3rd talk/ Miles Tierney: 3rd talk / 3rd talk

0:00 So for about 35 years I've been embarked on a program where my esteemed colleagues have shown a little bit of interest, but not very much. It's a program that starts fundamentally with geometry, or even more exactly starts from continuum mechanics and desire to understand that in a fundamental way. Since it seems most of the students are coming from logic, let me try to explain it from the angle of logic, even though that wasn't the original. So Cantor. What was Cantor's main discovery, according to Cantor? Not that I've read all of Cantor, but I have read some, but apparently most set theorists have not been saying the things they do. I get the impression that Cantor thinks that his main discovery... ...was the passage from Mengan to Kajinalzau. He discovered that, he thought, you know, there's a period early, he's quite happy, he's trying every possible thing to think, well, you know, the line is discrete and everything is totally, there can be these infinite discrete sets. We can do all sorts of things with it. And so somehow, I think if you, if you, you know, if you are the initiator of this idea... Then, well, it follows more or less as a corollary that there is such a thing as a power set functor, that you can do derivatives of closed sets, prove the diagonal arguments, all these, no doubt, very important monumental results of Cantor that we use all the time, they somehow sprung from this idea that you could have infinite discrete sets within mathematics.

2:30 In mathematics, not off somewhere else, in mathematics, in mathematics which consists mainly of mangan, or what I interpret, again, by an undefined assistive word, as cohesive, you know, in continuum mechanics, in geometry, in analysis, all different branches, interlocking branches of mathematics, the things one's dealing with always have some sort of cohesion. Stick together in one way, different modes, different degrees, and so forth. They all have this, and this is somehow what men are. But then Cantor said, well, in particular, within all that, we can make this massive abstraction, pretend that everything is not that they are, but that there is an underlying bag of dots, completely featureless dots. Now he actually called these things cardinal-solid. I believe I'm correct that what he calls cardinal-solid are not cardinal numbers in the sense that we normally understand it. Cardinal numbers in the sense that we normally understand it is also defined by Cantor at this point. The birdicide ring of this, in other words the isomorphism classes and so on. Now that may seem like a subtle difference but actually it's a huge difference because The cardinal syllables still form a nice category, whereas cardinal numbers don't. Now, again, my reading of German may be... It was actually Manio who first pointed this out to me when I was, you know, as I often do, lecturing on the notion of abstract sets,

5:00 the category of abstract sets, axiomatic descriptions of it, and so on. Manio pointed out that, well, this is also in Cantor, and he even gave me a page reference of his page 304. There's a page in Canto where this is described and there's some earlier writing. Since there was no available explanation, logicians typically say, well, that's of no interest or it's too complicated, it can't be done, or make some comment like that, and very few people have come up with it. Jumping ahead, the point of view of category theory is that, well, first of all, these men or cohesives, they must come in categories. There's not just one. There are a number of different categories of Mingen. So once you realize that, then I think one can actually start to explain it. So in one way, in one angle, one could say that the program is really to recapture this notion of Mingen and to fill out the historical story slide, I should say that Cantor's friend Dedekind, especially, together with Cantor himself, together with Hausdorff and so forth, They developed the methodology, which is school, as though this were the air that must have always had existed, but the methodology whereby we actually interpret particular ideas of cohesion in effect, although we don't think of it that way, but we interpret particular ideas like that as interpretations in totally abstract sets, specific concepts of structure. So, for example, already in a partially ordered set,

7:30 This is a model of a particular kind of cohesion that this was conceived of as a structure in the category of totally abstract sets, so the relation between the kind of structure, the background of abstracts, and then the particular interpretations as being the particular objects of mathematical interest, this is something that, as I say, I went to school before I knew that maybe did not always exist. So, okay, so liquid is a category of spaces, but the idea is that it's a particular flavor of cohesion. There's flavors, there's continuous cohesion, there's different flavors, in that sense there are categories. If we consider it sufficiently large, we're saying that not only algebras and orders and so forth, but even the cohesion itself, we find it inside saying, well, among these are the things that have no cohesion. If we say we take that flavor to its minimum possible, say it's no cohesion, as we'll see, it's a very bad mistake.

