 Yes, thank you very much indeed. Thanks for the invitation. And thanks in particular for being a workshop where much of the notions that I want to use have actually been introduced. This is rare for my experience. So I'll just jump right away into saying a bit what I want to mean by pre-quantum geometry. And you'll see in a moment. It refers to the well-established term that was introduced by Sorio in the 70s. It's not used that much, but I do want to use it here because it does make a point. So what did it mean in the 20th century? Let me start with this. So differential geometry starts with the concept of smooth manifolds. And I'm going to, as you can see from my title, I'm going to make some connections with physics even though this talk will be just about the math. But I do want to add this motivation part which is the first board. I say a few things about how it connects to physics. So in physics, as you know, this is how it became a big thing. Smooth manifolds represent spacetime, but they also represent phase spaces. The, if you wish, spaces of states of classic length with some extra belts and whistles, quantum systems, phase spaces. Now this also introduced in the 20s was the concept of Carton geometry, which is a generalization of Riemannian geometry. It also contains conformal geometry and stuff. So let me put here Riemannian geometry, Riemannian conformal complex, blah, blah, blah, all kinds of extra structure that you can put on a smooth manifold. And famously, this is now, I guess, the 100th anniversary of this insight, this in physics describes the field of gravity. And another important ingredient in this differential geometry story in the 20th century was fiber bundles. And in particular fiber bundles with connection. In physics, they represent two different things. So on one hand, these are the mathematical interconnection of gauge force charges. The famous charge quantization argument says that if you have a magnetic charge somewhere, then the electromagnetic field is actually one principle bundled with connection in the complement of the charge. Gauge force charges this kind of at the level of spacetime on phase spaces. And that's where the word pre-quantum comes in. If you have a certain line bundled with connection in your phase space that lifts the symplectic form, then this is what's what you're called a pre-quantum phase space. So you need a pre-quantization of a phase space to have a non-perturbative quantization of it as opposed to, say, just a deformation quantization in form and power series of h bar. So this is where this word pre-quantum comes in. And I'm not going to today talk about quantum physics, but just about I'll be careful about speaking sort of an ordinary differential geometry, but with this pre-quantum aspect taking into account. So this is a fiber bundled with connection. And then there is an addition, maybe, to Cartian theory. There's general theory, which is about finding characteristics of these fiber bundled with connection in the physics of the 20th century. This describes for physicists called instantons. And more generally, this is what they call trans-Simon's terms, trans-Simon's theory. I'm just saying these words for those who know it. For those who don't, I will not explain it today. It's just kind of as a motivation and related to this with Sumino-Witton terms. So the point is just here. These are important aspects of particle physics, and they're kind of the physics incarnation of this mathematics. Now something happened. It started in the 80s of the 20th century, but it's sort of happening now as we speak. So this is really particle physics here. And people thought it would be good to have something that should be called brain physics. Right. So if the brain is the one brain that's called string theory, but there are these higher versions of this, and we're concerned a bit about some of these things. So first of all, so something happens here. It could also have happened already here, but it just became noticed maybe fully in this context that spacetime is generalized to higher super orbit folds. So super geometry starts to play a role, and there's some group portal aspects that starts coming in. It's very important in this brain physics that we take spacetime to be an orbit fold. What's an orbit fold? An orbit fold is a special case of a smooth group point, of a lee group point. So there's a mathematical incarnation of this, and this is what I will be concerned about, axiomatizing these things that go in the right column here in this talk today. So the mathematical incarnation of what these things are, they are super ital infinity stacks. And I hope to, on the third board, I'll give you a complete axiomatization of how to do these things and how to work with them. Right, we've seen infinity stacks. It's our space value, if you wish, sheeps on some manifold, and we'll put a tau structure on them and make them super. So this is what you need to describe these things. Yes, as you know, super is a technical term. So what happened to gravity? So this you could have already done here maybe with a bit of imagination, but here's something important that happens. Gravity becomes higher dimensional super gravity. So what does it mean? The nice thing about the point of saying Cartian geometry here is that it gives us an immediate way to say what this is. Cartian geometry is the geometry that is induced from a group like the Poincare group with a subgroup like the spin group. And you build spaces that are locally modeled on the infinitesimal coset spaces of these. High dimensional super gravity just comes from using the super Poincare group instead and looking at super manifolds that are locally modeled on these. But the point is, due to the super aspect of this, it turns out that these super gravity theories intrinsically have higher form fields with them. And these high degree form fields, if you consider them globally, they are higher jerbs, which do live in a higher differential geometry, which goes in this right corner here. So there is something that should be called higher super Cartian geometry. I want to say a bit about what that is. The mathematical extremization of what physicists do in higher dimensional super gravity is flux fields. Then fiber bonds with connections, they go to, so here we have charges like the magnetic charge of particles. So now that we have particles replaced by brains, these are things called brain charges. And one of the big insights of the 90s was that they no longer just take values in just ordinary bundles with connection. They take values in what is called in generalized commodity theories. So they're represented by certain spectra and there's lots of belts and bristles. These are twisted and accurate in generalized commodity theories. And the presence of the connections on the bundles translates into the fact that these are differential commodity theories. And for this too, I will give you a quick extremization for what that is. There's a differential Brown representability theorem that will tell us that these are just those sheaves of spectra in the way we've already seen in various talks by Andres Rial and by Matjonell. Actually, we'll be using perma-tri-spectra in the tangent topos to get the twists, which are cohesive. So this is where this word cohesive comes in. So there's a certain axiom you can put on your infinity category or spectra, which ensures that all of them represent precisely these twisted generalized differential commodity theories. So lastly, Ternveil theory, I have lots of things that maybe I can't really say all this. So these transimus theories, so they are higher dim transimus theories here, transimus type theories that play a huge role. So one of them, and I'm happy to say this here, one of them is the very famous, the AKZ models, where the K is of course referring to Maxim. One can integrate them, they were originally defined on L infinity algebra, one can integrate them to things that actually live on higher stacks than they contain the, I'm just mentioning these words, will not mean much to anything, contain the Poisson sigma model, the Courant sigma model, the A model, the B model strings. So lots of stuff that that first is scary about this also. So these all come from quadratic shifted forms, whatever this means, but there's also 70 transimus theory, which is important in the ongoing discussion of the 60 super going from a field theory that's an 11 D transimus theory. All these come from higher analogs of the Ternveil homomorphism, and at last and at least there's actually string field theory, which is built by the same by the same means. So I will not say anything about this now, just for those who have ever heard these terms, just as a motivation