Sheaves and Topoi Workshop — Overview of ADG & SDG

0:00 The classical theory of its existence is crucially dependent on both the day-space S, which is usually part of our humanity, on the structure sheath A, which is usually in the structure of the fine sheath. Okay, the local form of D, the connection is you can split D locally into the flat differential, the connection 1-4, which is usually known, this process of substituting the flat differential, I have not shown, but it's still flat. this is that this process in physics is called gauging okay you can show that this local is of the phone the connections of the phone okay with these gauges and this is how you how this comes about i will not go into detail just that it's built locally into the flat derivation of the connection one-fourth. And that part of the connected gauge potential part of the connection and the local gauge transformations one can show is very well-known. engage theory that it transforms up finally another way to say that this is the important thing is that the connection is not a tensor here in a dg the connection is not a tensor because it does not respect the algebra coefficient remember it is a c morph is a constant shift is not a name it's not so transform finally the gauge potential this this term is not sorry this is exactly what makes the notion of connection not a geometrical notion not a tensor of course we can build a geometrical object out of the connection which is the curvature of the connection and in order to define the

2:30 curvature of the connection we need two forms first so we extend the the one forms to two forms by defining again similarly to the first differential I see linear Leibniz obeying shift morphism omega 1 to omega 2 to another shift obey modules which may be considered to be the two forms and obeying of differential from a from a to omega one the full the following which I know the following co-homologica new potency condition in the composition okay this just rely just as short to remark that this problem logical in your problems for the position is exactly marks the starting point in a at the round shape of homology. It must show that you still have an exact round complex. At least, by the way. So, in the same way that you extend the D0 to D1, you can also extend the flat connection, the curve connection, which one can put the superscript to 1 as well, from D1 to D2. And if the programmation is the following, for this year again satisfying section thus we can define the curvature of the a connection d by the following commutative diagram this is the curvature, from which you can read directly at the curvature position d. And of course, in the case you're using the flat connection, the usual differential, by the condition of the new potency, before you see that actually the connection of that differential zero. That's what we earlier called it a flat differential. Of course this homological

5:00 terms, essentially, yes, in connection with the last proof, it is exactly, it is exactly the, the capital D, the curve connects, deviation from your potency, that is to say from flatness, which in turn defines a non-vanishing curvature that prefer that prevents approach chain of a modules and shift morphos between them from being exact in a complex in turn the curvature of the connection provides a measure of the departure of the decode chain above from being an exact complex since section wise to have these relations In total, a non-virus in curvature means the same as the non-existence of a global gauge of global radius, which in turn is equivalent to the fact that our sheeps do not have global sections. It's sort of the majority of obstruction theory. And indeed, you will see how it is rather straightforward to see that the curvature of the connection is an amorphous of a module. So, unlike the connection, the curvature is an amorphous of salt. the curvature and respects respects the structures shape of coefficient so for this reason is a geometrical logic but the connection does not the connection is only c-linear is the dual to be a module omega one so in words the curvature are when a connection of vector sheath is a zero-costal local n-by-n matrices for entries over the section of the second order okay having defined a curvature okay local gauge transformations again if you perform a gauge transformation the curvature transform homogeneously as a tensor She theoretically is just it is just the fact that it is an amorphous. Well

7:30 While the connection which transforms non tensorially, it's not a she, it's not an amorphous It is It does not respect our Okay, yes, I spoke about this while the connection is merely a seamorphous, the curvature of the connection is amorphous. So, it's a fundamental distinction between a mensorial entity and a non-mensorial entity, which has some implication for physics. the free bottom line so baby G okay all differential geometry boils down to the structure sheath which doesn't have to be identified the role of a structure sheath in a G's view okay you can talk about as an origin or source space domain of recalls the big tellers that every commutative unit on the ring has a differential. And as a single target space, where our geometric objects take answers and take their values. Differentiability derives directly from the algebraic, but this algebraic candidate stalks of the sheaves in focus and not from the geometrical, locally Euclidean-based baseman, as of the classical theory. As in the classical field, a base space which in ADG contributes nothing to the essentially algebraic differential geometric mechanism, being merely an algebraic, an arbitrary topological space. Thus, with a bit coarse, we can claim that differentiability is independent of smoothness. Because so far, in one way or another, to do differential geometry, as Leibniz wanted to avoid, we intervened by virtue of space, of the manifold, of the base manifold. The base manifold mediated our calculus, our calculations, in particular the use of C infinity and as a structure she they gave us that structure about those

10:00 functions gave us the ability to differentiate differentially but here you do not have to assume that your space is though actually what you what you care about is again in a light nature way you you would like to do differential geometry with a geometrical object themselves do you mean by differentiability. Differentiability. Yes, smoothness. Smoothness but not smoothness, not in this sense. You mean differential structure? Differential structure, yes. yes all right I didn't want to talk I didn't want to put in the title general relativity I'm simply like that but instead of gravity because then because general activity because it that is concretes a theory based on the manifold and here alternative point is presented so all right even more slightly deeper is that in ATG the principal gravitational variable you would say is the connection and not the metric unlike in general relativity here there have been of course ideas to work with the connection starting from Cartan and Dyle, even Eddington. In particular, very telling are the ideas that Feynman had to regard general relativity as some sort of gauge theory. Of course, he wanted to approach general relativity, gravity, from a particle physics viewpoint, and we see the same tendencies in the work in the book by Weinberg. But there's some interesting questions. It says that Brian Hathield recalls in the lectures of gravitation, some interaction that he had with Feynman. So it's no surprise that Feynman would recreate general relativity from a non-geometrical viewpoint. The practical side of this approach is that one does not have to learn some fancy schmancy,

12:30 as he liked to call it, differential geology, study gravitational physics. Instead, one would just have to learn some quantum field theory. I don't know if that's easier for me. Probably that's more difficult. Quantum field theory. However, when the ultimate goal is to quantize gravity, Feynman felt that the geometrical interpretation just stood in the way. From the field theoretical viewpoint, one could avoid actually defining up front the physical meaning of quantum geometry, fluctuating topology, space, time home etc and instead look for the geometrical meaning after quantization Feynman certainly felt that the geometrical interpretation is marvelous and here is the crux but the fact that a massless spin-to-field can be interpreted as a metric was simply a coincidence that might be understood as representing some kind of gauging variance okay and then we have to we have the developments in this formulation of GR with hashtacar and new variables, but there is a shifting in our perception from the traditional metric-based presentation of general relativity. Okay. This is a very high-flying statement, okay, we can reproduce ADG, very important construction of the super-magnon geology in terms of magnitude, used in general relativity. Although we're using any base differential magnitude that supports the spatial scaffolding, there are some basic examples of the structures. First, we define a main matrix, a main matrix is the following shift morphism, which has some properties which is a by linear between the emojis involved it is symmetric signature for Lorenzi and it's strongly non-degenerable

15:00 This is just what Chris asked before, but we assume that an omega-1 form is the 0 of So you can define as the Riemannian-Christophe connection to be again, section-wise, section-wise. These are very familiar relations to the physicist. And you say that the connection is actually compatible with the indefinite, the metric, what is called a metric connection, or a torsion-free connection, if it satisfies this. So this is called the connection, we call it a covariant derivative. The covariant derivative of the metric is zero, so that means that the connection is a metric connection, is distortion free. These are fundamental relation with physics. And the analogous statement being ADG. Okay, so we're working with a metric correction. At which point did the vector sheet become finite rank? Finite trunk at all. Always. That's the definition. It's required. Sorry? It's required. It's a locally finite. It's locally AM. That's an accent. Yes, that's an accent. It is a locally free A module of finite rank. That's the definition of vector sheath. Okay. And then you can define the rich curvature operator.

