Examples of toposes / Tutorial session & discussions FW Lawvere, Freyd, Johnstone, Tierney et al.

0:00 So I want to start with the very first exercise. We know that, in general, if we have an arbitrary counter in the law of categories, then if we look at the Ritsch-Kilkos, we have, what we call the central geometric orphan, where we have, first we have upper star, which is the obvious thing, And then we have the Eppler star, together these form the geometric morphism, but moreover there's Eppler tree. Over this, and calculated in various spirits and ways, you can think of Eppler tree as a kind of direct limit. So if you know how to compute direct limits, and put some action things together, you can think of it as a tensor product, or a decon extension, and of course all these things are... It's really the same, but each one is a different way of computing it. Of course, the real way to compute it in most examples is by guessing. In other words, you start from the general theorem that there exists a unique answer to, say, f lower star of x and so on. After thinking about the example for a while, you come to guess, and you verify that the guess works. Or maybe it doesn't work, but having tried to verify, you get a second guess. That does work. So that's the usual. So I think just to have practice in this as well as, so I want to consider as my D, this picture, two maps, P and I, the relation of P and I with the identity on this other object, and that implies then of course that

2:30 Composite in the other way. And of course, this group category is the Marie Dick moment, which consists only of the identity map on this not-meant-to-mean. Two or two of them, more or less, this entire group, still has the same appreciation, in fact, the same representation in every category you can think of on the task list. In fact, there's another way that this is often called the notion of split epimorphism. The representation of P becomes equipped with a given section. And so the term split epimorphism is sort of ambiguous. It means there exists a section and there's a given section. It's a huge difference, but if you read it as a given, then again, this is kind of the third way of describing myself. Now this, if you might call this, and you look at it in the face of me, Dean. Ah, Dean would do it. That would be good, that would be good. Okay, so, this particular topos. Now on the other hand, as C, I'm just going to take the relevant arrow categories. And so there are two, well actually there are three non-critical functors that are forgetting about E.

5:00 There's a functor which is really the name of P, so I'll use the same symbol, hoping there's no confusion. And if you say that morphism in any category is just a functor of two, then this internal picture is the internal picture of, So here we actually have two functors, and so multiplying two times three we get six. So there are actually six functors that are produced here. You can find two more in PE as well. So we have six functors. The two inverse image punctures are just the obvious thing, they just amount to what sounds like a tautology, and if you have a diagram like this in the category U, P is a map and U, and I is a map and U, so those are such tautologies. Wait, all right. Sapinski-Mormoy is not the same as the Clichy's on that. Sapinski-Mormoy? I love it. Didn't you say there's a non-trivial sub-object of one? That's it. Well, if you look at m-sets for a non-right end, you're not going to get a non-trivial sub-object. Well, I did. What?

7:30 Because you have to do it in one. Oh, D? Here, I'm talking about two. Oh, two. I'm sorry. I was listening to you. It was D. Okay. So, one thing is that there are four slightly non-trivial factors here. So, the exercise is compute these four factors. I'll mentor you. It's so difficult to use both methods to actually compute as a tensor product or a direct limit or whatever you computed. You need both methods to really understand. I think this is an example of fundamental importance because you were taking an arbitrary map and then you were deducing four other things. These other things are the basic tools in analyzing that map. Analyzing a given map or picturing a given map. There are two. There are two standard pictures, and there are two standard grad bond combinations, 2x minus y, so these are important things to know about, because we use them all the time, but they also have this. The next exercise, why is D called D? Namely, I call this, real question, I call this the Dirichlet tool.

10:00 I was interested some time back in this concept of objective number theory, namely that a lot of the basic relations in number theory are really abstractions of something more concrete such as objects that are put together by coproduct and product associated to certain kinds of categories that are these rigs, which rigs are usually called. One hint is, if you make up a really bizarre notation, So n to the power of minus s, that means, the particular example of the kind of object here, it's something that's got n points, not distinguished points in a circle or a ring, it's just a pointed set, a pointed set of n elements. Analyze more and more, every object is uniquely a linear combination, very uniquely a sum of connected objects, and if you put together the repetitions of the given... All of these are connected objects in their representation. Any object is a linear combination of a set coefficients, basically. So if you represent those objects in this kind of expansion, and ask yourself, how do they multiply?

12:30 The ordinary is like each product. Each one expanded in that way. And how do you expand the... So, and going along with that, you see, the basic question of which... There's a certain object here which should be called the Riemann zeta function. This trivial category keeps coming up. The first time it came up, I remember, was Caterham. He was interested, like me, in describing mathematically the categories. For that, you know, really one of the first things is we've got to isolate this two. Which category is two? The first time I noticed it was as a counter and just some reasonable conjecture and then many years later I had some outrageous conjectures about the Problems of quantity and quality of the large and small and all that stuff I'm still working on. I had some outrageous conjecture. Peter Johnstone came up with a counter-example to my conjecture. And his counter-example was nothing other than the same old nearest way. And now if you look inside it, you see, as I say, you find that there is one object. And of course, everyone knows that the real one, the zeta function, is actually the Euler zeta function, or the Euler product form of the zeta function.

15:00 And the thing is that the Euler product form is true. We can continue this exercise and prove the Euler product form. When we say true performance, we mean in the objective sense. So there is some kind of isomorphism that you've got. Objects, and one of the objects is the zeta function. You know, and the closest objective approximation is the right-hand side. As a hint, the notation 1 over 1 minus x is really just a shorthand for the word algebra. All that because this is the way that we can express using infinite structure. This is a well-known equation that the pre-Monoite satisfies, but on the other hand, if you solve this, so to speak, formally on a great way, you get that. So obviously the objective replacement for number theory formulas, if they have an objective interpretation in which they involve this expression, then that's the thing to take. In some sense, a rather strange algebraic operation. On the one hand, the free product of one extreme. Another extreme kind of thing is the Cartesian product, say, extreme within the category of monoids.