10:00 These are spaces with no cohesion, but they were finite. So there was the idea of the finite discrete abstracts, just for a long time. So even more precisely, Cantor's leap was to say, well, even the discrete spaces could also be intermixed. The things which sort of clearly have to be accounted for in mathematics are really continuous phase on one hand, discrete sets on the other hand, and the interaction of those two. So the infinite discrete are some sort of, which doesn't quite show the people, but at some point, I don't know who introduced this swindle, called it the natural numbers. So that was the most basic thing. It's not natural at all. People started talking about the natural numbers, the completed system of 0, 1, 2, and 3. To call it natural was the first thing, really. Much more natural are continuous, infinity, and finite, x. So anyway, Cantor said that there are, and so for example, somehow, even more, there's an associated set of points. It's a puncture from the whole abstracted part. So u is abstracted from the flavor m.

12:30 Let's take the minimum. That's sort of an opposite, because they're not so well defined. They're usually very special objects. You know, these are particular examples of cohesive spaces on the one hand, but they also play the role of being arbitrary objects, because the morphisms, I'm assuming we have a category, domain in this A-shaped figures, or more generally in the category for complicated diagrams. The triangle is the basic guy. This category is the incidence of the figures and the incidence of the examples. Could you give an example? The point is that, well, in fact, many categories that aren't toposis can be analyzed in exactly the same way.

15:00 Toposis in particular, what is the actual method of analysis? We have this big category. We have these two abstract subcategories, which is sort of, you know, as I said, I mentioned cantors. So these incidence relations become interpreted in the abstract background in that way, that we define X of A to be, when you were thinking of it as an abstract set, but I'm now internalizing and formalizing or making finitary algebraic the very notion of abstract set by this functor. So x of a is gamma lower star of x to the power a. And if I have an incidence relation, it gets interpreted. That's why there's another name for this when gamma lower speed is understood. It's script m all of a. So that's the object of u. There are many true objects, and m is Cartesian closed, is functor. So in particular, the incidence relations become also maps, become maps in u. Now the structure, just repeating this the way you've heard it here before, waves, and it's about representing things in terms of pre-sheets. I'm underlining the fact, instead of starting with, he started with the world of mathematics, but had all this in the same way within each of those, elementary or not, never mind.

17:30 It will be in some sense elementary over u, if we could add some, but that's sort of the, we need to make more precise this. All of this is natural, is a horizon. Namely, well of course, he said he was included to them. That was the big leap, the first big leap, the zero big leap. That's called gamma upper star, with its left adjoint and so forth. One of the main purposes of this, from the point of view of geometry, is to look at connectivity. The left adjoint, this can be Cartier for example. Well, it's incredibly complex. You see, wood, you know, has fibers and it has cells and leaves. It's a very rich thing, a forest. But if you look at it from a distance, what you do is you count the trees. You just get a set of components. I'm imagining it being the connected components of this vastly complex thing, which is the actual forest. It's this abstract set of trees, which we can count. So in a way, you see, the character now is solemn. are the things which give rise to the still more abstract numbers, but they themselves are these imaginary. You've deprived the course of all this structure, but what remains over is that. So that's the puncture which is left adjoint. You see, there is a real possibility, very interesting category for purposes and so forth,

20:00 but the U contains only finite sets. I mean, surely the spaces in each of these categories are an infinite number of points. I mean, if you can't see that number as such in the category, it's certainly infinite. But at the same time, the number of components can be finite. Now, the first example where this was made very clear was the notion that it was real algebraic geometry. Then you have surfaces and things in all finite dimensions that are defined by Equations between rational functions of arbitrary degree and so forth. It's a very interesting category. Endlessly, one thing I've studied is, there's one thing about it, in spite of the fact that the objects are infinite, each one only has a finite number of connected components. So if I take the category of finite sets, consider each finite set as a trivial real algebraic variety. Think of it as the zeros of a polynomial with one variable. Which are necessarily finite in number, but not included in the left-hand drawing, finding the number of components. Pursuing this point of view has been taken up by a group of logicians known as O-minimalists. Following the logical significance, I mean the logical significance actually goes further, you could get a decidable theory.

22:30 But certainly you couldn't get a decidable theory if you had an infinite discrete set. That's essentially the content of Gödel's so-called incompleteness theorem. So this is a way of skirting around Gödel's incompleteness theorem. You can even get the completeness of the side abilities. So this is a vast program in itself, but, roughly speaking, the idea is that you can find richer and richer m's that still behave this way. Grotendieck's tame topology, the desirability of a category to replace the category of topological spaces, would contain, in some sense, all the interest in homological and even homotopical information, and yet definitely did not have awful things like a piano, space-failing character, etc. He never made this absolutely precise, but he essentially arrived at the idea that the thing must somehow consist of piecewise real analytic. This is a prime example of apparently contradictory desiderata in that I guess the first big leap in the old minimalist program was due to Van de Dries, who precisely constructed, you know, from a Lindenbaum category of some logical system, constructed a category that was basically piecewise revalidating by that number of pieces. That's key. I mean, it's a very... So then there had to be some, you know, special explanation of what happens in infinity and so forth.