that this is what the technology that I'm trying to present on the right now is going to be about. So here we can then build something that I called infinity transimus functionals, which you should really think as being the higher, the higher differential geometry version of the original Ternveil homomorphism, and similarly for WZW. So what is all this? This is clearly something, if you believe that there's motivation for this, and I'm not going to argue for this, but I think there is, then we do need the mathematics that does handle this. The 20th century mathematics does not handle this anymore, which is the cause of much of the difficulties that people see here, because it's like using matrix calculus for advanced linear algebra back then. Anyway, so this is kind of an inside joke, let me put 21st century here. This will have to be understood and developed to understand this stuff. So this is what I'm going to talk about. Why would I talk about this? You might just to maybe say, well, people are already doing this. Let me highlight something that is somehow maybe not as vivid in the brains of mathematicians in these days. The 90s were somehow different. There are big mathematical problems here that physicists ran into, which do deserve mathematical attention. One of them goes by this name. Let me just say this for those who know these words. So it's known that what is called F1 DP brain charges. So this is the big insight written in the 90s in other people, freed brain charges are in twisted, equivariant differential K theory. So there's some chief of spectral that represents twisted differential equivariant K theory. And it's important to use this to model the charges of these brains in direct analogy with the magnetic charges here. And it's known from kind of the physics law that this has to be lifted to, but this is open what this should be. This has to be lifted for something what's called M2 M5 brain charges in blah, blah, blah, some generalized homology theory. I'm not going to talk about this today. I'm just saying this is an open problem. There's a kind of a well-defined question here that finds something here that under a certain transgression to a circle reproduces this. I'm not going to speak about this today. Actually, it so happens that I will speak about this next two weeks in a lecture course at ASE in Vienna. So if you do have interest in time, come to Vienna the next two weeks at the average trading institute in Vienna. I'm just saying this to highlight that there's concrete open problems here. This is actually what got me involved. Some people got the impression that I'm somehow just interested in abstract formulations of geometry, but that is not actually the case. So what I wanted to talk about today is the following. So we saw prequantum geometry was the left column on the right way with something like higher prequantum geometry. And here's the issue that I want to highlight and solve. So doing this is hopefully kind of evident already from these keywords. It's not feasible if you do it in components. Not feasible, not tractable. It's just not human possible in components. So I started my PhD thesis doing this up to degree three for, you know, what's now called maybe two jerbs of connection in components and all the definitions and all components just explode. They just exponentially increase before you even write down any theorem, you have definitions that span pages of pages. It's just not possible. What is needed? Need a faithful, a good synthetic language, synthetic theory. So some abstract theory that doesn't excimatize these things by their atoms for their build off, but which does faithfully excmatize what these things should behave like such that you can prove the theorems that have to be proven without getting bogged down in just choices of basis. So the solution of this which I want to briefly survey is it turns out this is possible use infinity top bosses that are equipped with a progression if you wish. There's a sequence I'm going to write on the next board, a progression of a joint model operators. I'll say what this is in just a second. So first of all, yeah. And this is what gives to them. So this progression that will appear on the next board needs a name. So there was something that was suggested by LaVier, kind of in the Bond Topos theory called it cohesive infinity. So I call it not cohesive infinity topos, but it's a we do it. Yeah, cohesive. Thanks. Syntax error. And so we make we make with. Yeah, I don't know, you know, you have to choose somewhere. I guess in my book, I call it differential cohesive, but also at this super aspect, so it will be super. Anyway, you know, it's just words. The point is, there will be now a simple, a very simple set of axioms and and what was kind of surprising. And this is something I, you know, I didn't look for this. We, we learned this, we were trying to solve this problem. And at some point I noticed, oh, look, I can just do all these things that were really difficult, but just using a certain adjoint. And then, you know, once I had this, I started saying, oh, well, let's, let's check if maybe I can do more with more of them. So here's the, here's the deal. So let's move this up. This is my motivation for today. Find synthetic language that allows us to do a higher differential homology and differential geometry. So, so there's an absolute weight of saying this, but I'll do it sort of very concretely. Here's the site in the sense that was introduced for André Zoyard, or we call it by André Zoyard at the lectures at the beginning. The site for, you know, super differential geometry. If we do differential geometry and maybe with the super aspect, what is the grotto that we're working on? Well, it's very easy. Everything is modeled on, if you think about it, smooth manifolds are modeled on smooth Cartesian spaces. So this is what we, what we take here. This board is not very wide. Here's the red card space for the full top carrier of smooth manifolds on those that are R n's. It's the carrier whose objects are R n's for various n and the maps are smooth maps between them. This is the, the simple model space. These are the affines, if you wish. Or open set to R n or R n set. No, we can't just take R n itself because it's diffeomorphic to a ball. But it's, it's a minor simplification, right? You could take, just open set. I could take open set. We could also take, people like to take all manifolds. Then, yeah, we, I'm just giving one side. There are many different versions of giving this side. And, and so, so we, I want to define now something that is called, or that underscore called formal smooth manifold. So we can do formal Cartesian spaces. What is meant by this? Well, here's a quick way of seeing it. It turns out that the functor that takes a smooth manifold to its ring of smooth functions, regardless of an ordinary algebra commutative algebra over R. This is actually fully faithful. So you can kind of characterize manifolds by, you have this kind of shadow of algebraic geometry here. And so what we can just do, we can just say, as in, as, as Groten did back then for the formal schemes, we can just say, well then let's just take commutative algebras that do have infinitesimals. Think of these infinitesimals as being functions on a piece of our test space that is so small that some finite power of these functions will actually vanish. So, so I'll write this object as the object in formal Cartes space. I'll write it as Rn times d. d is my generic symbol for informal disk, an infinitesimal thing, something that is a point with a formal thickening. And it's defined by the following that I say we regard its algebra functions. So I'll define it this way. We define it to be the tensor product of the ring of smooth functions as an ordinary r-algebra tensor over r with something of the form with an algebra which is underlying vector space of this form where v is finite dimensional and nilpotent. Nilpotent for, well we can do this, you can filter this and look at a whole tower of variants of this but I'm going to take the case where there's some n, some natural number such that for every element there's some n such that its nth power is zero. Local ring with finite dimension of local ring. You could call this a local artyn algebra over r. It's real, it's real. It's real, yeah. As we discussed, Vale introduced this and it was called nilpotent. So the like Anascauka and Billaville called this a Vale algebra but the algebraic geometries know this as a local artyn algebra. So you should think of these, these things as being functions on the point kind of with their first orders of some Taylor expansion and then the space that they are functions on this z is so small that the physics, the