17:30 Of course, if you look at the commutative monoids, products and co-products are the same thing. In the category of non-commutative monoids, the Cartesian product of several free monoids seems like a rather strange... Because if you take a finite product, free monoids, then because each free monoid has a length function, but then, so therefore the product has a multi-length function to the multi-natural. Now if you put all these multi-natural numbers together, you get the free abelian. And the zeta function can be thought of as the free monoid with multi-length as opposed to length, and explain that in some sense. Apparently in a different way, it's sort of a statement that the two are isomorphic. So actually you get not just an isomorphism of sets or objects in this topos, but you get an isomorphism of a monoid. And I forgot one ingredient, namely, of course you take a finite family of primes, there's one number, but you have to take a limit in order to get the actual, which is a very simple object. So it's just the direct limit of its finite communication system, but this product of the three monoids generated each one by some prime, you see, if you take a bigger product, actually there's an injection map going up precisely because they're monoids, so they have these zero elements, and normally you'd think a bigger product projected down, so you can take a direct limit of these finite products, and that of course corresponds to the term on the other side, and that's the

20:00 So try not to sketch. So the idea you see is that when you see this infinite product, you don't look at the analysis as an objective, but as a quasi-inverse form of course. Perhaps one should interpret it not as an inverse, but as itself a finite product, provided you're equipped with absolute power. This is supposed to be a kind of illustration, but you can make an awful lot out of it, but apparently we're at a little, only beginning. Anyway, so I want to point out that, I mean, of course there's a unique hunter from any category to one. And so we can perform these objects and predictions there. And on the other hand, an object in the category is a hunter from one. Other smaller objects we get in there. So by taking all the con extensions, we don't get six of them. We get far fewer because there's a massive coincidence of the adjoins.

22:30 So there's one here, which is the trivial that gives you the identity. In the picture, the typical picture, the object is... There are some dots, which are the fixed points, and then everything else goes into that, all it is. So, Steve Shanwell started an analogy with this, you see, these are the, these are the catalysts, and these are all the methods that run into the catalysts and catalysts. And so, so essentially, well that's why... If you have this simple expansion, every object is a direct sum of connected ones, a connected one is nothing but a number, in a way, because, at least for abstract sets, it's more complicated to choose them, but for abstract sets, it doesn't matter which point you're choosing, it's a big point, so it's essentially just how many components are there, and then for each component, how many components does it have, and that's really easy to take. Which is both left and right edge-wise, with the same function, both left and right edge-wise. The left edge-line, you know, thinking of this as a special case in a more general situation, we should obviously call it the set of components. Here there are four components. On the other hand, the right edge-line is sort of a cantorial idea of extracting points. The set of components and the set of fixed points are actually in bijection, even if the picture still has structures that are as morbidly so, you have this collapse that all the adjuncts are absolutely the same, the left and right adjuncts are the same, if that's true here, it's also true there, so basically you just have these... So this, again, I want to...

25:00 Say that this is a typical situation, namely, quality. I previously, in other publications, I described all the stages and all the clauses that are central to mathematics. Well, this is what quality is. Now, what does that mean? You'll notice that it's, in a way, the fact that you have it one step close, a little sub-category with all these adenoids coinciding. It really amounts to us having a central infotainment in this category, where central, by the way, the center of any category, you mean the natural endomorphisms of the identity functor, and if you have a natural endomorphism of the identity functor, well, you could take its fixed points and so forth, and its infotainment, its fixed points agree with its infotainment. Anyway, splitting that infotainment. You actually create this other category in all these metaphors, right? So that's actually the central, the category that's carrying a central input. It represents a certain type of quality, of course, over the base. So, lots of toposes.

27:30 We can imagine that every topos is some kind of generalized space. And this carries on a long way. You can imagine that it somehow generalized the Alba theory. This carries you a long way. You can imagine that it's, well, how many different sides does the Alba have? There's probably more than it would. For at least every already noted side, some more blind wise men may come along and find other sides of the Alba. You can stretch it so that it really says something about every topos. Inversely, every topos is something that partakes of the intuition that particular one had found. But the mathematical way is to distinguish things according to mathematical properties. And so this is what I really want to think about is some distinguishing properties. I claim that there's some topos that really should be called generalized phases, but they're not supposed to be called topos. Many reasons for that. Others should be called categories of sub-spaces. Others should be called categories of qualities. I've told you the specific properties there. And of course, really, yes, all these things are really properties of morphisms, of course. When we talk about rotunding topos, We really should mean utoposes for an arbitrary Roten-Deep topos because Roten-Deep himself introduced this old technique to relativize.

30:00 So in fact there are some very crucial examples of Roten-Deep topos which should not be defined over, say, something like that. So the qualitative distinctions in that sphere. Now notice that I just wrote down something. If we have one topos defined over another, There are certain words that we use to describe this. In different contexts, different words. So really this map represents some kind of contrast between two. So if we call it global sections, which is very common, let's use the notation of these parentheses next to Y, the object of U, which you get by applying the internal exponential. Transporting, you can turn on how long the product is driven by the subject, which is enrichment. Global sections, so in particular, global sections just like that. What does the term global section come from? It expresses the idea that the objects in E are more variable than those in S. The ones in S are more constant, the ones in E are more variable.

32:30 This has to do with the intuition that, again, sometimes quite literally, and always in some vague sense, atopos is a variable set. On the other hand, we can call these things points, we can call the same function, the underlying points, fixed points or equilibrium points. And in this case, we're saying that the objects in E are more active. Well, again, it's a original example. You have a group, U, and E is the objects. So already, active and variable are obviously kind of related, but they're different. And so, one should be able to express, in terms of actual mathematics, why it's the right way. Well, the gross way is their life, and this is the gross example. Especially if we're thinking of geometry, algebraic geometry, topological totals and things like that, these are spaces that have underlying sets.