25:00 As a result, it all worked out very nicely. So there's a very, very rich as an answer to topology. I mean, it wasn't presented as an answer. In fact, it was I who told and resolved, really, as an answer to both of these. Whereupon I immediately changed the title of the book. It says, Topology. It's cut out that way. It's cut out of the study of decidability extensions of it. There are many people who work on this direction, but besides van den Dries, there's also McIntyre, for example, and several others programmed, I don't know whether it's an A or a big I and a Y. Here, this is an ongoing program, which I said, in a certain philosophical sense, is completely diametrically opposed to Cantor's fundamental idea that you could have components. Components are supposed to be at a speed stage, but they have the same number of components but have points. In a way, this program is quite diametrically opposed to Cantor's fundamental starting point. But now, first, the idea of contradictions like that in mathematics is that you resolve them. So, obviously, a very important program from my point of view, a program that I haven't really gone very far into, An O-minimum was categorized as a site for Grotendieck topos. The Grotendieck topos will contain natural numbers on there, but on the other hand there is this well-defined silk category, which does not, and which contains most of the geometry.

27:30 So to resolve this, apply the method you heard about in the last two lectures. Anyway, I'm going to point out that what I actually wanted to talk about was An axiomatic description of a special situation, the idea that these are cohesive spaces. I'm going to assume without further ado that him and you are, that I have, namely the components, the points, and that any one of the three punctures determines the others because, since it's a full inclusion, the composites are their identity down below. Already there's a very, so it's an essential or what we call globally connected. Geometric morphisms come down with. A very crucial point is that chemistry should preserve products, and this says that in some sense any object is a sum of connected ones, or the notion of sums, but a particular one is connected to the fact that this morphism is connected and so on and so forth. So I will just talk about a situation where I have two. I mean this implies you as a topos, and you have these elements, and then follows the view is.

30:00 Keep in mind the case where Ego is actually extracted from him by more specific means, but in the general theory it can also be found in some examples. Now it's quite important for the geometrical intuition of genuine category of cohesion that although this preserves products, it does not preserve equalizers. This cannot be localic. It cannot be localic because if it were localic, then preserving products would imply much exact, and then you would have this canonical point which would contradict the other properties. So this already sort of means this. And for the intuition of calculus and so on, the reason for that is simply this, that if you can have spaces X and Y, Gamma lower truth of X equals 1. Gamma lower truth of Y is evident in the picture. And then you can have two interesting maps, and then the equalizer of these maps in the graphs is a disconnected space. Gamma lower truth of the equalizer is not what you want. That's how we construct interesting examples of non-connected spaces. The quadratic formula gives us the possibility of constructing the category in the sense of the railings.

32:30 Namely, these categories have the same objects, I'll just use square brackets, are obtained by applying not gamma-lower star, taking the points, but rather taking gamma-lower tree. All you need to transport an enrichment of monoidal functions, You can transport one enrichment into another enrichment, here in particular the self-enrichment given by exponentiation. But of course, it turns out even better here that this is still Cartesian flow's category, so extensive, but emphatically not the idea of full-blank and you don't have them. It depends on M. There may be a further function necessary to obtain what Senator Quillen would call, particularly it's already pictured here, that there should exist a space

35:00 We'll call it I to suggest intervals. It has two points which are really different, so the equalizer is empty. The space of components of I is one, so in some way this expresses the idea of within these cohesive spaces, even a distant point is cohesively connected enough to this point that I can actually move from one to the other, apply it and connect it. So in some sense this is the possibility of global motion in some very way. Now, Grotendieck pointed out that this is actually equivalent to a specific object in a specific parameter. The thing isn't inconsistent. Gamma over omega should be... Well, this is something that you're going to get totally unheard of in classical logic, because in classical logic, omega is literally 1 plus 1, the most disconnected thing you can think of. Of course, and many are most hopeful of it also. This is not true, but in any case, avoiding logic rather than boolean logic can be true in some cases. Keep in mind, as I said in the first class, the example of reflexive graphs. This is certainly true in that case, because remember the omega plus this is degenerate. The gamma lower streak of a small graph, you just look at it. How many pieces are there? This implies a much stronger...