17:30 This is the encapsulation. And since it is matrix-valued, you can also define its trace, which is the Ricci scalar. Now we have a suitable background built to define a principal notion in this talk, which is the Lorentz-Einstein curvature space. A real Florentian vector sheath, which is a pair of vector sheath, and an A metric of indefinite signature, over a k-algebra space, such that, one, it is supported by a differential triad relative to which? There is a K-linear Florentian connection on satisfying the horizontal condition, the horizontal condition that is compatible with a metric. And furthermore, it is also a curvature space. A curvature space is a differential triangle, also a group augmented with a hierarchy of of this, by which you can define the extension of the convection, then the curvature, so you have a curvature. Okay? Supporting a known Ritchie-Scalar answer. The Ritchie-Scalar operator satisfies the vacuum Einstein equations. It's called an Einstein-Lorenz curvature space. As a brief remark, the vacuum Einstein equations can be deduced from the variation of an A-valued Einstein-Hubert action function, and the affine space of the A-connections on E, which I will not derive now, but people. And then one can explore the structure of modular space, excuse me, of equivalent connections, and then 1.2 differential geometry of this modular space. This is in the coming book now by Marius Gates series from an absolute differential geometric point of view. I don't want to dwell on that.

20:00 This omega star e, the locally finite e, is that possible for Poincaré-type lemma is true in a sense that it's always true? Yes, indeed. Poincaré is true. You mean the exactness of the neural complex? Exactness, as she is. Yes, yes. Okay. Another question is this. For the global case, triadula is like a fine type thing? But well, let me ask this question. For the omega i e, the cohomology group can vanish for if you choose the space nicely enough, like a Stein or a fine type kind of space? Can you specify the space like that? I don't know. I don't know. So Pankai is good? Yes, the Poincare conjecture, I mean, it holds by nature, by virtue of the nature of the space that you, that you have, you have a manifold looking. So, excuse me, one more thing, so it's, it's a, the Duramco homology is actually isomorphic to the, the first one, D, the kernel of D. Indeed so. Yes. Yes, you can prove the exactness of the RAN complex and you can, using AADG, actually you can calculate it. Sorry? I would like to talk about how the AADG fair similarities. I would like to begin with a sort of aphorism type of thing, but first, when one speaks of a manifold, one doesn't speak of anything else, but the manifold is the algebra of school functions. I would actually, I would like not to identify it, but I would like to write something like two directions are mindful well okay technically speaking the expression about this there the left to right arrow pertains to the covering to

22:30 the covering of the points of an otherwise structure this point set by by C-Infinity charts, while the opposite has to do something to it. We can find duality. Do you recall with the mind of the last spectrum of the topological algebra? C-Infinity, yeah. We've heard me the topological. It's about C-Infinity, correct. Pure algebraic. A corollary algorithm is that classical differential geometry and its manifold monopoly in some sense, all attempts to implement differential geometric ideas, general relativity, have involved or eminent in one way or another from the a priori assumption of a base differential space time manifold. classical differential geometries or equivalent by the opportunity to see yes but that the employment I'm going to be in CDG or CPMTM which of course are the labels and the smooth functions with the coordinates of the points of the manifold. In this sense, going via C-infinity M, C-infinity M mediates our differential geometric constructions. And this is more Cartesian, the Cartesian point of view, again, space intervenes in our calculations. And it gives us smoothness. The differential structure, we get it from this algebra. Now, this is an interesting, one of my favorite poems is Ihtakal by Kabafi, he says

25:00 there are lestrigonians and cyclopes, angry for sight, and such obstacles you will never encounter in your way as long as you do not carry them in your soul, as long as your soul does not raise them before you. Okay. You obviously never worked on pumping rubbish. No, the end problem is a bit. Yes, I don't think that Gavafi worked. I don't think that Gavafi worked. Yes, yes, so. Yes, okay. The basic axiom in GR is that space-time is a four-dimensional differential manifold, what physics is referred to as the space-time continuum. By this assumption, Einstein wanted to secure infinitesimal locality for the law of gravity, namely, that the laws of the gravitational field are a differential equation. Soon, however, he realized that the relativistic field theory of gravity based on the space-time continuum was assembled by similarities and associated on physical activities for the basic fields that we use as the smooth method or the smooth limit. singularities that he regarded as physically unacceptable a continuous field theory on the space-time continuum is not yet completely determined by the system of field equation should one admit the appearance of singularities it is my opinion but singularities must be excluded it does not seem reasonable to me to introduce into a continuum theory points or lines etc for which the field equations do not hold while he also believed that similarities in the classical theory also stood in the way towards understanding the quantum structure of matter is it conceivable that a field theory permits one to understand the atomistic and quantum structure of reality almost everybody will answer with no and at the present time nobody knows anything reliable about it. So, but we cannot judge in what manner and how strongly the exclusion of similarities reduces the manifold of solutions. We do not possess any method at all to derive systematically solutions that are clear of similarities. In fact, the existence of similarities in general relativity especially after the

27:30 celebrated years of Tolkien and Penrose is probably regarded, actually, as an unavoidable intrinsic fault or pitfall of the theory. One that is, again in a coherent sense, that limits the boundary of the capability and validity of the theory as it were from within. As it points towards a new theory, revising and possibly extending the original one, of course, to some physicists, the quantum theory of gravity will resolve the problem of singularities. It's very interesting in that debate between Penrose and Hawking, that Hawking asked him do you think that a quantum theory of gravity will do away with symbol acts? And Penrose was quite careful and he says, no, I don't think it can be quite like that. a quantum theory of gravity should be more clear about talking what is about telling us what is a singularity in the classical theory because we do not have as I said we do not have a very good definition of similarities. This auto catastrophic Popperian perception of similarities is very well said in this What about shortcomings of general relativity? Perfectly reasonable conditions at one time may lead to fill singularities at another, but looks as if general relativity carries with its own conceptual value the seeds of its own destruction. And the same is aired by Aztecar. So, um... And physical infinity, because, Aztecar says, even predicts general relativity, the occurrence of singular configurations in which physical point is becoming and physical infinities aside for a moment and even more graphically physically describes singularities as lucky in the space-time continue when the low gravity does not hold or ultimately breaks down now how can we understand what breaks down what I think my wild guess is that the Einstein was annoyed by singularities because he was reflecting the following way he said did didn't I discover a law?

30:00 What is the meaning of the word law if that law breaks down? If there is exactly, as he said, points, lines, surfaces, where this law does not hold? What kind of law will be, have I discovered a law after all? If that law, if there are some certain configurations that this law appears to be problematic, in what way? You know law, with the word law, that accompanies the universality, and this partial holding and partial not holding situation questions the very notion of law. What is the law? And in what sense? This is a credo, I mean, I would like to ask anybody here, do you believe that nature has a singularity? Probably. It is rather than ascribe singularities to, say, to the physical theory, as if, you know, the physical theory of course is defined by the dynamics which are here differential equations you know they have singularities and they break down rather than ascribe singularities to physical theory it is perhaps wiser to look for the component in the very mathematical framework within which these physical theories are conceptually formulated and technically expressed, we are more inclined to look for the pathogenic gene, GR's conceptual and technical genome, and come up with the following methods, which again in a way follows from the previous two when identified the smooth functions with a manifold. All singularities in GR are due to our primitive conception and assumption of space as a differential manifold. Effectively, they are due to our assumption of this, the usual algebra of spin functions. In one way or another, singularities are built in the sink, and rather than regard them as shortcomings of a physical theory, as if a law breaks down, ascribing to the very way we employ differential geometric ideas to model those physical laws.