35:00 So in that case, what we're saying is really that E is more cohesive. Cohesiveness was probably intuitively the original idea of the capital of Romania. Cantor's really big move was not only to figure it out, but he himself thought that the best way to discover it was that you can actually extract and abstract things without no cohesion. Well, that turned out to be undecidable, but at least qualitatively less. So you have more or less cohesion, and the term otherwise sets one off. There are these words that are used in this situation. So, what are the actual mathematical properties of the topology, to be even more exact, of the understanding of the sheet? We could go through a similar discussion for the adjuncts again. Again, the adjuncts would be on a constant sheet. That's very cool. On the other hand, there is a trivial action. In the case of cohesion, of course, there are discrete states that are viewed as special cohesives, discrete topologies, which, as a subcategory, have a tremendous content of the sort that they don't have this as a category to take. So, generally valid conception of topology. Sounds like great geometry, really.

37:30 This is really a series of branched out and graded down. I like to annoy people by saying set theory, but geometry, set theory, is clearly a special form of a totalist degree, but certainly that's part of geometry. So, but again, while this really does have some content, it could get much more content if you look at totalist unique special properties. And so, in the lecture that drove me to the 1970s period, I wasn't there, but I was there in another form. Notes by Federico Gaetano are available. It's part of them. We have part one and part two in July. It starts off by asking this question, what's special about algebra and geometry? And the answer that he comes up with is what you translate into our modern experimentation by saying that it has to do with lex-tensive categories. Now, of course, here I mean categories of algebraic spaces, not the petitotals or another, again, another story. That's what I'm talking about.

40:00 This is in fact something that we crude colonials concoct. It's a conspiracy between Australians and Americans. Nonetheless, there's by now a lot of papers that use this term. It's actually a combination. Again, this is even more abominable. Lex and Extensive based on the idea of I say X twice. Australian slang for finite limits, the lex-tensive category has finite limits, but one need not say that this is related to the equality, it turns out that there is a lot of relation between the two. So the lex-tensive, you know there are no problems, and the no problems have a special property which, well it's the way they answer to the following thing. There are two different questions that are the same, but suppose you have a site C. Category C defines products, and you look at the category, and therefore you can think of C as an algebraic theory, so you take the algebra, the product, simply the product of C into the, that's an algebraic category, which has various, you can ask, when is an algebraic category also a topos? That's a certain condition on how the products get.

42:30 In the same sort of, we should have called this maybe A or something, in the interview, if you have a category C with products, you can ask, well, when is it possible to embed C into some totals in a way that preserves the sums, because these are essentially good, and in that form, you basically know that the answer is that the sums are disjoint and universal. So in all these cases, we're trying to... They don't have to repeat universal or effective or equivalence relations two times in the hypothesis of the theorem and once in the hypothesis of the theorem, but the word extensive is not just in order to have a shortening of this, but it expresses, in some sense, a third aspect of it. And it's directly about the category rather than about the species or the alabas. It is that if you, so it's a category with coproducts, and if you slice it by a coproduct, that will obviously relate by a function in this way. And if you've got something over A and something over B, add it together, it comes out over A+. And so the extensive axiom is that this is equivalence, and if you want you can change it and throw in.

45:00 So the reason this is called extensive is because of the role of extensive quantities and thermodynamics and the desire to at least in some cases interpret these objectively. Objective numbers are things that we mostly know about, just like what's the logic of them, Multiplicative and additive, like the frame of subsets, addition and multiplication, but there is also co-variant, co-variant like the contrapedia that's being talked about, it's a kind of extensive as opposed to intensive logic, truth-telling logic, but there's lots of different, objectively, this says that we perform the Cantorian abstraction, looking at the set of co-variants. And we have assigned to every object A these measures. These measures are coherent, additive to a story in thermodynamics as we recognize an extensive quantity by the fact that if you divide it in two parts, it adds up. So it's linearity. This is linearity. It's usually shown, again, as an exercise that you have in mathematics already, that knowing that this is an equivalence, It turns out that the inverse of it can be described in these ways, that actually it's an extensive category. It has some pullbacks. It doesn't have multiple pullbacks, but it does have pullbacks along co-product injections.

47:30 And so, in fact, that's really how this is done. Anything over A plus B, you have the co-product injections. It happens that co-product injections are horrible, and so you get the C sub A to the C sub B, and then it turns out that the C is actually the sum of those two things. In other words, any math that's co-domain is the sum, uniquely splits as a sum of two maps, where the two domains are intrinsically determined. So this is sort of the most primitive form of the idea of the additivity of the measure. The thing over the base is expressing the size of something. So, all right, so in particular, we have many expensive categories. We call g, the functor is x, so we just satisfy this exponential law, right? And all the contemporary countries of X, small, small, small, small, small, small, small, small, small, small, small, small, small, small,

50:00 And so this, you could say, this is a morphism of extensive categories that preserve clients and the same co-products, but both sides are the same. This is where things blow up. What probably begins, because things, say in algebraic geometry, is orbit spaces are new factions. And then algebraic room backs in an algebraic space and you take the orbit space. Or you want to glue together schemes to obtain non-alpine schemes. So, gluing together, again, the typical sites, there's this notion of pre-topos, and a pre-topos could be defined as something which is simultaneously exact in the sense of a bar, and also extensive, just splitting the two aspects of quotients and sums, so... So, of course, the toposes, pre-toposes, are nice because they can be embedded in topos, and they're practically equivalent to the coherent toposes, and the coherent topos, if we sort of harmonically extract the study as well, is a pre-topos.