way I was taught calculus in my physics courses becomes actually true. At some finite power these things actually become zero, right? It's a finite dimension quotient of the power series. Yes, yeah that's another way to say it. So this is how to find these formal Cartesian spaces and then there's of course this embedding here which just regards the Cartesian space that's having no infinitesimal thickening and it will be kind of important for us that you can easily check if this is co-reflective. So my joints will always go this way. Left joints are on top of right joints and just for the sake of it since it's causing so much happiness from the audience I'll also consider the super case, super formal Cartesian spaces where we do just the same thing I just add, I just allow these algebras to be super commutative. So there's z2 graded and commutative in the z2 graded sense and I simply continue this, let this still be ordinary ring of smooth functions on Rn but now this is allowed to be a super ring. So there's an obvious thing and something that will be kind of important here is that there's of course the also embedding here which simply embeds everything into the even graded piece and this is actually two reflections and we'll speak about this. One takes the bosonic body, the other takes the even former piece. Yeah so we'll come to this and if you wish of course there's a point which is reflectively embedded here. So this is the side we're going to use and now I'll apply the sheaf infinity function that was discussed in the lectures at the beginning of the series to define the infinity toposes of super differential geometry, the grow one, it's differential geometry. So let's start on the very left. So what is infinity sheaves on the point? Maybe you know pretty much this notation, I hope this was on the board a few times. So this is the same thing as what I will throughout call infinity group points. I know there's a, and it seems to be very popular here, there's the habit of saying spaces for this so some of the speakers before me wrote spaces for this. I do want to caution about this and you know these questions came up, it's a really important thing of this is a purely combinatorial thing because there will be lots of other spaces appearing here, extra geometric manifolds. So calling this spaces is really applying a trick which one should be fully aware of when it's really looking at topological spaces and there's simply localization at the we comma to be equivalences. So it's kind of a trick that allows you to speak about infinity group points as being a localization of topological spaces but it's really important that at this point as far as geometry is concerned there's no geometry here, these are disc, so these in particular contain the discrete group points, the discrete sets there's no geometry here, these are not topological group points. So infinity group points are combinatorial in some sense. Yeah so you can also, maybe I should also write, so this is also the same as, you can also take simplification sets or if you wish you can restrict to current complexes. I mean this is really category theory, you know current complexes localized at the, at the homotopy, you know you can actually localize these at the homotopy equivalences. Okay so this is kind of our base topos and now we want to equip this with geometric structure and the way we do is that we simply consider now infinity groups that are parametrized of these smooth guys by the function of points perspective which I guess in this context, in this conference I don't need to say much about, so let's see how much space do I have, one, two, three, four, let's leave a bit of space. Chiefs infinity on cart space, this is what I called it which, and which I'll ask you and invite you to think about as being the smooth infinity group points. In particular Lee group points, they are a full sub 2.1 category of this here, SR for instance, you know simplitial group points and simplitially group, sorry, not yet super, so we get to this, so you know we, we will now do the same thing, keep going, so we can do the same thing on formal Cartesian spaces. These are now smooth, these are now infinity group points that are equipped with a geometric structure that is seen by testing function of points wise with formal Cartesian spaces, so they will look like formal smooth infinity group points. This, the one topos truncation of this is what is famous in the literature is the Cayet topos, was one of the first topos that were faithful models for Lavier's synthetic differential geometry, so this is kind of infinity, the standard model for infinity synthetic geometry, and then we can do one more which also has a vague precursor in the literature, but yeah anyway it's not, not a big deal to write down these things, a big deal will be to see that the joints between them are actually all we need to know about them, super formal Cartesian spaces, so this is now, you know I just keep adding these edges, so you should think of this as being super formal smooth infinity group points, so then these adjunctions induce on the pre-shift categories, they use adjoint quadruples or even adjoint quintuples by a left and right carn extension of functions between these, these shift categories and the thing one, one needs to check the slightly non-trivial statement is here, which of these do pass to the sheeps and it turns out for the first case all of them actually do, which is, which is a strong statement about these topos of Cartesian spaces, so there's the, let me, let me go down a little bit here, so there's a global section function here as for every infinity topos, since this has a terminal object, this is just a valuation on the point, a valuation point on the point means that we forget kind of the smooth structure, we forget how to test our smooth infinity group points by mapping open balls into them, we just know how to map points into them, so this is just forgetting the smooth structure and we get an ordinary underlying infinity group point, yes? Just to understand why we call that smooth infinity group point, if I have a lee group point, can you, how do we embed it there? Simply by the function of points, so the lee group point, which, which let's call it G, bullet say it's a system of two smooth manifolds, you take it to be the, the, the sheave of group points that to each test manifold u to each rn assigns the group point, which is c infinity u comma g1, you embed it using more or less unitary, a lot of the same, okay, and in particular if you, instead of having the lee group point, if you had the kind of can complex of manifolds, yes, exactly, like Gesla, Gesla is looking at, can't simply manifolds are the geometric objects in here, the geometric infinity group points, this was essentially proven by that special case, yes, important special case inside here, but I don't want to get into this, but I have in the notes you see a big diagram that filters this, but all the stuff that sits in there, lots of classical stuff you find as fully faithful as, as full subcategories in here, yeah, particularly the, the can't simply manifolds that many people are, are into there in here, but I'm following Grotenig's perspective, I feel that before we focus on one of these that we do may want to concentrate in the end, but which do not have as nice, which do not form a nice category, I focus on this very nice category, I set my theory in here, I then know how to do everything in here, and then in each particular case in the end I'll check if any given construction actually lands in one of these subcategories, I think that's the way to go, because otherwise you, if you restrict yourself too early, you lose the nice theory, there was, I think one of Grotenig's point of introducing anyway now. So this is in there, and so this functor has every, I hope this might have been mentioned at some lecture, maybe I forget, so this has this left exact left adjoint, which often goes by the name delta, let me call it however disk, because what does it do, it takes an infinity group out and regard it as an infinity stack that is actually locally constant on the site, but by our interpretation, so it will assign the same value to each Cartesian space, but that thing of the Cartesian space is being, you know, the points that we map into our group, it just means every map from a Cartesian space into our group has to be constant, so this equips infinity groups with discrete smooth structure, the only smooth maps into them are the constant ones, and now it turns out due to the fact that we do have this adjunction here, this does, so this is what you have for every topos, there's a further left adjoint here, which happens to preserve products, so this is Lex