32:30 Of course, granted that you would like some notion of locality to be obeyed in nature, so in some sense you would have that primitive plan to model physical laws by differential equations, and since perhaps the manifold is so far the only way we know how to differentiate, then this manifold monopoly, in some sense, at the initial, like in Cavafi's verses, it's exactly our assumption of a manifold, a modeling space-time after a continuous, a smooth continuum, that, from the start, brings in the singularities of the pathologies. It is the way we affect our differential geometric construction via M, via the algebra of smooth functions. That is at the root of the problem. Exactly. They are intrinsic faults of the mathematical framework that we use in the first place to model the theory and the laws defining. I will be very quick in the, in my last, ADG has been applied to, has been applied to similarities in GR. Yes, another rhetorical question was such an almost religious commitment to the differential manifolds. Again, I don't want to be simplistic or even smug or anything, but is it simply because CDG called this analysis the only theory method and technique which was just for doing And if you look at the traditional approach to singularities, you'll notice the following trend. First, okay, we have the original work of Finkelstein and Kruskal. And at the root of singularities, people have tried to pinpoint singularities. What are singularities? give a precise mathematical definition.

35:00 They have failed, and they tried, rather, to define singularities by elimination. At the root of any attempt to define singularities, the notion of analytic continuation, or a smooth continuation of the space-time manifold past a singular point. And here we see, we get that by elimination. Arguing is from the prologue of the book by Chris Clark, The Analyst of Spacetime Singularity. He says, the central aim of this book is the development or results of the news needed to determine when it is possible to extend the space-time through an apparent singularity, meaning a boundary point associated with some sort of incompleteness in the space-time. Having achieved this, you shall obtain a characterization of a genuine singularity as a place where such an extension is not possible. Thus, we are proceeding by elimination. Rather than embarking on a direct study of genuine singularities, or trying to give a direct definition of true singularities, we study extensions in order to rule out all apparent singularities that are not genuine. So, another way to say, when you encounter a singularity in a classical theory, you try all your analytic techniques to extend your space and pass the singularity. If such an extension is possible, then you say that the singular point was just a coordinate artifact. In fact, namely, you changed your algebra of analytic functions or Swing function in a way that actually Finkelstein and Kruskal, then subsequently Kruskal, changed the coordinates to show that the exterior similarity of the Schwarzschild solution is actually not a similarity, but it's just a coordinate particle. But yet, the theory, calculus, breaks down when you're trying to deal with the interiors forges, so singularity, namely when you try to describe the gravitational field right at the point mass source. Okay. And here we see again the mindful conservative attitude is we're going to try all our analytic techniques, and when they fail, then we declare that point point a true singularity. So, again, singularities are breakdown points that limit the domain

37:30 of applicability and validity of calculus on differential geometries. I would like to talk to you more about the actual applications of ADG to smooth singularities that's actually currently my work with Manius. But I'd rather, of course we have no time left, I would like to close with some remarks about questioning very basic concepts in physics, which I think is suitable for our maverick or iconoclastic gathering today. Yes, yes, it is Einstein in his autobiographical notes, he made some remarks about questioning the basic concepts in particular about the space-time continuum, and here's what he said. Concepts such as the space-time continuum which have proved useful for ordering things usually assume so great an authority over us that we forget their terrestrial origin and accept them as an outer move facts. They then become labeled as conceptual necessities, a priori situations, etc. The road of scientific progress is frequently blocked for long periods by such errors. It is therefore not just an idle game to exercise our ability to analyze familiar concepts and to demonstrate the conditions in which their justifications and usefulness depend on the way in which these develop little by little, okay? And very recently, the other day, I read from the Tufts paper, and here is to satisfy some, perhaps some philosophers that they are here. Tufts states clearly that the problems of quantum gravity are much more than purely technical ones. They touch upon very essential philosophical and And these are some of the people that I would like to thank for helping clarify some things. Thank you.

40:00 And I just realized that I forgot to tell you. Oh, it's okay. Any questions? I always ask the same question, forgive me, but it's an important question. Do you or Malia, or has anybody else worked out any simple actual examples of this structure, showing it to work? Yes, we are currently... Will you actually solve the Einstein equation? Not for the case of singularities, which is very complicated, but some simple final model, say, for space-time, you actually see all the actions are satisfied and that you get out things from Michael's, say, solutions to science and science. But if you can see that done, it would enormously enhance one's understanding of the significance of this. Is that possible? Has it been done? Abstract differential geometry, Chris, has been applied? No. The direct answer to this is no, so far. In this paper that we are working now, we are not giving solutions to Einstein equations, that some misconceptions having to do with the interior singularity of Schwarzschild can be resolved by ADG. ADG has been applied to singularity theory of the most robust kind. For instance, you can use a a sheaf, very singular functions, even distributional, distributional characters, such as is an algebra, differential algebra of generalized functions, goes under the name, is called Rosinger distributions, and Rosinger distributions, for instance, have embedded in the Colombo distribution. Yeah, these are all ideas that actually are based on manifolds, after all, in the case I'm sure you are improving the notion of singularity, but it's certainly coming to the manifold. I suppose you really took the idea seriously as a alternative models of space in general. The reason I ask this is my question about this assumption that Leibniz will satisfy because even the simple difference operation on a finite lattice does not in fact satisfy Leibniz at all.

42:30 So it's not at all obvious to me, actually, that there are, or how it will construct, examples of this particular academic scheme, which all we actually really are satisfied, which, for example, are, say, discrete models of space-time. I can understand a very powerful way of extending continuum notions of space-time, as you do with theory of singularities or distributional things and so on. But suppose you really want to take the attitude that they're returning from others of space-time. Can everyone do that, or is there something fundamentally wrong by the assumption of the light in the beginning? If one has a concrete example and you know it works, of course. I would rather counter this. As far as I understand it, ADG is not a question whether it is a continuum or a discrete. written, really. The nature of the underlying space contributes nothing. No, I appreciate that, Yanis, but the thing is, if you want to construct concrete examples, like in the physics point of view, obviously it would be to take discrete numbers of space, all I'm saying is, on the face of it, it would be clear that this, in fact, wouldn't work. I'm just wondering what other type of simple model is. The simple model is not a very complicated model. It's actually a very complicated mathematical structure. Is that a very simple model that one can see the thing working? The simplest that we have worked out is applying the ABG technology to, you know, the finitary substitutes of reference working, you know, very basic constructions on spaces such as, zero pose, it's already Alexandrov Nerve's associated with coverings. This is an application of ADG so far to discrete spaces in that sense. No, there's a limited application so far and results but in this direction that you are asking Chris you had a question I think no

45:00 If I'm understood you right, your toolbox of abstract differential geometry is meant to be sort of plugged into any kind of physics theory that was being done before, like general relativity and three years of singularities. In other words, generalization is just a sort of test case, because if I'm writing that, you can practice on something simpler, like, because you know how even Newtonian mechanics have recently been discovered to have these bizarre, special cases, you know, Newtonian guys have fundamentalists, you can have these very contrived cases where all of us have to run off to infinity and find that in time, and then you're like wondering what's the universe like after that, is it just empty, I mean, you know, so these are a sort of singularity, so you could maybe plug in your form of calculus to boring old Newtonian mechanics and see if it makes the singularities go away. You could practice on a simpler theory than generativity. If I'm right, it's meant to be a universal singularity-destroying meta-theory. Is that right? Is that what this sort of meant to be? Could be. Because that means you could practice on a simpler underlying. It's just a thought. Yeah, for sure. The entrepreneurs are always required to be commuted. If I spoke about possible relations with non-commutative geometry, exactly one could, we have discussed with Freddie, I mean, one could use a structure sheath non-commutative algebra. Yes, but in ADG, the structure sheath is of commutative unit . But a non-commutative version of ADG would have to start from something in a non-commutative differential trial. Yes, and I would have a problem even with the quaternions. The differentiation is not linear anymore and things go completely wrong. So maybe it's the wrong idea anyway to have differentials. I don't know. But would there be a physical motivation for making them not committed? There'd be a physical, very good question. I don't know how, look, again, I don't want to, it is, it is okay if you are a maverick, but you do not say so.