52:30 So, from the point of view of constructing, from the theoretical point of view of constructing toposes, pre-topos is, by the way, a set. It's very bad because you can't understand very well exactly which pre-topos it is that I'm talking about, because you started with different sort of data, the different sort of data, but what it has in common is precisely what's in here. So, in a way, the whole fight, why are there so many topologies? So, why did those arise? They arose precisely as attempts to struggle with this problem. How can I understand, how can I have very good quotients, like in a topos, and yet understand those in terms of the site, which is definitely not a free topos, but in some sense, an opposite kind of extensor category. Now, this is, you could say this is an exercise, but I don't know exactly how to explain it, because The thing is that the other examples of work-sensitive categories are also good categories in a way, but good in a quite different way. It doesn't match up. So the typical example of an extensive category is this. You take K as a rig, A sub K, which is a category of primarily presentable K rigs. You can see what a rig is. It's something that you can join in the end and get a ring. Now, this is joining in the left edge line to the algebric infusion.

55:00 And again, this is again one of those... The idea is to have a shorter term than commuting a semi-ring with a unit. Yes, and then so-so. So, in the general sense, algebraic geometry is about extensive science. Exactly in this form. And I say rigged advisedly because it's a mistake to think it's all about rings. Committed to algebra in a way began with ideals. Why are they called ideals? They're ideal properties. And yet ideals don't form a ring, but they do form a rig. And in fact they form a two-ring. The grid, which is in some sense obviously most unlike the ring, is the one, of course, the only elements it has are, it only means zero to one, but the addition law is that one plus one equals one, not as in the range of two. So this is, of course, the modulo of that is just the seminal. But the rigs are quite interesting because we didn't say anything about the... In fact, you can take any convenient monoid and make it, embed it as a part of the multiplicative structure, even of a k-rig for any k, including, of course, just a set of finite subsets of a given multiplicative under union as addition to some of this element-wise multiplication. Problems, I think, in the future, where groups are still not really well investigated. Most of the books about, I mean, there may be exceptions, but you know, the books that talk about commuter, semi-rhythm, unit usually do not take the approach of abstract algebra systematically in modules, projective modules, free things, quotients, all the things that we know as well.

57:30 One of the standard things in an algebraic category is to look for simple alternatives. Now, we know that the simple rigs are just heels, and then those remain, of course, simple rigs as well. What other simple rigs are there? Everything is common. There's an algebraic geometry, there's an arrangement, there are 10 rigs here. So what are the simple ones? Well, it turns out, and not a very easy proof yet, that the only one is two. Every point is either a field value point or two, but what we still don't know is what are the sub-director irreducible points, or sub-director irreducible, but regenerative points, again for rings, that are clearly what those are, they're very special local rings, and one would like to know what those are for two rings. Anyway, the point here is, you see, that since these things are constructed, From K-modules, you don't have to say semi-module. If K happens to be a ring, then the K-module automatically will be grouped. But the thing is that these categories, and it's very likely that when you're out there and you don't know, it's just that there are these other examples that really are ethically related. I define with respect to that tensor product. I mean, they're defined separately as well, but there's a multiplication of the tensor product.

1:00:00 The tensor product can be used with the byproduct that came out of the galactic categories. The products and co-products are the same thing. So, the tensor can be used with the co-product because it's a left-adjuvant to how. But that co-product is also a product, and when you pass to the algebras, it remains. And that's why the distributive law is true, but also the relativized distributive law, because you can take any, you can vary K as well. And so that's the basic reason why, of course not A-K itself, but A-K-A is an extension. It's due to the role of inputs. You can detect whether something is a sum or not. And since we don't have subtraction, we can't just say it impotent because it impotent may not have a complement. Now, the complementary pair of the impotents in the quantities is equivalent to the co-product of the components geometrically. And since these equations are mapped out in a proper way, we can verify that this is what it is. So therefore, if you look at the classifying topos, If you look at the toposmorphisms over u from the arbitrary e into this, then this is the all k elements, but it's not necessarily pi and e.

1:02:30 So now, this is all, now comes the part where we try to deal with the quotients. Therefore, we somehow have to deal with... Now, a notion of coverage... Oh, so in particular, notice that means that sometimes the whole discussion of coverage can always be reduced to a single map. We're interested in finite coverings, but the finiteness can be taken care of by taking a co-project. Which single maps are we going to count as epimorphic? The inclusion to the restrictive topos should take those kinds of maps into epimorphism. And there are some purposes that I will discuss next time. There's sort of a, for me it's a preferred one. I don't know if this is a good idea or not, but if we have a map, in reality, if it has an actual section, then it's got to be. But what can happen is that you have what I call stochastic sections. So stochastic section means, including the very point on the base, you get a probability measure on the top, which projects back down to the Dirac measures below, to the trivial. You don't have precise choice of something in every fiber, but you have a definite distribution.

1:05:00 Now, this idea can be expressed in the linear algebraic context by saying that if this is spec A, oh by the way, spec is nothing but an A, it's the opposite, spec is just an A, this is spec A and this is spec B, and it means we have an actual homeworkism of A raised to B. And so what a stochastic section is, say P, for example, if P is an A-linear, because of course the given morphism makes B into an A-logical, so it's an A-linear math, which is a section, or sorry, it's a retraction now. But I guess an averaging process is a way of reducing variables to constants. So whenever you have this situation, the functions of A on E are more variable than those on A. Now, I really wish I could say this also in the smooth context, in the analytic context, because it would have a sense, but it isn't really a coverage, so it wouldn't work. The reason that it works here is a remarkable thing. We really go back to the linear algebra subject. The same tensor product function is the one that corresponds to the full-wax geometrically, but it's the same one that applies to all any logic, and so the tensor function applies to this G, even though we can full-wax G.

