perturbative QFT. And so"
},
{
"end_time": 9535.759,
"index": 371,
"start_time": 9506.766,
"text": " So the question then is, how do we even go about formulating M3? Do we write on a Lagrangian or is it just another matrix model? And 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."
},
{
"end_time": 9563.951,
"index": 372,
"start_time": 9536.203,
"text": " 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 I very much agree with this sentiment and I guess what I've been showing here is kind of suggestive of possibly being parts of an answer in this direction."
},
{
"end_time": 9592.312,
"index": 373,
"start_time": 9565.538,
"text": " 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."
},
{
"end_time": 9621.391,
"index": 374,
"start_time": 9592.773,
"text": " 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."
},
{
"end_time": 9648.387,
"index": 375,
"start_time": 9621.903,
"text": " Thank you so much for coming on. I appreciate you spending three hours with me."
},
{
"end_time": 9673.251,
"index": 376,
"start_time": 9649.326,
"text": " Thank you, Kurt, for inviting me and for doing this. I appreciate it. That was a 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."
},
{
"end_time": 9681.886,
"index": 377,
"start_time": 9673.677,
"text": " new update started a sub stack writings on there are currently about language and ill-defined concepts as well as some other mathematical details"
},
{
"end_time": 9710.299,
"index": 378,
"start_time": 9682.09,
"text": " 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 physics, philosophy and consciousness. What are your thoughts? While I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. Also,"
},
{
"end_time": 9737.602,
"index": 379,
"start_time": 9710.435,
"text": " Thank you to our partner, The Economist. Firstly, thank you for watching, thank you for listening. If you haven't subscribed or clicked that like button, now is the time to do so. Why? Because each subscribe, each like helps YouTube push this content to more people like yourself, plus it helps out Kurt directly, aka me. I also found out last year that external links count plenty toward the algorithm,"
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"text": " Which means that whenever you share on Twitter, say on Facebook or even on Reddit, et cetera, it shows YouTube, hey, people are talking about this content outside of YouTube."
},
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"end_time": 9771.937,
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"text": " which in turn greatly aids the distribution on YouTube. Thirdly, you should know this podcast is on iTunes, it's on Spotify, it's on all of the audio platforms. All you have to do is type in theories of everything and you'll find it. Personally, I gained from re-watching lectures and podcasts. I also read in the comments that hey, toll listeners also gain from replaying. So how about instead you re-listen on those platforms like iTunes, Spotify,"
},
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"end_time": 9796.049,
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"text": " podcast catcher"
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"text": " You also get early access to ad free episodes, whether it's audio or video. It's audio in the case of Patreon video in the case of YouTube. For instance, this episode that you're listening to right now was released a few days earlier. Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much."
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}No transcript available.