47:30 Look, I don't know how well physically motivated is, say, non-commutative geometry. For instance, I remember some time ago David Finkelstein told me, ah, do you think that non-commutativity is a necessary characteristic of quantumness? Do you think? Non-commutativity is characteristic? Well, why non-commutative? Non-commutativity, okay, non-commutative features appeared with, you know, Heisenberg's analysis. Is non-commutativity associated, I mean, if and only if, quantumness? but it's more logical I mean it's much more restrictive to assume commutativity exactly non-commutativity it's put wrongly it's negating something that really ought to be assumed it's actually just the general non-commutative algebra is just general algebra look at these I thought that's not general physics but it is general algebra Commutative means we can do it. Yeah. Yeah. Sometimes. Well, function composition is not, you know, map composition is non-commutative, it's the general, expect categories, you know, to be commutative, I mean, they're extremely special, commutative categories are an extremely special case. Should we break for tea, because commutative tea. Thank you.

50:00 I'm very pleased to be here. All right, well, what I plan to do, I'm not much really adept at using transparencies, but I do have some here, and I will put them up when it's necessary to go into equations and so on. But I have been masquerading as a philosopher, at least officially, for the last 14 years, and so philosophers, of course, you know, tend to read their papers. In fact, may sit down, because I won't, and I'll spray you that particular type of performance. But anyway, I'll put the transparencies up when it will be easier to refer to the equations and to some, as Ioannis put it, aphorisms, sometimes called slogans. At least the category theorists like to use those terms. All right, well. Well, so let me begin. Traditionally, there have been two methods of deriving the theorems of geometry, the analytic and the synthetic. Well, the analytical method is based on the introduction of numerical coordinates and so on the theory of real numbers, and, of course, on some kind of numerical presentation of the idea of continuity. The idea behind the synthetic approach to geometry is to ferries the subject with a purely geometric foundation, in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Bizarre in the 17th century and revived and developed by Carnot, Paul Slay, Steiner, and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and more generally in mathematical analysis, to be introduced to the geometry, leading in particular to differential geometry, which, of course, developed independently, at least initially, of synthetic geometry. The term differential geometry, incidentally, as I'm sure some of you know, was introduced

52:30 in 1894 by the Italian geometer Luigi Bianchi, and the, of course, much work, early work in differential geometry, or work in the development of differential geometry in the 19th century was done in Italy by Levi Civita and others, and of course this was the, it was the work of Levi-Civitat that Einstein used in developing general relativity, although, of course, the original idea of that goes back, of course, to Riemann in the mid-19th century. Anyway, given the fact that differential geometry does seem to rest very strongly on the, or rely very heavily on the use of the calculus, and this This is a point that has come up, of course, in a previous talk. And indeed, there's going to be a bit of an overlap between what I have to say and what the honors was saying, but that should be all right. So if that's the case, the idea of a synthetic differential geometry seems elusive. How can differential geometry be placed on what might be called a purely geometric or axiomatic foundation when the apparatus of the calculus seems inextricably involved? to hear Joanna speaking on abstract differential geometry, something I really never, this is the first time I've heard of it. So the only examples I had before this, as I say, this was to my previous knowledge, there have been two attempts to develop a synthetic differential geometry. The first was actually initiated by the mathematician Herbert Busaman 40s, building on work of Finnsler, who was known on other things for working in logic. He's a very versatile mathematician of the 1920s. Anyway, here, in Boozman's approach, the idea was to build a differential geometry that, in its author's words, requires no derivatives. The basic objects in Guzman's approach are not differentiable nanofolds, but metric spaces of a certain type, in which the notion of a geodesic can be defined in an intrinsic manner. Now, I'm not going to, I don't know very much about this approach. I did sort of investigate it briefly when I was sort of interested in the history of

55:00 the subject, but I won't have anything more to say about that here. The second approach, that with which I shall be concerned, was originally proposed in the 1960s by Bill Overe, who was in fact striving to fashion a decisive axiomatic framework for continuum mechanics. I don't know how well it's remarkable that at least part of the origin of topless theory lies in the fact that in the 1960s, Lovere really was looking for some way of axiomatizing continuum mechanics, which is what he had originally been interested in as a student. Anyway, his ideas have led to what I will call synthetic differential geometry. SDGs, also known sometimes as smooth infinitesimal analysis, SDGs formulated within category theory, of course, as we've been here, a branch of mathematics created in 1945, officially by Eilenberg and MacLean, and of course category theory deals with mathematical form and structure in its most general manifestations, at least as far as we now know. As in biology, the viewpoint of category theory is that mathematical structures fall naturally into species, or categories, as they're called there. But a category is specified not just by identifying the species of mathematical structure with cognitive objects, one must also specify the transformations or maps linking these objects. Thus, we have familiar examples of categories. Category of sets, for instance, where the objects are all sets and maps, all functions between sets. Category of groups, groups and group homomorphisms as groups as objects and group homomorphisms as arrows or maps. Topological spaces, with objects all topological spaces and maps continuous functions. And the category that we shall be interested in here, the category of manifolds, smooth manifolds, with objects all Hausdorff, second countable if you want to put these conditions on, smooth manifolds, and maps all smooth functions, so C infinity functions between them. Now differential geometry lives, in some sense, in the category of manifolds in man, and one

57:30 might suppose in that case, one might then suppose that since it's the category of manifolds that in some senses provides the framework for doing differential geometry, that in formulating a synthetic differential geometry, the category theorist's goal might be to find an axiomatic description of man itself. It would be natural to suppose that it's happened in other situations, that you identify a structure, and then you look for some way of characterizing it axiomatically. However, and this has also been pointed out by, well, some of these points have been made also by Ioannis. In fact, the category of manifolds, it has several deficiencies. In fact, it has some deficiencies. Ioannis mentioned, I can't remember, five or six of them. There are others, which I'm going to concentrate on here. Man has a couple of deficiencies which make it unsuitable as the object of axiomatic description. Actually, it's interesting. The whole problem of axiomatizing categories seems to be rather difficult. One tends not to axiomatize categories as such, but rather general features the categories possess. For example, after the axioms for categories were developed, the first, probably the first further axiomatic development, one of the first, was the development of the axioms for abelian categories. It's true, there's a kind of generalization of the category of abelian groups, but it was not characterizing the category of abelian groups as such. It was isolating the features between the category of abelian groups convenient for doing certain kinds of algebraic operations. OK, what are these two deficiencies? And perhaps I could venture to put this up now.