1:07:30 Once you have a stochastic section, you can full-wax it, and so this is the... Now, is there a standard name for this topology? I don't... Is there a standard name for this? I've seen this concept. This often comes up as an example. Does it have a standard name? It has a role in what I'll say tomorrow about the infinitesimal. There's actually a somewhat slightly more general thing than just all kinds of presented behaviors where they can have the same effect. There's this notion of an algebraic function, where you refer to an algebraic theory in another, and there's a substitution along that, which always has a left-hander. In various cases, it also has a right-hander. So if you have an inclusion of a variety, or however you think of it, an algebraic function that has a right-hander, then of course it deserves the... So, the thing is that sometimes there are subcategories of these big categories. Which are not in themselves of the form all L rigs for some L. That's, of course, the standard way of doing it. But, for example, with two rigs, since these, even two modules are singularities, there's an intrinsic notion of less than or equal. F is less than or equal to g, and f plus something is equal to g. If you look at those two grids which have the property, the next is less than or equal to 1. It turns out that this right element of the algebraic structure, I like to call it the core.

1:10:00 There's a core of any two grid consisting of these kinds of elements, but if 2 is a two grid, it's closed under addition and multiplication. So this is one of those examples where the core exists. Since the eluded preserves everything, this still has the same tensor, the same pullback, and so on. So it's also most of the same expression. Another one is, in some sense, the opposite. If you want, x is always less than or equal to x squared, functions that are even zero or bigger than one, sort of. That's another example. Again, inside any two-ring, the elements are set by this. On the subreddit, there's the core, and there's the category, and all the other things. On the side, if you intersect two core varieties, you begin with the core varieties, and of course what you get there is the distributed lattices. So the setting of distributed lattices and its subtopology, like superficial sets, is also part of that, it's your excellence. I mean, geometry, what we need from these things are these two rings where everything is less than or equal to one. Parts of the cubes that are formed by equations involving them. So that has become popular lately with the name of tropical geometry. Tropical geometry, at least to a certain extent, is very, very popular. I think we should go directly to the ocean, right? In the 15 minutes, you know, I could be on the ground for a while. That's fine, okay.

1:12:30 There's so much to go on. Thank you very much. Now, actually, there's a certain reason why I'm so open, because in one it will stay in the other one. Now, that's just had new batteries in it, but those are all new batteries, both those and the other one, but that should be the 20. You'll have batteries beginning to go flat. When that, like there, starts to flash orange on and off, when that happens you need to replace the batteries. Hello, I'm very disdainful as I'm doing this talk and for lunch I'm leaving about a quarter past two and I've got all your details here which I was just about to give to Davide, including a spare key to my house.

1:15:00 And all the way to details. So why didn't I give that to him? We'll just save some time. Because I imagine... That was amazing stuff. Yeah, lots of... I might get a chance to talk to him much longer. No wait, okay, yeah. These are the 120 minute cassettes. As a matter of fact, I've got a couple of those, so I'll stick one of those in now. One of those will record four hours, because these are one hour per slide, but we've just remembered that these are half speed recording. So that will record two hours per slide, and it should reverse automatically, so you don't need to turn it over. But be careful at the end when it does reverse to take it out, because if you start recording once you've switched off, Then, when it starts recording again, it will start recording... Yeah, you see what I mean? In other words, you have to be careful not to record over what you're recording. The advantage is that this records a hell of a lot in one take. So that is now... That's now recording. I've got it on... Leave the setting between about 6 and the 8, and that's all, and it's on. This is the speaker. This is the recording time control. You want to leave that on double. Just leave that where it is. And this is the voice operator. You want to leave that off. Definitely off. Don't switch that on because then it only records if somebody is speaking directly into it. So leave that switched off. Leave that on. And that should be just about right at the end of the speech.

1:17:30 You don't need to change anything at all except to change the batteries when they start getting flat. All you have to do is to press the red button. For heaven's sake, believe it or not, I have had extremely intelligent people, including John Maybrick, who I think you met, a very bright guy who recorded some lectures for me, only went and pressed that one bloody button. So it's the one with the red button. Yes, that absolutely takes it with it. And you can tell it's recording. So that's now recording. And that will record almost to us. That should be fine. We've got plenty of spare batteries. Yeah, you can put it in the front. Yeah, that would be fine. That will record perfectly. It should be able to record from anywhere in this room. Yes, leave that pointing. Although, well, it's a directional microphone, so leave it pointing towards the speaker. It will pick up from anywhere in the room. So just press that, and you've got everything here in the way of replacement tapes. That should be more than enough for the whole meeting, so 1, 2, 3, 4, you've got 4 more 120 minutes, no 5 actually, and if you run out of 120 just go to using the 9s, but don't forget that you will need to change the boundaries once in a while, and then in addition to that, sorry don't say that I don't try and think of everything, I'll give you a spare recorder as well. Yeah, no, no, no, there's nothing like safe and safe in case. This one, again, is, that's only got a 90 minute take in it, so that will record while you talk. This is also a half speed double recorder. But this one, you do need to change over. This doesn't automatically. And okay, you just press the button. So the next, you just, you just, you just, it automatically, when it reaches the end of side A, it will reverse and start recording on side B. But obviously you want to be sure, so in other words, that should be able to record the rest of today's talks, including the, but this one is, in case anything goes wrong with that one, you've got this as the backup.