here, and this is something very special, and let me be this my first example, I'm going to call this pi infinity, and I put the examples here maybe, so let's start seeing what some of these things do, so there will be lots of examples here, so if X is a smooth manifold, for instance regarded as sitting in smooth infinity group odds, under the sort of extended, you know I'm betting, no I can't unfortunately, aha, but I can do this, yeah, then, so pi infinity, this is a theorem I need to prove, actually it works also, this was proven by Defka Kiddie, it works also for Frechier infinite dimensional manifolds, and a few more things, but anyway, so this will always produce the homotopy type, this produces the fundamental infinity group of X, so if you wish this is the, this is the singular, simple complex of the unaligned topological space of X, regarded as representing Kahn complex in Henson infinity group, this is why it's called pi infinity, this is the etal homotopy type frontier, it sends a scheme, which in our case are manifolds, to its homotopy type, there's something one needs to prove, and the fact that these are adjoint, if you think about it, it gives us already a synthetic version of Galois theory, because it says it relates, it relates locally constant stacks to the homotopy type, maybe I'll have time to talk about this at the end, there's one more adjoint here, so it turns out, also this turns out to be fully faithful, there's one more adjoint here, which you should think of as being equipping stuff with a co-discrete structure, so what does that mean, in order to analyze these things, and this is where these modal operators come in now, see what I really want to be working in is all the way in here, this is the big infinity topos that I'm going to talk about, so I always call this H, this is the home for my generalized commodity theory, so the big H is a reminder that we're doing commodity in there, so I do want to replace all these adjunctions that go between different categories as just anamorphisms of my one big category, so let's try to transform it this way, all right, so we can go down, if we are here in smooth infinity group points, since this reflects into them in these ways, I'm going to consider taking first the global sections and then re-embedding discreetly, so this is just the underlying discrete infinity group point, but thought of as sitting in here under this canonical embedding, and this needs a name and yeah, it just so happened I'll call this flat, for why is this called flat, well here's another example, take there will be lots of boxed examples here, so g a lee group or a simplicial lee group, if you wish, then you can, then it turns out you have the following, then maps to, there's the internal delooping of the lee group, so the delooping of a group object internal to the top, this is the classifying stack for g principle bundles, so this is a principle bundle, this classifies a principle bundle, and then you have this flat operation here, you can form flat bg, and since it's, there's now a coma or not, there is this canonical forgetful map to bg, and you find that lifts here their equivalent to flat connections on that g principle bundle, flat connection, that's why it's called flat, it sends modular stacks of principle bundles to modular stacks of local systems of flat connections, these are little things one can observe, so this is the beginning where one sees that these these adjoints actually encode interesting information, so then I'll give a name to the other one, so the other one as we take global sections, but then re-embed code discreetly, this seems to be an exotic operation, I'll call this sharp just because that rhymes on flat musically, and sharp happens to have a nice interpretation too, suppose x is a zero truncated sheaf, a set internal to our smooth infinity group, so suppose it's in just plain sheaves on card space, regarded as fully faithfully sitting in the infinity sheaves on card space, then you have this factorization, you have the unit of sharp, so this sharp x is something weird, but here's the thing, consider it's, I mean from a classical perspective, consider it's image factorization, we factor this through an epi followed by a mono, it turns out, so I'll call this sharp one, it was the first post-nicole stage in this image here, sharp one of x, this is precisely the defiological space completion space underlying x, so defiological spaces, we're also introduced by Sorio actually, there's Patrick Iglesias-Semur, he's written this big book on defiology, they generalize smooth manifolds and they are what this is saying here, defiological spaces are precisely the concrete sheaves in here, those that are separated pre-sheafs for the La Viettini operator, sharp, so just to mention this that we see how these junctions begin to see actual manifold geometry here, and now we have of course one more, so we can also go down, we can also go down with pi infinity and then re-embed discretely and this after a long discussion with Mike Schulman we ended up calling shape, why is it called shape? Because as you already saw here, it sends an object to its shape in the sense of shape theory to its homotopy type, so for instance back to this example, if you have the internally looping of a Lie group, the modularistic of G-principle bundles, then shape of it, it's an ordinary infinity group, an ordinary homotopy type constantly re-embed this turns out to be BG, so without the bold phase, my bold phase B is to indicate that there's extra structure, this is the smooth version and this is the ordinary classifying space of G as a topological space up to weak homotopy equivalent, whereas flat BG, just to emphasize flat BG is what in standard textbooks we would call KG1, so all these things you see are now seen by these adjoins and yeah, so you saw that you want G is considered as discrete, yeah maybe yeah but this is the yeah, so maybe let me put this down, this makes everything discrete, that's how this comes about, whereas this really takes the geometric realisation, so generally if you have a simplicial manifold, shape of it will be the fat nerve of it, the fat simplicial realisation as a homotopy type, so the fact that there's flat structure appearing here shows us that this starts to know something about differential comology and this will be one of the main theorems that I hope to put on the next board, but let's maybe complete this and yeah, so I don't know, we can complete this to these squares if one wishes and without continuing to give names to all this, let me just indicate that from these further adjoins you have, we get no further functors here, so we get the embedding of just smooth infinity groups into those that also have formal thickening and there's a reflection to this and what does it do? Well, let's look at it, so again it turns out there's four functors here that are mutually joined and those from this adjoint pair and I'll give them names, so in the same way the leftmost one that appears I call re, re, this will be the reduction operation, yeah let's let me write it down, so this is the one that comes from the top, what does it do? So if you have a formally thickened, a formal manifold then this is its reduced part in the sense of algebraic geometry, you, you get, they get the object whose ring of functions is the same ring of functions but without the nil-potence and it has an adjoint which comes from these two adjoint functors here which you know to use the kind of dual notation I call im and im also has a classical name if, if your imx in algebraic geometry is called the diram stack of x, another way of saying is this is the, the quotient of the two projections out of the formal neighborhood of the diagonal of x, that's what it comes down to, so all this is kind of a, I'm making statements, this is not supposed to be entirely obvious, I don't know, so in particular what we will use, I'll, I'll say more about this on the next board, it kind of, it follows from this, well I mean this needs a bit of, but if you slice over imx then this will be the category of PDEs over x for infinity stacks, so the object in there will be bundles over x equipped with a sub-bundle of their jet bundle that defines a partial differential equation, so this is at the end of the talk whether the field theory comes in will be able to do variational calculus just, just using non-linear PDEs, yeah, so if you just look at, you know, in algebraic geometry people like to consider just the quasi-coherent chiefs on the ram stacks, they have the d-modules on the original scheme, they're kind of the linear PDEs, but we'll actually, if you want to do some serious variational calculus you need non-linear ones, so, so this has a further, further