1:00:00 There we are. OK, so there are really two features that make it. There are several features that make it unsuitable. But from the point of view of axiomatic description, it has these two, what you might call, divisions. It lacks exponentials. It's not Cartesian closed. Space of all smooth maps from one manifold to another in general fails to be a manifold. Of course, this isn't the only category. Category of topological spaces has similar problems of making these things, trying to produce Cartesian-closed versions of these categories has been a certain amount of effort has been devoted to that. Anyway, even if it did, and this is really the crucial feature here, it also lacks infinitesimal objects. So in particular, there's no infinitesimal, what I like to call, incredible shrinking manifold. for those of you who have seen the old movie delta as I'll call it sometimes called D for which the tangent bundle in the usual sense of an arbitrary manifold M can be identified as the exponential manifold M to the delta of all infinitesimal paths in M of course it is intuitively the tangent bundle of the manifold It's just, of course, that in classical mathematics, one can't provide a rigorous definition of this in its original terms. So you can remark that it's really this deficiency that makes the construction of the tangent bundle and the category manifold something of a headache. Of course, you get used to it. If you read Spivak's book, for example, he has a very nice account of the construction of tangent bundle. He gives several equivalent definitions and then tries to relate it, of course, to the intuitive definition, which is where it came from. And indeed, the early workers in the Verlis and Cartin did use, of course, something like this idea as a sort of intuitive notion working in their own work. Well, Lovere's idea was to enlarge the category of anathons

1:02:30 to a category which we'll call S. Category of so-called smooth spaces, or smooth category, which avoids these two deficiencies. Of course, admittedly, perhaps it's stretching a point here, stretching a point. I know, I know, it's quite unintentional. I really don't, no punks to it, but what really regards this as a deficiency, of course it's a deficiency in some intuitive sense, But then classical mathematicians had gotten used to the fact that they'd thrown out infinitesimals in the 19th century, even though mathematicians had long since ceased to say that we really need them. They always, infinitesimals are always a bit of an embarrassment anyway. Of course, they're back. They're back with a vengeance now. But for a long time, they were regarded as somewhat dubious. So right back to Newton, trying to find some way of getting rid of them. But anyway, so whether you call this a deficiency or not, classical mathematicians would just say this is an impossibility. I mean, they wouldn't, you know, I like trying to do six impossible things before breakfast. I mean, you can't actually do them. So it's hardly a deficiency to demand that the fact you can't actually do them can hardly in regard to the deficiency, but never mind. I'll include that as a deficiency. So his idea was to enlarge this category, the category of smooth spaces, to voice these two deficiencies, admits a simple axiomatic description, and at the same time is sufficiently similar to the category of sets, which, of course, is the category which has been used for some time, at least implicitly, as the kind of framework within which the geometric construction of a lot of other mathematics, most mathematics, has actually been done for mathematical construction. It does have a certain simplicity. It's arguable, actually. We can talk about that later. But anyway, speaking myself as a sort of ex-set theorist, it's sort of impossible to get rid of sets altogether. Anyway, or indeed set theorists, it's originally similar to set for mathematical construction

1:05:00 and calculation to take place in the, what is this answer, the familiar way. In other words, by the familiar way, I really mean what the advantages of set theory were that you don't really have to think about it very much. It did have certain advantages. Half the mathematicians, you ask them, well, ask them for some axioms of set theory, they Why would they bother? They just use the notion so implicitly and worry about it. And you have to say it, except that he did have that advantage. Of course, it also conceded lots of difficulties, as we know historically. But anyway. Now, the essential features of a smooth category are these. Enlarging man to this thing, S, no new maps between manifolds are added. Now, the point is that the maps in the category of manifolds, of course, are all smooth. There aren't any discontinuous ones in particular. In particular, there's no blip function. Now, when you enlarge, yes, all maps in S between objects of man are smooth. So, in other words, you don't throw in any new things that are dependent, if you like, on the discreteness of the underlying sets. Now, of course, this is not the case when managed and lodged a set. I mean, usually, if you think of the category of manifolds, it gets embedded as a subcategory of the category of sets, right? Where you have all kinds of, not only do you have lots of discrete objects, you also have, of course, arbitrary maps between those discrete objects, So this is going to be a different enlargement, because the traditional enlargement, I mean putting it succinctly, it entails enlarging man's set, whereas here we want to enlarge Now, of course, when you go from here to here, you satisfy this, Cartesian quotas, but of essentially. There are various reasons for it, one reason being that the existence of infinitesimal manifold that satisfies the appropriate axioms for this entails the failure of the law of excluded

1:07:30 middle by a rather simple argument, and so it has no models in the category of sense. But anyway, that actually is an incidental feature for us here. Okay. The second Additional features, then, are Cartesian closeness. Of course, what I already mentioned, it contains has products and exponentials of its objects in the usual sense, essentially in the same sense as the category of sense has. And this is going to be the decisive principle that gives us the existence of the infinitesimal metaphor. What I've called myself in my little book, the principle on the subject of infinitesimals, what I call the principle of microstraiteness, this is also known essentially as the technical Mark-Lauvier axiom, geometrically it means the following, that if we take the real line or the line, the object in the cutter that is the counterpart of the real line in the usual sense, a smooth line, a fine line, and we consider it as an object of man, and we can also consider it as an object of S by, since we're thinking of this man being a subcategory Then there's a non-degenerate segment of R around 0 which remains straight and unbroken under any map in S. In other words, delta is subject in S to Euclidean motions only. This is what we mean intuitively by the idea of a geometric infinitesimal or, if you like, as it's sometimes, a generic tangent vector. The idea being that if one takes a curve and one looks at a location on it, then there's a small portion of the curve, you see here, right, which I've lost mine. It isn't very good there. But when one has a map, say, of the line into any manifold at all, then although this thing becomes a curve, this local part of it remains straight. In other words, this object is subject to Euclidean motions only, or you might say that

1:10:00 it actually has, but it's not degenerate. It has the feature of possessing direction and location, but not extension in some sense. Of course, classically, this is only possible, but it's consistent in intuitionistic logic, which I'll cut to. Anyway, intuitively, this was the idea of a generic tangent vector. It's an object that measures, that indicates direction, but nothing more. Delta is an object of S. Delta is an object of S. But it maps into R. No, it's actually a sub-object of R. It's a part of R. A sub-object in the S sense. Yes, yes. In other words, if we think of R as, in the usual classical sense, okay, This thing, when it actually becomes, of course, I'm regarding man as a subcategory of S just for convenience, but what actually happens, of course, is that when you have an embedding of man into S, then what happens to R when you embed it is that it picks up this part that wasn't already in man. There's a monic mapping that goes into R. Actually, in the sense that, yes, as a category, yes, but we don't need to think of it that way because we can use the language of ordinary inclusion and so on in the, that's technically quite right. There actually is an arrow from this into R in S, but I'm going to use the usual language of inclusion and so on because in tapas theory, this thing is what's called top us a category and there's a the ordinary language of set theory can essentially be used there so we'll just use it as if it really is a part of r yes yes yes of course because you could do it by translation of course because by translation does it happen around every point in R, as we understand it, in A, or just that I know? No, in S, yes, yes. Every point, for every location, there's a, right, there's a part. In fact, for any curve, there's a part around any location on it that is locally straight, yes. Okay, so even around one of these points that doesn't exist in classical R. Oh, yes. Well, actually, as it happens, yes, now that's an interesting point.