37:30 Namely, that there exists an endofunctor, a kind of transformation from the identity, and next tilde has a remarkable property of being, by contractible, I simply mean any space which becomes one over here, which becomes the terminal object in this category of comatose classes in math. Probably it's one of the reasons why Hilbert invented space-based. You can live without it, but you can literally say things in a reasonable way. So, in a way, contractible means that, because just look at the definition of the column, I mean, this is going to be equal to one point if and only if for all x one component, so it means that gamma over straight to any power a, the fact that the straight being contractible in the first instance, it quantifies over all the domains a here, but in fact you can check it by just looking at the... Not necessarily a wide field, but any space wide. It operates just in its own self function.

40:00 Maybe I have to ask if it has a point. But in any case, that implies that essentially you can use... This is a monoid. It's a monoid with zero. The action of any connected monoid with zero works just like a divider connects up things in such a way that the action of a connected monoid is zero. So, this one we divide to the y. Of course, axon is y to the a, and that shows the criterion for contractible function space. Now, why does this follow from merely saying this is connected? Well, first of all, omega itself is zero. This Vick's fact is well known to have just zero as follows. So, saying that omega is connected implies that it is contractible, exponential of it. Any space maps into the power set, but the power set is always contracted. So you have a natural monomorphism from x into p of x, satisfying these conditions. The program made this contractible, and it's natural. It's natural. We started off with some existential, essential thing, but in this way it became also factorial. In this classifier of something, you can insert a picture like this. There's a copy of the original state here. The point falls here, but then there's a connection between every point and zero, and this connection goes by the path known as omega. So omega itself is a parametrizing path that connects up this whole thing down here. This isn't any total. But if omega happens to be connected, well then, so that's sort of the minimum.

42:30 From all this geometry which follows comes some very simple assumptions. When I said that M can't be localic over U, the precise theorem is that if I have both of these axes, product preservation by components, and, let's say, truth values are connected, compact way of saying it, then M cannot be localic over U. Because if it were, it would lead to various non-localic totalisms to express this idea of global cohesion. In the case of the directive graphs, which, as I said, is a surprisingly useful example to always keep in mind, there's also a further punter down the upper street, and that in turn determines for sure, and a graph is obviously an unsolved. To think of this as a sort of natural corollary of Cantor's, it seems like the lecture has that, it surely has that. On the other hand, it's hard to prove. In general, this exists, so the techniques, the instructions, you might support them.

45:00 So in those four left sluts, assume we have those four adjoints with those two axons. The left most one is the first product, which wouldn't follow from adjoints alone. And that the truth value of it was that there's lots of connections. See, this idea here, by the way, you can also kind of imagine this. X might be a disjoint sum of things, like it might be Ireland and England. The fact is, that's included in something larger, which is connected and can actually sail across, so the idea that we're all, all of us are connected, even if momentarily we look disconnected, this is somehow part of the, part of the intuition behind this. I didn't say, in fact, that you could sell those two things and put them together formally as a coproduct, you haven't done too much, but now if you put that inside a connection, you have them. The possibility, if you live in Philadelphia and somebody else lives in Cambridge, these points are far apart, and you think, well, now there's actually a real world in which people are building airplanes, flying airplanes, operating airlines and airlines' tables, blah, blah, blah, blah. What we really care about at certain stages is simply the fact that there do exist just flights and somehow this abstract idea that this union can be embedded in something. Connected, and that something connected can be covered by a more detailed story, isn't it? The first step is to know that they must be connectable somehow. I always worry about how to implement it. But again, that approach only works with strange action. Truth is connected.

47:30 Truth can be transformed into false. Classical logic doesn't. Now, because I, well, you could say there's a third action, mainly that this is a full inclusion, and therefore this is a full inclusion, and therefore these three things, if I ignore the counting the trees part, these three, this is an example of a unity in identity and adjoining opposites, as in my paper on that subject, and it's worth thinking about from that point of view as well. From points to components. Every point is in some component. I did a unique component. So there's that map. It just follows from the adjectives. And dually, I see, the discrete space maps into the cotascrete one. If you take the graph which is just the dots, and the graph which is just the same dots with everything connected uniquely, then there's an obvious inclusion one way, so no maps the other way except constant ones. But there is that map. So typically, these maps don't have sections. That is, given a component, there's maybe no definable way to put a point in it. This may not have a retraction. But I do want to consider, partly, I'll try to cite it when I say so, the further action that I call the null-stellar dots. Null-stellar dots, goes back to Hilbert, this concept, is the idea that a non-trivial algebraic space Must have points, possibly defined over some extension field and so forth, but somehow must have points. So to say that this map is epic, the natural map which exists automatically by the headphones, to say that that's epic is a kind of internal logic expression of the idea of every component that epimorphic,