adjoint which I call it, but I'm not going to, going to talk about it, it exists, it's important for, yeah, some, anyway, so let me just give these names, and then here is the same thing, so we get, so we get these two ways, so first of all formal manifolds, now embedded into super manifolds is the bosonic piece, and there are these two ways to projecting back, take the bosonic body and take the, the fermionic piece, so I'm, I don't know, looking for some notation here, maybe this will cause some excitement now again, so I called, I called, I decided to call this guy that takes the bosonic body like so, because it produces the symbol for bosons and Feynman diagrams, right, so if you have X a super stack, super manifold, then, then this thing will be the underlying bosonic body, what's called the bosonic body, whereas, and if, if I made this choice, then the left one, it produces even number of fermions, so, so if you, I didn't denote it by double arrow, this is the formal space, formal space, you know, of even degree, so you don't, you don't throw away the, the, the odd graded guys, but you keep just their even pair so that they, that they're even graded, and then it has one further adjoint which, for reasons that I won't talk about, I call, reonomy, this appears in supergravity literature, some, doesn't matter, so we have, we have this, and if you wonder why I, I counted to 12 in my, in my abstract, so, so what we're really doing is, you know, we're starting, but I don't want to talk much about this, there's an initial adjunction, sort of, in every infinity to top of us, you take the, the monad that is constant on the initial object and the co monad constant, or the other way around, constant on the point, they are adjoint, this guy is flat, this guy is sharp, meaning it's in the, these are idempotent, since these are faithful, this is important, right, these things are all faithful here, so we get all these monads and co monads are idempotent, so we can ask, is sharp of the point, again the point that it is, and it turns out actually, sharp of empty is also empty, so, so these are included, these are included, these are included, and these are included, if you just want to, kind of, you know, end this properly, graceful exit, you notice there's also this trivial adjunction, and that's how you come to 12. Anyway, so this is just some notation, and it's, it's not super hard to prove this, I mean, this is maybe the, the thing where one has to work, the rest, the rest is just making the right observations, but the point now is that I don't want to emphasize this, which is not obvious, at least it was surprising to me, you can now ask, well, you know, let's put it this way, if you, if you take a random top in infinitopos, you will notice, it does not have many more adjoints this way, beyond the first two here, so these are strong conditions on infinitopos to have this long sequence of adjoints, you know, it's like having an adjoint quadruple this, why I shifted it up, and then it doesn't quite continue to a quintuple, but after you resolve a bit, there's an x one, you resolve a bit more, there's yet an x one, so there's a strong structure, and, and you could say, well, if, if the context we're working in has this strong axiomatic structure, that means this axiomatic structure must know at least a whole lot about what we actually do in components in this thing, so there's a hope that using just the adjoints, you can actually do differential geometry, and the, the running claim that I want to make is you can actually do a whole lot of just traditional differential geometry and lift it all to the infinity context, so I do want to close now by, or, you know, the last part, I want to say how this, I want to give you some theorems how this works, so in this box here I'll do, we'll do synthetic differential commodity using just these adjoints, in this box I'll do synthetic, well I would call it synthetic differential geometry, but of course this already is used much by Laverine Coch, and what I'm doing here will subsume this, but let me maybe, let me maybe emphasize that, so we'll actually do this carton geometry here, synthetically, so we'll find, we find etal infinity stacks as internal manifolds, and we will be able to equip them with conformal Riemannian blah blah blah structure, super conform, anyway, and if time permits, but probably it won't, then yeah I should, you know, to come back to the, the first words of my abstract on this board, if time permits I'll speak about doing synthetic, I'll give you just some main facts about synthetic variational calculus on jet bundles, but we, we kind of do it in this pre-quantized way, so it will actually be, so ordinary variational calculus is kind of the mathematical home of classical field theory, but since I will not work with forms, but with our differential code cycles right away, so this will be, this will be this pre-quantum field theory that you see in the abstract, so I'll just highlight the, the first, of course the board won't contain the whole thing, but just the first key statements, and in the unlikely event that I'll still have time after that, just for entertainment purposes, since I'm, since in this context here it might find some friends, there's also, there's some nice, yeah let me just, so I could, if you insist, say something about synthetic, just some simple facts so that I kind of neatly come out, so let's, let's start here, yeah I was a, you know, I started playing with this in my habilitation thesis, and then this main statement, the way I'm saying it now, unfortunately I didn't, I didn't quite have myself, this was formed by Bunker, Nikolaus, and Falkel, so this is one of the, I was very fond of this statement, so let me call this the differential, differential, brown representability, sorry, brown representability theorem, so the brown representability theorem says that every homology theory, generalized homology theory in the sense of Unberg's statement is, is given by maps into some spectrum, and the question is, is there a generalization of this for differential homology, if you know differential homology this is an open question, if you don't know differential homology you can take what I'm saying now as the quickest easy definition and then just go from there, so this was proven last year by Bunker, Uli Bunker, Thomas Nikolaus, and Michael Falkel in Regensburg at that time, so it says the following, yeah you know I'll just, yeah there's limited space here, but let me try to get, get the gist, so we consider now that the, so I'll write th for the tangent topos in your sense, tangent infinity topos, is that okay for right, I forget which notation you use, maybe some different notation, and so, so a first, a first, yeah so this is parametric spectrum objects in our topos, so first simple observation is, this is again cohesive, if, if we start with any of these toposes and apply th of it, then the result will have the same structure, you simply observe that, you know it's, yeah so I'm calling this here cohesive, sorry this structure, if I do have these extra, these extra things with this fully faithfulness, so this structure I call cohesion, at topos with this structure, this is fully faithful and then it's equivalent to this being fully faithful, they're both, so I call it infinity topos, cohesive, if the terminal geometric morphism has the properties that the inverse image is fully faithful, has a further left adjoint that preserves products, and the direct image has a further right adjoint, and on one topos this is what Levere had called this, so I first wrote this down, and then there was some, Peter Johnson alerted me that this has been considered by Levere, and he called it cohesive, that's why I picked up the name, and then if I have this extra structure I call this, I call this differential cohesion, one way to motivate this is, so this cohesion here comes from the fact that if you think of this topos as knowing what it's, how it's pointing together, how in an open ball the pointing smoothly together, this is exhibited by this shape which contracts them away, one can show that this the ramsec functor is like the infinitesimal shape, it contracts away infinitesimal close points, in this sense this is differential cohesion, and then this I don't know, you know super differential cohesion, okay, yeah sorry it's just terminology, it doesn't really matter, choose another terminology if you want, so look at parameterized spectra, and consider for the beginning just a spectrum over the point, so just a sheaf of spectra over Cartesian spaces, let's call this E, so the theorem effectively says that any such, so these are, these cohesive spectra are the differential homology theories, what it