1:12:30 But actually, I'm not really going to get involved in that. But in fact, in most models of synthetic differential geometry, most versions of this, no really new points are added in the technical. It's not really like non-standard analysis in that sense. what happens is that the that the logic changes and the you know the the the around each point kind of gets spread out but not to new points well this is the thing but because in because simply of the fact that one doesn't have this So, you see, because you don't have this. You have to talk about the... Decidability of identity. No, I'm telling you, it's okay from up here. You don't have this. You don't have decidability of identity. That's right. And you can't convert this, you see, into saying that they're, sorry? The boardwalks in front, along the front wall. Oh, I see. This is a, great testing of my skills. Multi-medium. Well, what, what, what, what, multi-medium stage management, nil. Anyway, because one doesn't have, well, one doesn't have this. You can't convert this, for example, into the assertion that there are two points. there is an x and there are y such that both x is equal to y and x is not equal to y well the thing is you can't convert this you see when you research you have a point that corresponds essentially to introducing an existential quantifier so if you wanted to say there is a point x that has such and such a property you'd use this but the point is that in classical intuitionistic logic in the internal logic here you can't assertions of this kind, which are universal assertions, well they gave a universal assertion, into existential ones. So in other words, although it's not the case, for example, that the infinite test, it's not the case, for example, so if we take zero, there are going to be put that you can't prove, right, well this

1:15:00 This assertion is correct, but you can't then say, ah, well, there is a point which is both not equal and unequal to zero, because that would be simply a contradiction. So in that sense, no new points can edit, but generalizations about variables ranging over, let us say, i, those change. Even though you're not going to be able to assert explicitly that there is a point, right, for example, which satisfies the condition that it's neither equal to zero or, I don't know, I'm sorry, both even, you know. Anyway, let me come back to that. This is not, here's another great way of putting it. What happens is, the non-standard analysis, I don't know if you're familiar with it, non-standard analysis is essentially an enlargement. If you did this in non-standard, well, you know, there is a version of this in non-standard, not of this, but you could do theory of manifolds in non-standard analysis which uses classical logic and in which you actually get new points and when you have a manifold and you enlarge it and you use your Robinsonian enlargement or something like that what starts a point and you get out this a monad, you know, points around it which are genuinely new points that really weren't there this is not what happens in general in this case what happens here is that have an actual point, right, in the usual classical model. And what happens is that it gets smeared out with a penumbra, right, of something, which aren't actual points, right? But nevertheless, the original point is still there. So this is put roughly. In non-standard analysis, the monad of a point actually consists of a bunch of new points. In smooth infinitesimal analysis, it does it. It actually, a point gets stretched out into a generic tangent vector. It's a different, the construction is quite different. Okay. Anyway. Just to clarify, in Robinson's non-standard way of doing it, would it in fact not quite true that you have microstreatness? Because although you have the person in the original real sort of called microstreatness, nevertheless, ironically, a Robinson- Quite right. No, you can't have microstrainers because it's in the path of classical logic.

1:17:30 Well, you can see why. I mean, I don't know why Leibniz, one would have thought Leibniz would have come up with it. He declared nature and non-fac insult us, right? Nature doesn't make jobs. He and lots of people thought that the universe, or the way of thinking about nature, should be that it was continuous. That's interesting how that shifted in the last decades. But anyway, and the way there are two, of course. But if you just consider the, you know, the blip function, that's one at zero, zero everywhere else, that's discontinuous. Quite incompatible, of course, with the existence of this. with this, because every function is clearly going to have to be continuous, at least in the intuitive sense, if one has micro-straiteness in this sense. I mean, it's immediate consequence. And therefore, even if all functions are actually just continuous, never mind that you don't need the full strength of that principle, that blocks the existence of this very simple, the simplest discontinuous function. And what causes this to be a function? well namely this to be precise what enables the blip function to be a function defined on the whole of the real line of course it's equivalent to this principle here in other words the excluded middle so if every function is continuous immediately Leibniz didn't I mean it surprises me the way I've been of it the connection of course connection between logic and mathematics was I think in a way rather superficial before this sort of work nobody would really see that going before the before intuitionistic logic I think this developed connection seems rather superficial or comparatively superficial despite the efforts of Boole in Leibniz too because here you can see there's a connection immediate and a lot of excluded middle and universal Anyway, okay. Now, in fact, you can easily see, I'm going to run over time if I don't get on with it. In fact, I can use my, we can look at this curve and determine exactly what the,

1:20:00 what is, well, I don't even need it. If we just take this, let me go back to my friends' parents. I don't seem to have a diagram for it. But anyway, this is all right. In fact, you can see easily that if you take the parabola y equals x squared and then look at the intersection, if you like, of the curve with the x-axis, you could call this thing is the locally straight part and in fact you have a this is what it is actually this part of r around zero anyway can be identified with the set of points or the part of r that consists of what are called the dual squared infinitesimals the set of part of r X is for which x squared is zero. And that gives you the basis for doing... This is, of course, the idea that infinitesimals are one version of infinitesimals, an old version, which goes back quite a long way and still used sort of by engineers, is that an infinitesimal or a differential is something in two square you could ignore. You can do it routinely, except it's rigorous here. All right. Anyway. to prove that all the x squared equals 0 on x squared equals 0? No, no. None of those principles will happen. And of course, it's also compatible to have, you know, to higher order infinitesimals. Things whose cubes are 0, but you can't prove their squares are 0. And so anyway. Now, it's quite easy to see that, let me just, since I've got it all down here, as I say, my impulse to start scribbling on the blackboard is going to have to be restrained. So there it is. I did have a diagram. I've forgotten it. And there's the definition. If we use a letter epsilon, time-honored symbol, for a small quantity, I mean, we use it for a nil-square infinitesimal, what I call, what we could call a microquantity.

1:22:30 Then we can show the principle that we, the basic principle, and indeed, this is the, what I call the principle of microrefinance. This is really the version, the original version of Coughlin-Logier. But if we have any map from, this expresses the idea that every map on delta is affine, that delta can't actually be bent or broken, that for every map from delta to r, there's a, for some unique member of r, there is for all epsilon that it behaves affine. In other words, f of epsilon is equal to f of 0 plus b times epsilon for some unique b. So in other words, that just says, essentially, that if you have, let's say, delta, and you subject it to some kind of map, And the only thing that will happen to it is that it will get translated and rotated. That's all. There's no other effect. B is then the slope of the, and it's this sense. So in other words, so if we can, sorry, here's a diagram. there's what happens, right, so here we have a curve, a usual thing, y equals fx, and delta gets mapped to f0, which is up here, and then rotated by a certain amount, and that's all that happens to. So the principle of microfinance asserts that each map from here has a unique slope. This reduces, and I'm not going to do it here, you can read my little book on the subject if you're interested. This is how the differential calculus gets reduced to simple algebra. It's very beautiful doing the calculus. Now, more important for us here is that the principle of micro-refinance actually asserts, and there's a way of putting this in making this formulating the whole principle behind all this in this way, micro-refinance. If we take the space of maps from delta to r, right? that's r to the delta, then essentially it asserts also that if we look at the map, right, that goes from here to here, which assigns to each f the pair consisting of the value

1:25:00 of the function at zero and the second coordinate at the slope, then that asserts essentially that this thing is an isomorphism. It's actually an isomorphism. It can be turned into an isomorphism of rings by giving this r cross r ring structure called number structure, and you can formulate the Coqlovere axiom or the microfinance axiom to assert that it is this, in terms of this isomorphism. Now, of course, this already introduces one of the fundamental ideas. R to the delta, of course, is the tangent bundle of R. I mean, if the tangent bundle take any manifold, the idea was that the space of maps if we take a manifold or a space M then this thing should be the tangent bundle of M the space of maps from delta to M. Well, of course R of the delta is isomorphic to R cross R, but of course we know from classical the use of classical differential geometry that R cross is just the tangent of Buddle of R in the classical case. So we do have the right identification there. Now interestingly, so this is another interesting point, which has been emphasized lately by Bill O'Leary in some of his writings, which are interesting. You'll find some of Bill's writings are trifling impenetrable. It's better to hear him talk about it. nice account I think recently there's some things on the web and I've already talked about it but of course if you if you there's a sense in which Delta generates all the objects or can be considered to generate all the objects that we normally get in Euclidean I mean the usual sorts of manifolds Euclidean manifolds, anyway. Because if we take, it's easy to see, that if we take delta to the delta, the space of all maps from delta to itself, and we write this for the part that consists of just those that preserve zero, so essentially what is happening is that when it's considering delta and f of zero equals zero, so what's happening, it's just considering all the possible ways