1:20:00 This is not such a, quite, quite such a good recorder, so you want to use that one by preference, but this one will still be perfect. Yeah. You've got, you've got, so I'll give them to you both. Okay, yes, you've got it. Yeah, and take this, and the only thing I need to do now is, everything is in there, when you want to see it. If you can just, when you've finished the, if you can just make a note on them, just put 1st of June, p.m. or whatever, and just say it, which is, that would be brilliant. And the last thing I need to do is to give you this, which is the, I can't remember, how well do you know Paris? How well do you know Paris? Okay, you can't know, I'm trying to remember, I should remember which metro line it is you take to get from Garda North to Garda, well, I think it's metro line 11, it's not, I'll just have to, okay, this is just a note from Bill, and so he's got, well, I'll just leave it in here, and tell him that there is a spare key to my house in the cockpit of this case. So, in the event that I've dropped dead or disappeared, he can still get into the house and let himself. Okay, Davide, thank you so much for this. Oh yeah, you're welcome. Just give all of this, just put the recorders, give everything to Bill. I mean, it's a bit heavy, but I'm sure he can manage that. That's it. That's fantastic. Thank you very much indeed. I can't thank you enough. And I will make sure that you have a pocket of everything. Give me your address in Roman. I'll send you a copy. It'll take a little while, it'll probably take about two or three weeks, but I will, I promise you, send you a recording, a copy of, not only of all the recordings here, but also of the conversations with Pierre Cartier and McIntyre and the others in Fougere. Right. Yeah, if you just scribble your address on there. I'm really grateful to you for doing that. This is really an important scientific work, I think, to make sure that the records are preserved. Well, I did, of course, record everything.

1:22:30 I think this is new. Absolutely. Well, you know, this is actually, well, this is sensational, but this is all in the line that grows to the degree of this huge scheme using, essentially, ring classifiers to do away with logic, to completely unify mathematics in terms of this huge scheme. It's an absolutely incredible vision, and obviously if you could only give us a little bit of your ideas, there would be so much more to do. If he starts having really interesting conversations with you, I would be happy to ask him. Okay, I'll get back to you. No, thank you. I'm going to leave after lunch. I'm going to listen to Lucas talk, I'm going to be laying at the hotel. This is the train 10 to 3. And you're actually going with Bill to Brussels, I think, on the... You're actually going with Bill to Brussels on the last day, aren't you, on the 5th? Not the... Oh, okay. He said something to me. I think he was planning to go with me to Brussels. So how are you going to get back to Roma? He stayed in Brussels that night, reminded me of him, just reminded to give me a call. So you've got everything now, I think. Are you including the bit of paper with the address? Okay, that's great. And the key? The key is actually in there, so keep that sitting up, because I don't want you to lose that. Yeah, absolutely. Okay, and best of luck with the taking.

1:25:00 I'm going to stay around. You've probably seen me at the lunch interval. Today has been a real rollercoaster, as I said, I've heard him talk a little bit about that in private, when he was talking about his understanding of distributions, which is, but I never heard it. It's absolutely fascinating. It's fantastic, because he started very quietly. He basically always does that. To deviate attention, I don't know, he always starts very, very, very low key. And you actually think when you're hearing him for the first time, when is he actually going to say anything? And then it builds and builds. It's a bit like listening to a great symphonic performance. It always starts. I'm going to just use the little, I haven't really got time now. Okay, you've got all that stuff. Propositional theory. So a propositional theory will have actions with propositional letters that actually will form three lines of a suit where the words in English are not made by propositional letters, but are represented. Here is enough to just take a look at it. These models are projections where S and E are fixed sets, and S is infinite and E is non-entering.

1:27:30 So, we've got L, the propositional letters, but this has to be a compositional theory. The propositional letters F, S, and E. But this is not a formula, this is a letter. I should have written it in B sub S comma E. You say, maybe not strictly saying what you say, but you say that if capital S is E and capital S is E prime, then E is equal to E prime. I'm going to be racist. How do you say E is equal to E prime? Well, you say it's true that E is equal to E prime, but if it's false, then E is not equal to E prime. But if you are in an ancientistic context, or if you are inside a compost, it doesn't have to be a soup. You have to imagine it as a soup. The soup of the family of propositions, all true, are the subset, the subset of vowels. This emphasis is difficult to define, and this vowel is difficult to define, and somewhere in between, we don't know. This is not finite, it's not infinite, it's an axiom. I want to say that it's a certain axiom, true in class, for any B and E.

1:30:00 For any B and E... Yes, it has a value. And then you have to say for every S that there is an E, that's the value, which I think you might know. It's a certain axiom, and then you have to say it's a certain axiom. It's a simple proposition too. You can, it has come up in various books in a specific way, but you can put yourself inside a corpus and do this for any cool object in the corpus. You put yourself inside a corpus, you see, move your corpus, such that e to the t, the mathematical arguments, are a project of correspondence. Thank you. But in many variations, all of them are dispatched from one to the other. But in me, you don't have the sets S and E anymore. So you have to get this from somewhere. And you use it every couple. It has a canonical organism of what is called gamma sets.

1:32:30 So, A and E and they don't never outstand S. So, the cover in this, in this side that I constructed last time is non-empty in a strong sense. So, this topos, it's non-trivial in a quite strong sense. Even if you have an element of E, perhaps, it's much bigger than it has. This corresponds to the Nobel Phenomenon of Cardinal Collapses and Set Savings, for example. So we can do this, we can do this. Inside, come two topos, and we get where s and t are objects of t. Now you may wonder what it means for s to be infinite, but in the application, s will be of the form, s will be of the form d. And what you get is how Vt, a different index, can emphasize that the word in order V will be a topos. It's a crypt, i.e. there is an F. This characteristic property will have to be expressed by geometric morphisms between toposes all over V.

1:35:00 Peter, this Peter, mentioned in his photograph that The open levels are characterized by the fact that all the colors are inhibited, so this will be a homo-conservative theoretical theory, a propositional theory, which is constantly related. I want to return to the slogan which I mentioned at the end of my first lecture, namely, every theoretical theory has a conservative and a conservative biopropositional theory. I think this segment should probably be a bit more required because we should know that p is consistent or something. So I'll simplify it because I'll assume that p is single coordinate and the laws of p consistent concepts consist of one set plus all the relations, functions and so on. And also I'll assume that p proves that this and that is x and y.