50:00 Now that's entirely equivalent. This one is mono. Think of examples where you have all these functors, but this natural transformation is not mono. If you have every morphism between a pair of left adjoints, then you'll get a monomorphism between the corresponding pair of right adjoints, and conversely. So the middle-stellar massage is actually equivalent to... Going back to this, I introduced this idea of quality. The situation where the adjoints coalesce into two. I just have two punters, but they're both left and right-handers, and of course it implies the action of the camel over the street, and of course the idea, it applies not just to topos, so what I want to say is, somehow the desideratum of homotopy is that it should be expressed in schools, so with this, the quality is kind of like K-theory, in the sense that, let's say if you ask yourself, what's the dimension of a vector space? Well, it's a certain natural number.

52:30 So if you look at modules over a rig, even a ring, projective modules over a ring, you have to first decide what's going to be the rig of values of the dimension function, namely it's K0, right? And then there's the math that assigns to each object its value. So in the same way, this definition of quality might have sounded abstract, but then completely, you can extract that from a particular situation by a functor, and along that functor... Each space will have a certain, will be measuring the quality. So the idea of the homotopy type is to measure the quality that's known as form, but in fact, this tends to, and the classic case does, satisfy that axiom. There's a pair of counters to sets, which are adjoint on both sides. Now, of course, in some sense, Triguli's truthful points, because you put x equal 1, If you think that the points means maps from one, then certainly that is the right edge of an inclusion, x equals one, and the points of y in the sense of this category is just the components of y, so you've got the full essence of a... and then the point is that through constability or finiteness or various deeper regions, anodyne extensions, etc., etc., to actually achieve that... If not for this, it depends on what M is, but at least for a further thing, so that oversets, well, over U, thinking about the desideratum of this category, but the fragmental, non-relativized, these two functors are added on both sides, so it becomes a quality over U, and the homotopy type, which is this functor, measures, specifically, extensive qualities. Why have I said extensive? Because it seems that associated with any such situation, which is almost obvious, if these two functions aren't equal, let's take their equalization.

55:00 Well, more exactly, let's call it M sub epsilon. Epsilon is supposed to remind you of something very small. This is actually a connected surjection of toposes. This ought to be true in much more generality than I can state it right now. We've said if the notion of Codas breed exists, then gamma lower star or points functor have two co-limiters, so there must be a right edge on it. But now, to show that the inclusion is left exact, we have to show that these x's are closed under, or opposite, closed under products also. We get that this is a geometric morphism. Geometric morphism is being intensive quality, being a quality where this has, by definition, it has the problem, but you just have to check that the two counters are still the same two counters.

57:30 I proved that this is a geometric morphism in a much, much, much stronger situation, mainly the Null-Skeleton Sots. The Null-Skeleton Sots, mainly that this is always epi-morphic, sub-capillary words and isoms are very strong problems. M sub epsilon, as a subcategory, is closed under arbitrary suboptics. Nice exercise, just assuming that you have, in that way, used math is then actually opposed to things that are fixed and preserved under arbitrary suboptics. So that in turn implies that's much, much stronger than saying that you have a left-hand joint, that the canonical math is surjective, right? It's so-called epireflective. So epireflective... Offward theory, which I joined, we actually get an essential, a locally connected morphism, we interpret that as being the intensive quality of the arbitrary spaces. What do I mean by that? Well, clearly, in this situation, I don't just have the two functors, so really, I should call it the whole natural transformation.

1:00:00 These, again, these lifted things have a natural math. Epsilon, lower star, Epsilon. And so, if you like, that's the full measure of the intensity quality. Or, again, let's not insist on that I define what substance and form are, but in a certain limited context it's a useful contrast, because what the homotopy type does, you see, through an interesting space, is it throws away entirely what it's made of. It looks only at how it's external form. We want to study, we need to study that as part of one aspect. On the other hand, what the intensive quality does, it doesn't care about the gross form, but it insists on, in some sense, what it's made of. The sort of thing that it could be made of is plumb catastrophes, in that context. Intrinsicism is part of a space. That's the same point. So that sort of thing that we play. So, can you recover the original space knowing this function is not consistent with the left edge joint and the right edge joint, but the left edge joint is not, I don't think, and maybe it's true that in some cases this function has an edge joint. To finish off here, I want to go back again to the directed graphs, but all this makes very good sense there and it seems to contain some information.