1:27:30 in the sense of rotating, sort of like a speedometer needle, of rotating the delta. I mean, this is put intuitively, because you need already the idea of two-dimensional space, right, in order to be able to realize this idea of this, to present delta to the delta, right, sub-zero in this way. But nevertheless, it's, I think, helpful. And then it's rather easy to see that R is itself actually isomorphic to this. And again, you can see that intuitively because, of course, the ways in which this thing can be rotated corresponds to slopes. And the slopes are, of course, just real numbers. So here's a way R, you can generate R this way from the infinitesimal, from the generic tangent vector. You can get all Euclidean spaces this way. There's a sense in which the whole of this, the familiar part of this world, what I like sometimes to call the smooth world, is generated entirely by category natural operations on an infinitesimal object. So we can think in some sense that this is really a, I like to think of this as a realization. Einstein may have been somewhat unhappy. This was interesting quotations that Uranus came up with, because I've been looking to further my propaganda for the continuous over the discrete, because I didn't find any joy in what I heard earlier. It was Herman Weill, of course, who championed in many of his writings the idea of the infinitesimal, that physics is infinitesimal physics. I mean, he says so in several places in space-time. There's a very nice sense of which, of course, one has to say that this is, of course, an idealization. And the interest for physicists, of course, is whether it's really, let's say, a correct idealization. It may be correct for the notion of the idealization, but whether it actually corresponds in some way to the future of nature. The idea that the infinitesimal geometry and smoothness and so on, it was thought indeed to correspond to an act, it is a natural idealization, would seem to work very well, as we all know. In the past, quantum physics does pose serious issues for the extent to which this type of idealization will still work.

1:30:00 But I still think it's got a fair way to go. Anyway, so that's one thought. Now, for any space in S, we do take the tangent bundle to be this exponential. OK, so elements of n to delta are then the tangent vectors. Now, then, of course, the, so a tangent vector, right, and this is all, this is what makes the whole theory very, really, very nice. A tangent vector is just, and with base point, let's say, p, that should be a p here, not x, is just a map from delta to the m where the zero is taken to this base point p. So a tangent vector to P is a micropath in M with values in, with base point P. Now, I haven't given myself much time, haven't I? Oh, well. Important point to make here, then, is that if we identify each tangent vector with its image in M, then each tangent space to M, so the tangent space, then, of course, is the set of tangent vectors, point with that particular base point and in fact you can in some natural sense identify tangent spaces as actually lying in the manifold so in this sense each space in s is infinitesimally flat I'll come back to that point later because because there was a quotation that I wanted to to introduce. Anyway, so I'm going to have to cut this short to get to something more. John, I think that you think that you have 10 minutes. It looks like I... But no. I think I think you're right. That communication doesn't hold you. It's very interesting. That's okay. In fact, you think that you have it doesn't mean you haven't. Well, so how much time do I have? More than a quarter of an hour. Okay. Well, then I'll cut out some of this. Anyway, so you can then see that the definition, right, the compatibility, if you take the definition then, so r to the delta, and when you make the computation, right, just check the, it's a natural thing, you check the tangent bundle of r to the n, and you use the usual rules for exponentiation in a category,

1:32:30 and then you find the tangent bundle of r to the n and you go through it, turns out indeed to be r to the n cross r to the n, it's the usual computation now here is the real point, at least from the category theoretic point of view, standpoint of this, that if we take that if we take we assign we have a thing called the tangent bundle functor, which of course is the usual familiar thing, that assigns the tangent bundle to each manifold M or in space M, and we get a thing called the tangent bundle functor, and here's the definition. We don't need to worry about the action on arrows for a moment. But the whole point of synthetic differential geometry was to render the tangent bundle what's called representable. Tm becomes identified with the space of all maps from some fixed object, in this case delta to M. And of course, classically, this is impossible. And this enables you to simplify and make quite intuitive many constructions in differential geometry which I'm not going to I'm not going to go into here because I don't have time but for example well I'll just make one mention here if for example if we look at the idea of a vector field a vector field is a just a map or from from a from a manifold M or space M to its tangent bundle so what it does of course as usual thing, it assigns to each point of M, it assigns a little arrow, right, a direction on it, and that's the definition of a vector field. Well, it's rather easy to, and you can do the same thing for higher order cases, the tensor fields and so forth, and I'm not going to go into the details here, but the simplest case is just to look at the vector field case. And the Cartesian closeness, of course, of the category of spaces enables you to correlate vector fields with all kinds of other natural structures which are difficult to, or more difficult to, whose identification is more difficult to carry out of the classical case. So for example, let's look at this one. You use, there's a natural correspondence as the familiar Cartesian encloses what's called the usual junction.

1:35:00 Maps from, if you have three spaces, s, t, and u, then maps from s to t to the u correspond to functions of two variables, if you like, or functions on the product, s cross u to t. This says, of course, that the Cartesian product here is a right adjoint to the exponentiation functor. Sorry, left adjoint. In the usual function argument notation, then one has the familiar thing, the transpose, if you like, of a function, let's say, of two variables to a function-valued function. It's a familiar thing. We're all familiar with it in set theory. And of course, it all works here, too. Now, in that case, just to give you an example, bijective correspondence applied to vector fields here enables you to identify vector fields of what we call micro flows or I have an actual term for it flows that is so if you have a vector field from let's say xi that will correspond by by this adjunction to a map from m cross Delta them and these where you take a point x in the manifold you take some infinitesimal epsilon right and that tells you that thing is goes to m it tells you to move x to push it forward by some infinitesimal amount these are actually of course the bijective correspondence uh and then one could do the same thing with what i call but i don't want to go into this here i don't have time but by similar you can then you can then use cartesian closeness to or the adjunction to move the m back up here And then this gives you a correspondence between these things, microflows, and what you might call micropaths, that is, maps from the generic tangent vector of the infinitesimal around zero into space of maps from m to the n. Okay. Well, I, now the point, one point that I did want to make here that I, okay, yes, is the following. So anyway, enhanced vector fields, microflows, and micropads are all equivalent. Classically, of course, this is a metaphor. I mean, we all, I mean, anybody who works in this does know that in some sense they're the same.

1:37:30 But now, an interesting point here that I don't know, doesn't seem to have been made much of. The category of smooth spaces is what's called an elementary topos. And I'm sure that you're familiar with the idea. It's a Cartesian-closed category that has what's called a sub-object classifier, an object of truth value, so that it enables you to do logic in it or do mathematics in sort of more or less the usual way, except that the logic is intuitionistic in general. It's not that intuitionistic, actually. It's very interesting. I can't do the technical details here, but actually the deviation, the logical deviation between the usual models of synthetic differential geometry and the category of sense is much smaller than one might suppose at first. It's true that the law of excluded middle fails for quantified In other words, and you can make this quite precise, if you take an object, here's another way of putting it, if you take an object in S, then most objects, it is, are actually in decompositive, and certainly the ones correspond into connected spaces. A connected space, of course, is one that classically has the property that it can't be split into two open sets, disjoint open sets or disjoint non-trivial open sets or closed sets. That translates into the property of indecomposability, usually in these, mainly for the corresponding spaces in the category of smooth spaces are indecomposable in the sense that they can't be split into two non-trivial parts of any kind, open, closed, or whatever. that is telling you is essentially that the law of excluded middle fails for properties. In other words, in general you can't, that law is, in general if you have a property this