1:37:30 The theory is just like any other constant, for example. T prime, the models of an infinite set. T prime, these models are triples of the pairs M, where M is a model of F, the surjection, and S. Models in sets, let's say, are artistical. In some sense, I remember to describe a theory The problems of n which are expotions. You replace this theory with another language in the main session. So replace x1, x1, playing around with e, by propositions, by s1, sn, for s1, sn, s. You could actually just do this for the atomic formula, so there's a community of disjunctions and conjunctions of thoughts. It's not that important. But I want to emphasize that one formula is replaced by lots of propositions, they're parametrized by one another. And then there are some new propositions.

1:40:00 Sorry, this is for all little s out of big S? Yeah. You're doing it for all symbols? Yes. So a current formula with n variables is replaced by s to the nth nth propositions. Okay, and then there is one, so maybe more importantly, to rewrite the theory as just a theory with relations and... I have a whole composition of proofs of relation in order to do it. I've also introduced Fs equals Fs. So again, this is not a formula, but this is a composition. So this is, this code, how the model M is a quotient of the given set S. And then the action of the, oh, it's higher up. The action of the proofs of relation. The number of actions for root of 0, root of 1, 0, 0, 0, 0, 0, 0, 0. Then, the model is not defined. It is quotient. For this, so it just means that if you have phi as 1, as n, and f as equal, f as 1, equals f as 1 prime, and f as n, equals f as 1 prime, this implies phi as 1 prime.

1:42:30 And then, finally, all axioms, from x, phi x, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's, to the y's. I assume x is a sequence of x1 over xn and t is a single sequence of t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 over t1 Of looking at this, maybe I don't want to give a detailed proof, but you can see that it's quite plausible, for example, in the final case, in the first case, just by a very large problem theory, the language has a commonality, so if you take S with the commonality of the language, then automatically any model can be explained to be a model of S. If S is smaller than the length of the language, you can use the problem of collapse, right? It doesn't use that as far as possible. And certainly, there are some ways of expressing it on the scale, and we will formulate it in a second. So, we get the code. We get the classifying combos.

1:45:00 Bt, Bt', Bt'. Any model of, there is a Witten-Wilson model. Witten-Wilson model of Bt' which was described. Now, this model is a model of T plus a subjection. So this is a model of T plus a surjection, so I'm going to do a little bit of math, dT' and then we also put for emphasis, we're going to have two sets, gamma and gamma dT' so that 2T' is of the form gamma prime upper star, sorry, S, surjection, this is what we think, the interpretation of that, to dT' of U of T. Because the model of VT' is really a model of T plus a subject of form S. So if you forget that second part, you get a model of T, so let's call that the model of T. So the model of T is the T-upper star of the model of the universal law. Also, I will try not to use the terms of blackboard. To relate this situation to the previous example, VT' being constructed relative to VT,

1:47:30 Gamma-compositional theory is the same thing as B inside Bt or S by T-double-priming, but by T-double-priming the theory, the gamma-compositional theory models, you can do sets inside Bt, so the role of S is now played by gamma of Bt, which is gamma of S. And the role of P is kind of described by Newton, whose models end up, from his models they output propositional hypothesis, at least between the two of them. So P is this, Bt is this value given by propositional theory in the absolute sense of a step. It's also given by a much simpler propositional theory in the relative sense, seen from Bt. It's actually a simple theory. If not, you rewrite the theory in such a way that it can be a theory. The sum of our all sorts, and you use predicates to separate them. Or you do a variant of this where you have lots of surjections, not really a story. So, now I'll make a little jump. For specifying purposes, the same thing as...

1:50:00 You could take as a definition of what is called the organic topos, a topos which is a form sheathed on the side, where the side is just a partially folded set. If you have a propositional theory, then this category sin t, the syntactic category, would just be partially folded set. And there may be a equivalent that is defined. It's very much like sheathed on space, except that it's much better behaved if you work inside the topos. We start with the theory of surjection from S to E, and the cardinality of E is much bigger than that of S, the space of surjection from S to E is infinite, but the locale is only a propositional theory, so the locale will contribute. This in my case would be one of the advantages of working in such a class. D-Defined is a propositional theory which is given by a locale D-SET and this is also given by a propositional theory which is given by a locale in D-P. And the fact that we can use these functions to determine this theory means that this thing is an open circuit. Properties of this injection. Now I can't take everything into the locale.

1:52:30 All of these terms can be used to define the It is used very much in qualitative, alphabetized, quantifiable topos, and you can also use it as a universe in which you do theory of mechanics. Since every topo is a classified topo, I have proof, and this proof, which was originally proved by Diokonescu working directly with Seitz, and in this form, in the memoir of Joël Le Pen, more exactly in this form of this theory, So, I bring Y with Z and an open subjection E from Y of E, and for over, this is a consequence of what we have explicitly here, for over, there is, for which E from Y is the same as E, inside E of X, and now to say that this is an open subjection, in terms of the case of E, it means that...

1:55:00 So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. So you can work out that from x to the modulo in space of e is an open subjection to ejection. To approach the situation systematically, I'm not going to give you a definition of OpenMAP, but I'd like to give you a few properties.

1:57:30 So, OpenMAPs are, consider a composition, isomorphism, forming more and more compounds. Isomorphs are open. But we have a subcategory containing all isomorphisms, identities. If x to p is open, so is y equals px to y. So the open maps are preserved on the fullback. And also, for our first sound, if p is an open subjection. So open maps are preserved on the fullback and then we kept it down. There are quotients, which we have seen in Hong Kong, there are quotients here. Subjection, the co-equalizer. The kernel pair is co-equalizer, stable, universal, and you preserve given K, B, X.

2:00:00 And also, if R is an equivalence relation of locales with R log, then it's characterized Because co-equalizing exists, co-equalizing exists, and this against that. Properties are enough to develop the descent theory, or this is enough to develop the descent theory, and in a very easy way, as I will show you.