means is the following, one shows that if you take any such guy, and then look at what its projections are under this pair of, or this triple of, a joint that we have, then these projections fit into a big diagram, and you observe that this diagram is precisely the diagram, the hexagonal diagram that people used to write down to excretize differential homology, so here's two aspects of this big hexagon, so first of all IE, let me, here's how it works, so if you consider the shape of E, you get an ordinary spectrum, a spectrum kind of is a sheaf over the point, constantly embedded as a constant sheaf of spectra, a locally constant sheaf of spectra, and this is the underlying, is the underlying homology theory, generalized homology theory, no more differential, so for instance I don't want to, you know, because we use up all my time, we can do for instance the deline, the deline complex, deline homology, regardless of sheaf of spectra, the deline complex, and then if you do shape of it, you get hz, right, you find that it's ordinary integral homology, in this sense deline homology is a differential refinement of, you can take, you can take the sheaf of vector bundles, the stack of vector bundles with connection to its infinity group completion, this is part of what they show, it gives you a sheaf of spectra, its shape will be ku, the ordinary k theory spectrum, so in this sense, you know, infinity group completion under direct sum of the stack, the smooth sort of infinity group out of vector bundles with connection is a differential refinement of ku, and so forth, and so here's what is nice, so one can, one can see, one can also form the co-fibre of flat, let me call, let me call this flat bar, so this is my notation for simply the co-fibre of flat e from the co-unit of this monad goes to e, this turns out to be the home of the curvature forms, differential forms, with coefficients kind of in your differential homology theory, so if we, if we take ordinary deline differential homology, this comes out as being the sheaf of closed forms in the given degree, the ordinary sheaf of smooth closed forms, if you put in k theory, you get the sheaf of forms are all even degrees created by the bot generator, if you take elliptic homology, you get something wilder and so forth, so there's another thing, if we took the co-fibre of flat, we should look at the fiber of shape, so let me, let me call this shape bar of e, the fiber, the homotopy fiber of course of, of e goes to shape e, well these are the things, these are the differential cos cycles, this underlying bundle if you wish is trivial, these turn out to be the, these are the globally defined, globally defined connections in your differential commodities here, again if you do it for ordinary differential commodities, this comes out to be the chain complex of differential forms, you know in lower degree, in one degree lower than your curvatures are, and then you get the canonical map here, this comes out to be the deram differential, again on ordinary commodities is the deram differential, in general it gives some map that behaves like the deram differential, in the following sense, so here's another theorem, they approve, so this d, so abstractly from using, using nothing about what e is, just adjoins the fundamental theorem of calculus holds, yeah, so you know for, for generalized valued forms in the sense that there exists a map, there exists a map integration from 0 to 1, from the mapping stack from r, or real line which into, into these curvature forms, these closed forms, back to the coefficients of the, the connection forms in one degree lower, so this behaves like, so one can construct this abstractly, the map exists, this is the statement, it exists canonically, and it satisfies the condition that if you pre-compose it with d, so if you go this way and then integrate, then the result is the difference between evaluating at 0 and 1, the difference taken with respect to the abelian structure on spectral, and again, I'm not writing it to the board, again part of the theorem is if you do put in ordinary deline differential commodity, then this comes out as being the ordinary integration of differential forms in the ordinary fundamental calculus, a fundamental theorem of calculus, but you can put in differential co-bodism and then you know now what the, the fundamental theorem of calculus is there, yeah that's what, what fits on this board for synthetic differential co-modulation, this is actually, one can also, if we now take our spectrum, not in the point, but over, over say peak of KU, of peak of some, some E, then we get twisted differential co-modulation just, just for free, but it doesn't fit in there, so this actually solved an, they solved a problem, that's why they, why they did, they solved a problem that was, that was raised by Simons and Sullivan in 2008, and so this increases, again this was not, this was not, so these authors did not introduce this because they wanted to play with axioms, they wanted to solve for concrete problems that were open in differential co-module theory, and it turns out you formalize it axiomatically and then it drops out, and so, let's say something about synthetic differential geometry, if I have time, so, so fix, fix sigma in, in our, so, so now I'm, I'm working, yeah h is my ambient context, and then consider, consider the base change, depends on, depends on product context and base, base change if you wish, slice over sigma, slice over, it's the RAM stack, so then from, from the unit of the im monad, we get this adjoint triple here, the unit I should call eta im for sigma, yeah it's a bit cumbersome mutation, pullback, it has a right adjoint, eta im sigma push forward, so the point here is just, we have this God-given structure, now we have, we have our monads, they all come with their unit and co-unit maps, and as you can see here, we're just playing with the structure, I'll find, form co-fibers, now I'm pulling back and forth along these units and co-units, and there's of course the left adjoint here, which should be called eta im sigma schrieg, and so this induces again, I always transfer back to monads, adjunctions to monads, so this induces in a joint pair of monads and co-monads on the slice over sigma, and I'll call this T infinity sigma and J infinity sigma, for reasons that will become clear in just a second, this will be just, this will be formal tangents, so this now goes from the slice over sigma to itself, right, so this goes first down with the left adjoint and then back and the other one goes first down with the right adjoint and then back, so the jets we need for the variational calculus, here's the, here's the statement that we will need for the differential geometry of manifolds, you observe the little fact that T infinity is sigma of sigma itself, is, yeah I already almost mentioned it, is the formal disc bundle, so it can only come to map down to sigma, it's a bundle over sigma and its fiber is the formal, its fiber over a given point is the formal neighborhood at that point, and using this we can do something, what can we do, so I'll say this and then also observe, actually the following statement is something that I learned from a paper by Maxim Konsevich and Alexander Rosenberg on non-commutative stacks in very different language, you have this characterization of formal italen maps, if one unwinds what's happening in this paper this is just, what, this is just three of the adjoints that I'm using here, so in that very nice paper it was observed, it's about non-commutative geometry and then suddenly there's a side remark saying, saying if you have a map in such a context between two spaces then you know consider it's the naturality square of the imunit of this deram stack functor unit, and you can ask, you can now ask that this is a pullback, so if this is a pullback, this being a pullback means that this map is formal ital, this, you can take this as being an extremitization of formal italeness in the sense of I guess what Andrei Geryal did way back with somebody, with Ika actually, so that's one can check, one can set up formal axioms for what open maps should be, and then one can check this, this satisfies this, but so this is a good extremitization, but part of the statement is if you actually put into smooth manifolds here are two ital stacks, which I haven't defined yet, then these match actually come out as the local different morphisms, these are the correct formal ital maps, now if we have this, so I think I'm lacking one thing I wanted to say, oh no it's okay, so using this we have now a good definition of ital stacks, so given V a group object, a group object, so an infinity group in H, a super formal smooth infinity group, so it could just