1:40:00 law will fail. However, when you write not in front of you, it's provably absurd or nearly... Well, it's just in the internal logic. It means yes, yes, it means that the assumption you serve this and you get a contradiction. Yeah, usually. Of course, that's also true in classical logic. Not that before. No, no, no, no, but when you put a negation in front of you. Okay. On the other essentially to in decomposability where P is the property of being let us say some part in some part of a connected space. On the other hand for propositions you have a law of excluded middle. If you have if you have just propositions right sentences then the law of excluded middle in most of the models, as has been verified by people working in the area, I mean, that, in fact, the law of excluded middle does hold, it's in the big, the models are what are called the bivalent or two-value. In other words, you have the law of excluded middle at the level of sentences, but it fails at the level of property. So in some sense, you're not, you like at the level of the points, the original classical points. In some sense, the logic hasn't changed. So it doesn't deviate. It deviates much less than one might suppose in the classical case. Although even this tiny deviation is enough to open doors to what appear to be these sort of miracles, like the representability of the tangent bundle factor. Now, one further thing here. This is this little thing here. With the appropriate choice of arrows, each of these things here, vector fields, microflows, and micro paths they form the objects of a further top us the top of differential structures if you like over the original object oh the objects in the original space and they're the first or first order differential structure so it's quite interesting that the smooth manifolds themselves turn out to be the objects of the tapas in a natural way. So do the first order differential structures. Now, as far as I know, and this has been noted,

1:42:30 it follows from a general feature, it's a general fact, consequence of general fact about tapas is that it is itself a tapas. But as far as I know, nobody has really sort of worked in that tapas of differential structures. In particular, nobody seems to have thought axioms, by told, there. And indeed, it is quite a natural place to work. It's often said, well, really the objects of interest come up again in Giannis' talk. I've heard this before from a remark, something like that from physicists before. And again, I think it's reported out long ago that in some sense, what is really interesting, let's say, in general relativity isn't the points. It's not the points of the manifolds. It's actually the differential structure structures. That one should really be working there. That's where the action is. So it would seem rather natural to me to think about working there. And indeed, in higher order structures, it turns out that there has been some work trying to establish that higher order differential structures also have this property. And I think that's what some work of Koch and Reyes. But further point, John, something very quick. So the representability of the tangent bundle functor operates at the level of predicates or properties, not... Oh no, what it means, yes, what it means is that the representability of the tangent bundle functor implies the failure of the law of excluded middle for property. In other words, because it gives you in decomposability. It says it's just like an example. Yes, yes, that's what's like. Weill had this idea, Weill and Brouwer. Weill talks about real continuous, like an exagerate. You can't chop it as if with an axe. Weill says that quite a few times in his writings. And the interesting thing is that this principle is realized here. And yet, at the level of propositions, you see, in some sense, classical logic has been retained. It's really interesting. But you can put the line between those. I mean, simply by talking about representability of the tangent. This is the connection, you know, I think a beautiful example, the connection between mathematics and logic, which was just not imaginable before, I think.

1:45:00 Anyway. Well, I'm not going to have time to talk about it. Well, I'll just briefly mention it, because it is very interesting. in some, there's a further rather remarkable fact about this object delta that in fact not only does the, you know, actually have a left adjoint, it also has a right adjoint exponentiation by delta. And this is a fact that was, nobody seemed to know what to do with this, although it It is a rather extraordinary non-classical feature, if you like, of these models. So, for example, what it's saying is that these maps, if you take R, for example, here, maps for here are differential forms. So if I take N to the delta to R, these are differential forms. And these correspond actually, right, to quantities on end, to some funny space which you could call r to the 1 over delta. It's a non-classical feature of these models that is yet, I think, is beginning to be exploited now. A rather remarkable fact that somehow, you know, essentially a vector, right, a function, a differential form which assigns a number, if you like, to a vector, right, can be converted into a function that acts only on the points, into some curious larger space, right? and this is one this opens up of course the general question about what the real relationship is between synthetic differential geometry and classical differential geometry it's not it's only given it can't be given by the usual adjunctions and limit processes Because these are always given by the, you know, you get the product and so on, and this thing is actually, this is a right adjunct, which is not present in the usual Cartesian closed categories. So it's not going to be expressible in terms of, in general, in terms of the usual operations that you mentioned.

1:47:30 For example, exponentiation of products or taking sub-objects and so on, or limits. It's entirely new. another feature is it happens it's not describable of the internal logic makes it tough for logicians like me to understand what's going on well, alright, well finally since I have a little time left, let me let me get to, oh I did want to make this one point, actually two points I may never even get to, I did have some remarks on space time but I'll never even get to the minimum. I did want to make one little point. So delta is sort of tiny. It's a very tiny object in some sense. It's very small. And there's another sense in which delta is tiny, which has led me to perhaps probably an idle speculation. And to do this, we have to look at order. In synthetic differential geometry, R carries an order relation. which differs in certain respects from its classical counterpart. It is the order relation, the usual order relation looked at in this setting, particularly fails to satisfy trichotomy. This law that for any points, for any x and y and r, either x is less than y or x equals y, y is less than x, of course, is actually refutable. Now, the order relation behaves more oddly on delta, epsilon and eta's variables ranging over delta, you can show that even though you can't prove that delta collapses to zero, in fact, that's another consequence of this. You can't prove that every one of these infinitesimals is actually equal to zero, even though you can't actually isolate one explicitly that isn't equal to zero. But nevertheless, it's true that this is also the case, that for every epsilon and eta, it's not the case that either but they're incomparable. It's not the case that epsilon is less than eta or eta is less than epsilon. So in particular, if we define equal to or less than relation by the negation of the usual definition, right, the negation of less than with variables reversed, it's true that for all epsilon and for all eta, epsilon is equal to or less than eta, and eta is equal to or less than epsilon,

1:50:00 but of course you can't then conclude that epsilon equals eta because the thing doesn't satisfy antisymmetry. Now, in particular, the members of delta are all simultaneously equal to or less than and equal to or greater than zero, but can't be shown to coincide with zero. Now, this suggests a speculation, which I, as I say, probably idle, but never mind. I'll give it to you anytime. Let me find the... Oh, here we are. In his recent and, I think, very nice book, Just Six Numbers, Martin Rees, I don't know if there's a familiar to everybody here, makes some remarks concerning the microstructure of space and time and the possibility of developing a theory of quantum gravity. In particular, he says the following, some theorists are more willing to speculate than others, but even the boldest acknowledge the Planck scales as an ultimate barrier. We cannot measure distances smaller than the Planck length, about 10 to the 19th times smaller than the proton. We cannot distinguish two events or even decide, and this is the point here, we cannot distinguish two events or even decide which came first when the time interval between them is less than the Planck time, about 10 to the minus 43 seconds. It strikes me that on Ries's account, Planck scales are very similar in that case in certain respects to delta, our infinitesimal object here. In particular, the sentence one, that's this thing here, above, seems to be an exact embodiment of the idea that we cannot decide if two events, so to speak, in delta, which came first. In fact, it makes the stronger assertion that actually neither comes first. Could delta provide a good model for Planck scales? Well, it's certainly small enough. It's the smallest infinitesimal that's been considered. It's very small. But if so, it would be remarkable, since because delta inhabits a domain in which everything is smooth and continuous, while Planck scales live in the quantum world, which if not outright discrete, it's hard to say, it's a total of a mixture, isn't it, of continuous and discrete, is certainly far from being continuous. If Planck scales could indeed be assimilated to micro-neighborhoods in smoothing for the testament, smooth synthetic differential geometry, This would suggest that the quantum micro-world, the Planck regime,

1:52:30 smaller, in Ries's words, and I quote, than atoms by just as much as atoms are smaller than stars, in a nice phrase, this world is not, like the world of atoms, discrete, but instead continuous, like the world of stars. Well, I think this would constantly be a considerable victory for the continuance that has long struggled with the discrete, but I leave that as a speculation. Now, finally, well, I'm not going to, I don't really have time to, I can show you later. I have this diagram as to what happens when you look at Lorentz's geometry in the elementary in SDG, but I'm just not going to have... Thank you.