2:02:30 Now, if you have any class of maps, which you could call OpenMap, and they have this property, then you can use the descent theory. But the checklist of properties are the highest and the lowest. So I think one of these occurs again in the same type of theory, and the other occurs in the same type of theory by Morita Itudović for locales groups, which you can find in the library. I'd like some connection of sheaves to the topos of sheaves on a locale, so if also one of them in this context is formed, a locale X, the same thing. Open, and, I have an open banner, which is open, I mean it's a copy of that. This one is open, this one is open, it's a hectare, for a hundred more people, not for a hundred more people, I'm sorry, I'll have to keep these open. Xml of locales, x to the b, and f. And when I look at locales on the b, that's from some locale, x to the b, the common category,

2:05:00 And there is a pullback function f to the locale over x, and I want to somehow characterize the image of this component. If you are given the locale over x, when can you descend it to e? When you descend it so that it pulls a pullback or something to the top. If this is the case, then e should be constant along the fibers of this matrix. And this constancy is expressed by some action. And the axioms are constant along the iris, and expressed by theta, which is the convention for the left-hand direction of quantum mechanics. So what theta has is the following. Theta x, here is a little point x, here is a little point y, all of which are different from e.

2:07:30 And they are at the same point v and v. But it's also like, if you're in a differential movement, it's also a little like a connection. You can move these points a lot more towards all the paths. And this connection, if you want to keep the terminology, should be flat. So it should satisfy something like this. If you start with E over Z, then you move it to Y. You get this point, theta y at t, and then you move from that point to x, like now you have here, and then you move all the way to x, theta x, theta y at t, which is actually the same as moving one step from z to x. And also, if you move from z to z, you should not do anything. Theta z e t f t of z. So, for the final capturing, maps are matched with respect.

2:10:00 There are two such, e to the e prime, equivalent to the state of that state of prime, and that is the method e to the e prime over x, which have a Riemann diagram involving the state of that state of prime. The descent theorem, again from the mentioned reference of the theorem, Then, F-alpha-star, let's say F-alpha-star induces the equivalence of categories from locales B to the deaths of an academic. So, to say that this can be induced by F-alpha-star, then we should write down a function. So, here we have locales X, locales over time, locales over B, locales over X. So I'm saying that if you start as an account of B, then there's a canonical such data. I want to prove this theory real, because I want to emphasize that it's really completely based on this theory.

2:12:30 All these actions are open maps. You copy them from open maps. And you put them in the directions of the functions. So you can perform them based on the open maps. So you can copy this proof for any class of maps which is a syncopated. For example, Coppola. So, because of the categories, don't put this in your name. We have a pullback, x cross b, x cross b, b prime, and then we have a pullback, x cross b, x cross b, b, b prime, and here are pi, 2, 3, and pi, 1, 3, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi, 2, 3, and pi.

2:15:00 Here is a graph of everything that has been tried in this F, precisely the polynomial that makes left-hand squares for u. This is a mathematical failure which I can spell out, because one of these is too bad, and the other is too good. So a map in S is just a graph which makes these triangles for u. This is the pullback of the calculus of ejection. You should really pull those. Rejection on the first and third axis doesn't happen. Peter is right. He really is right. Okay, so you have these rows and these maps are pullbacks of the calculus of ejection. So the rows are co-equal axes. Rows are symbols of equal axes. And that G is in the scheduling process known as the base of these two squares of mu, and the fact that these squares are not equalized, any such G that makes these two squares of mu, was actually uniquely here. So G, the factor is uniquely required. So this means that M is true in this case.

2:17:30 You might think that it's not so much that easy, but now this would have been essentially subjective. So it goes as follows. Take, say that, In the descent of M. So, in form, x goes between theta and phi2 to B. This is a full-blown moment-of-conference injection. And that is really isomorphic to phi2. The state is just a diagonal shape. Both these maps are open subjections. So, you can form the code equalizer. So, this is stable. Theta and phi2 are open. It's a whole B, because everything is a whole B. And I claim that P is the same thing as... that P is the plane of the function of a movement. But this is an experiment which is fun to catch up on, where the p is the same as, the same as, of course, x goes to the q.

2:20:00 We like p and x goes to the q. Well, there is a map from p to x goes to the q. That's already in life. So let me give you some maps. So this is q, and this is f, h. So here's a map. Here's a map. And this is a map, p to x goes to the q. And in this observation it is h comma q. So now, construct a map out of x cross b, q. Well, you know that q is a co-equalizer, and it remains a co-equalizer because it's x, because it's stable. And then it becomes easy to construct a map out of it. x cross b, x cross b, e, x cross b, e, x cross b, q. Here we have an algebraic theta, here we have a monomers theta, and pi over 3, so we get a unique matrix, theta bar, because you can check that the diagram concludes, possibly replacing the 1 by 2. So I'm using the topogram, the Q. You get this net theta bar, and then you just do a diagram change.

2:22:30 We try to execute it using pi over 3 here, but this is the end of the proof. Excuse me, can you stop the poster? It's a completely soft proof. The proof works in any category where you have a class of quotients which you have to remember. Such as the over-selections in this case. I want to draw an epithelial subjection, an epithelial subjection, that Sharpe also induces the category of schismatics to perfectly fit into this. This is just proof of the strict punitive rules.

2:25:00 While the properties of all of these are listed, it follows that the equivalents are preserved on the pullback and the reflected time of the pullback around the whole subjection of the equivalents of these two concepts. So, let me just close by saying that in my next lecture, I will briefly reinterpret this. Combining this with the theory about the orderly subjection from a locale, As I said, under every common sense, technical issues can work out, and then I will look at some variations on that. I mean, turn off. Stay in the field. Stay in the field. Thank you.

2:27:30 Thank you very much. Thank you for your attention. Thank you for your attention.