be a Lie group for instance, I'm writing V, you're really supposed to think of it as being for instance just a vector space with its additive group structure, and we want to model manifolds on this vector space, but it could be any group, then I say then a V manifold, this is now sort of the internal definition, this comes out externally as being a V et al infinity stack, but sort of in this internal logic we just think of everything as being sets, a group of structures, so a V manifold is, so it's an x equipped with an atlas, so within a one epi, an effective epimorphism in the infinite topo sense, out of some U such that this is et al, formally et al, so this extremity is the fact that you have a cover by open maps, but then we want to say this is all locally equivalent to our fixed V, so we just say that U looks locally like V, so there's a formula et al map from U back to V, it's a correspondence if you wish that transfers the structure on V, the local tangent structure on V to the local tangent of x, so let me take this as my definition and then the statement is, sorry, it's in the spirit of captain geometry, we'll see captain geometry in this little square up here now in just a second, so what I do need to extract to get captain geometry, I do need to prove that this is sufficient to get me frame bundles, of which I can then reduce the structure group, and this is, it's not entirely trivial, but it's a simple, so I mean part of the thing is these proofs come out, it's kind of, they flow, so using my Vittorius arguments and stuff you can prove the following, you know what I mean, call it proposition, it's a little exercise probably for people like Andrea or something, so you check that, that if you have a group object then it's frame bundle for the group object is canonically nically trivialized by left translation, by left translation, which is, you know, if this is a 17 group stack, this can be, this will be tricky in components, but you can just synthetically deduce this now, whose fiber, so the fiber, the typical fiber of this trivialized formal disk bundle, whose fiber is of what I will call DVE, just the formal neighborhood of the neutral element, the point in V, so this is the first statement, and then B, you know, this is not sufficient to transfer this to X, X won't have a trivial formal disk bundle, but it will now have a locally over U trivial formal disk bundle, then there's another theorem that tells you that in infinity in top of every fiber bundle it's actually associated, so it follows that for every, for every V manifold, V et al super infinity stack, then it's formal disk bundle, X, then it's formal disk bundle is a DVE fiber bundle, infinity fiber bundle if you wish, associated, so there's a theory of principle bundles, infinity bundles, and then infinity top is associated once into force, so it's associated to a what I want to call a GLV principle bundle, where GLV I take to be, it's the internal automorphism group of DVE, the linear automorphisms, right, so this will, or it's the jet version if you take all formal things, principle bundle, principle infinity bundle, and this of course I call the frame bundle, frame of X, and then from there we get carton geometry, I mean from there it's easy, now you have the frame bundle, you can now ask for any group homomorphism from any group, infinity group G to GLV, and can lift the groups, the non-trivial statement is maybe to really do carton geometry you want to do torsion free G structures, so you need to do one more thing, you need to reduce the structure groups and then say that again formal neighborhood wise, they're torsion free, so that formal neighborhood they're trivialized, but trivialized with respect to this trivialization, this turns out to be, this turns out to give everything that was ever done in high-dimensional supergravity, for instance it turns out that the equations of motion of what's called 11-dimensional supergravity is equivalent to taking the right V here, we're certain super infinity group that is canonically defined, and then just looking at just the torsion free 11-dimensional carton geometry of this, this turns out to be to give 11-dimensional supermanifolds that solve the equations of motion of supergravity, so that you find this and point us to it in these notes that I distributed, okay so I guess I'm out of time is it right, but I'll just keep going until you stop me, so yeah let so just quickly here how do we do jet geometry and yeah let me just it takes two minutes to fill this board, so let's let's fix a bundle now over some sigma and think of this as the field bundle, so what will happen is like this, so I will now put ourselves entirely into the slice for this board, we work over space time sigma or wealth volume sigma, and I'm asking you to think of like follows, here's F regarded as the guy in the slice, and here's sigma regarded by the identity map in the slice, so this is the terminal object in the slice, so a map here will now be a section, simply a section of this bundle F, and if we axiomatize physics, if you open the first book on any mathematical physics textbooks, so sections of a given bundle, this is how you axiomatize field configuration in physics, so we think of this now as being a field configuration, right take F to be the bundle of you know rank two tensors or something, then this will be this will be a field configuration of gravity and so forth, and so this is a standard stuff, this is what what you're supposed to think of this F, but now we can do the following, so in H we have differential homology here, so H over sigma won't be cohesive anymore, but we can kind of compare, so what we can do is now we have the canonical map that shifts everything up to differential equations, so we can you know take any guy in H and regard it as forming the fiber of the trivial bundle over sigma, and then we can further go to regard this as the trivial partial differential equation over sigma by, this is really, so this comes here from the back theorem for, this is really defined as being the, the Einberg Moore category of this jet core monad, then by back theorem you notice it's just a slice, so what we do is we push forward along this thing, this is the same thing as producing three differential equations, we just regard the bundle as being the differential equation which is trivial such that all sections of the bundle are solutions, so this is a way to canonically embed our cohesive context now into PDEs, and now up there we have, we have a splitting, yeah there is horizontal splitting, I, one can define this, I'm not, I don't have time not to say this, so you can take our differential homology theory down there, let me call this map sigma, we shift it up in this PDE context, we regard it now over sigma, and then there's a canonical projection to its horizontal piece, I call it EH, the horizontal piece is roughly that which is actually seen by sections, so this will give us a structure up here in PDE theory, but if we pull back along sections then part of the structure will disappear because the sections will only see a horizontal structure that's the standard terminology of jet bundles, so we project out that piece that will be seen when evaluated on sections, and then it follows that, yeah so this is projection here which one can define, and then it turns out, and this is explained a bit in the first six sections I think of the nodes, then you can axiomatize field theory just like so, so here's our field bundle, and now we have a map, but we're in this Einberg-Moore category, we say we have a map to EH, but this map being in the Einberg-Moore category between two free guys will actually be a closely map, so it will be a map out of the jet bundle of this, it will be a map in H out of the jet bundle of this into our differential coefficients, and these are precisely Lagrangian densities, Lagrangian densities and standard literature are differential forms on jet bundles, and here we get now differential forms pre-quantized to differential cosigates in some, so this will be a Lagrangian density which defines which physicists think of defining as a physical theory, and then we have the curvature map, this was this one here, remember this they send a generalized bundle with connection to its curvature, and now we have kind of the horizontal part of this curvature map gets us to the horizontal part of these differential forms, so this is the coefficients for differential forms now made horizontal on the jet bundle, and it turns out this is the, this will be the, this is the Euler variational derivative, it comes out as being the variational derivative of this Lagrangian, one uses the fact the the bi-complex structure on jet bundles to prove this, it's kind of, it gives kind of a prankery lemma adapted to jet bundles, and anyway so this composite we can then, this is like the the variation of L in the physical literature this is called EL, it's the Euler Lagrange form, I mean if you feed an E to be ordinary differential homology in some degree this comes out as the ordinary Euler Lagrange form is in the standard textbooks, and then it turns out forming fibers here of this thing, the kernel of EL is usually called E, this is the shell, this is the, see this is now an object in PDEs so it defines some partial differential equation on sigma and this turns out to be the Euler Lagrange PDEs of motion that that physicists associate to a given Lagrangian density, so once you have this you can then say well then these sections here, so if you have a PDE then the maps out of the sigma into it, these will be the solutions, so these are the classical solutions, so all these can just be, you know I'm just using the jet comonode and I'm using that I know what differential homology is and that they are compatible, and again the same is if you put an ordinary delinco model this comes out as the ordinary story, prequantized, if you put in differential k theory something more interesting will come out, yeah and so forth and one can do the prequantized phase space, yeah and now I guess I have to ask you do you, one minute, did you say one minute or did you say stop now, so here's something internal sheaves is that there's a nice definition of internal etal topos, take an x in a cohesive, in a differentially cohesive topos then you can make the folding definition, right sheaves, internal sheaves, I mean, yeah here's what I mean, I define this to be the slice over x but on the on the formally etal maps, the full subcategory of the slice on those maps that are formally etal, so this is like in the in the standard algebraic context discussion of the of the etal topos and that we're just using there's no finiteness condition of my fibers, it turns out to be reflective and co-reflective by etalification, this turns out to be so one can prove using the x and so this is a topos, this is actually where this x-erider join appears to prove that this is a topos one needs, so we can think of the guys in here as being sheaves over x and now you can do the following, suppose let me let me do the quick nice notation, suppose we have a map now from x, let me do it like this, to to the discrete image of say small infinity group points, smaller multiple depths with some small just to make it an object in our topos, then you know over here there's this the univalent universe, you can I mean there's just this is really a star right univalent, this is stashf, he observes the universal vibration for the connected components of here back in the 60s I think right, so this is the let me put a hat on here, this is the the direct sum of b odds of f's for homotopy types f's and this is the direct sum of homotopy protons f mod odd f, this is really was known in the 60s, so we can pull this back form this pullback here and I'm I'm claiming this this will in this from this perspective this will realize x-head as being a locally constant infinity stack on x is being so I need to show that that it will be in here but you just factor it like this so here's yeah I shouldn't do this now I don't have time I have minus minutes right so so this will be actually one can show from just factorizing and using the pasting law that this will be a stack over x and then you see now here's the beauty or one of the beauties now use the adjunction between disk and pi and you notice that locally constant infinity sticks on x the left to join to disk is pi infinity the etalon monotopy type function x by the way is anything it doesn't have to be a manifold it can be any stack any high stack so so we learned that we learned this one line that locally constant infinity stacks on x are you know give us the homotopy type of x in that the kind of infinity permutation representations if you wish of the etalon monotopy type of x classifies locally constant stacks I thank you for your attention and your patience a couple of questions yes first of all in the thing for plane physicists this is kind of disc verical to zero it's important question would be BRST or BV is your Lagrangian formalism automatically embedded to some BV well yeah that's thanks for the question so the way I set it up this is this guy this just gives the BRST part right the le algebra to get to get BV there's there's footnotes in the notes on this you would have to replace your replace the Cartesian spaces by derived Cartesian spaces which is rather straightforward and one has to be make it be a little careful to have them still be cohesive but then it just goes through and then yes then then the infinitesimal version there's also formalization of lead differentiation and taking these guys and lead differentiating will give BV BRST because it's in solid state matters people have knows also this is a very spectra looking out from you know probably you mean in the symmetry protected phases and topological phases in yes so so in particular I mean this is where this I mean freed and more considered k theory but I think what seems to be most interesting to me is that when the the reason the researcher to who sees idea goes back I forget the first name he he he claims he claims in his original article that higher wzw models for a given legal g that they actually classify the symmetry protected phases of metam what's kind of curious for me is so I was interested I was I'm using these as you can see in the notes to describe super p-brands green schwarz super p-brands and he's kind of claiming that the precisely the same mathematics should or the same broad mathematics actually controls the symmetry protected phases of matters but I have to say when I look at his articles I see that he claims this I don't really see that he proves this but anyway it would be interesting so you alluded to the fact that also phase space yes you can model do you have a Hamiltonian formalism yes so yes so there's a the point of setting this up for us was that we wanted to to have a classical field theory or pregon field theory kind of de-transgressed to all co-dimensions so this is what happens sort of in co-dimension zero your values on all of sigma but but it it keeps going down next to choose a Cauchy surface or a boundary of sigma inside sigma and then just put that boundary here and then you need to adapt your notion of horizontality then a bit more will become horizontal since you're testing with fewer sections and then you you get other jerbs here which are called LePast jerbs well and then you can do the same diagram and now you get then you get the symplectic you get the multi-symplectic forms on the covariant phase space which which which I was so common called the pre-symplectic currents and then if you transgress back to co-dimension zero these will become the the symplectic the canonical piled bracket symplectic structure on phase space you can see this in the notes if you've got a copy of the notes the notes are circulating are they have maybe they're all used up your colleagues have notes yeah yeah so this was a big motivation for us you see there's this gap in the or this you know there was this big breakthrough with luri's proof of the cobaltism hypothesis that quantized topological field theories are known it not we know what it means to do them n categorically but there this is gap that it wasn't known you know the way physicists produce field theories is always by quantization from a Lagrangian but there was no formalism that would give you these extended quantum field theories from a Lagrangian you know there was a gap the the quantization step just happened in co-dimension zero but we we learned that the output actually happens in in full co-dimension n categories so this is the motivation here we this this sets up a formalism of classical and pre quantum physics that goes all all the way down to full co-dimension and I have a link to an article where I this has been working for us where I sketch how it seems to be possible to quantize these by pool portion generalized comology to extend in field theories but but this is I think this one of the big question you know there's some people thinking about it I think Dan Fried is thinking about this and and maybe maybe if I wanted to one thing is I wanted to alert maybe to a group of high-powered topos theorists that there is this this is a big open mathematical problem actually which which topos theory is something to say about and if one could make progress there there would be this I think huge impact on people who don't actually care about topos theory as such but about physics