1:02:30 So, first of all, what are these? It's got dots. If there's one point where it's entirely of loops, then if you get the point, you might even, just to keep something in mind, call it an atomic number. But then it has this extra, important extra feature of a certain number of loops. Different points in the same body may have different numbers of loops. So again, this is just a sort of suggestive way of thinking about it. So now, if you take an arbitrary graph, there's this right edge on it that just extracts the various parts that looks like that. In other words, it just takes all the loops that there are and forgets about the rest. The components function, of course, you know, it takes the components, but it keeps, at this level, it keeps all the original, because they have to be reflected. All the original arrows remain, but of course they've all been turned into loops, so if two points are on the same component because of some arrow... Well, that becomes one point, and that arrow becomes a loop. So there's a huge increase in the number of loops, and that's, I'm talking about the epsilon lower street. So, thinking about this, okay, let's, for example, the value object is actually a code generator. Every graph is a sub-object.

1:05:00 If we apply the two factors to it, that is epsilon, what we find is that, well, in the first case, there are two points. But this point still has an electron. It shows that these two points are no longer connected. And on the other hand, since I did have those connectors, and on the other side I'm going to do it at just one point, because this is connected at just one point. And so one point is, so to speak, the original loop. Well, not so to speak, because if this canonical map picks up the original loop, So again to continue this vague physical it's like the you know energy becomes mass where the forces are really particles in certain contexts the connecting arrows themselves become on the other hand it is very suggestive well going even further with this suggested there's always a there's always a problem in the way you understand lower street, country star, blah blah blah the connotation it's very confusing so you need a suggestive name. In this case, the suggested name for left and right adjoints is hot and cold, hot and cold. If you have this, so this original body, if you superheat it, so the idea of superheating is that the atoms have become so far apart that they no longer interact. On the other hand, if you supercool it, a huge drop of, because there were many atoms before, but as long as they were interacting, one big drop.

1:07:30 And so this morphism between them tells you that it's not determined by this thing. You have to go back to the original body to see what's going to happen. Go from your superheated experiment. There's going to be that map there, but that involves additional information, knowing the nature of the body. So now we're trying to see whether that analogy continues to have some sense in more complicated cases. Of course, on the one hand, everybody knows that these are truncated simplicial steps. What doesn't seem to be widely known is that these are the canonical schemes on delta. Canonical topology was introduced. We haven't talked about it much because it's a topology on the site. Now we talk about topologies on the topos. Well, in any case, there is the idea that you could have a subtopos with the property that every object is a quotient or is covered by things in the subtopos. When you say covered, it means some direct limit, but that direct limit is computed in the big topos, but in the small topos it would be out. So if every object is a direct limit in its own category of things, we can say that this is a subcanonical. At least subcanonical, if not canonical, has a very clear meaning. And in some cases, the smallest subcanonical makes sense, and that's, in this case, the canonical topology. It just gives you this. The directed graphs appear as canonical sheaves on lots of different model categories, and maybe that's why zero to proximate is u, u to the, if you go further, on different methods.

1:10:00 And then when they come down by gamma in sets, this preserves all the structure for a regular category. Look at the collection of all such things by slicing them by an arbitrary object, not necessarily full support.

1:12:30 This is a collection of functors, sets, and archetypes with finite limits and co-limits that all the sentences satisfy for the category. You can much simplify that definition, but that's in the spirit of the definition, right? Once you know the things like the additive structures is also part of that. And you find out that you don't have to worry about these things. Just take one of these things in the sets, represent it as a regular category. You take an object in your abelian category, it has a group structure therein, actually it's called an addition to A, and that is the abelian group structure here, and that tells you how to lift this up to the Gippo function here, which faithfully preserves everything in sight, and there it is, there's the full, not the full, the exact compelling theorem.

1:15:00 It turns out, you're not using the distance of... The technical fact finally is very useful for a certain point, before you know it as the problem is, you don't know yet that it actually has images. What's the technical definition? The technical definition is not because they're interesting. So, the quick and dirty ones, in this case, the first of them, take a limit, delta plus or zero, and this doesn't finish. This gives you what you might call a relative capitalization of a little single underline of points in here. Countably iterate this and that's what you're... Ah, the construction of the slice categories that we've done, all this goes to a wonderful theorem.

1:17:30 If you apply it back here to the helium category, that you can at once for every small helium category take in groups which now acquire some functorial properties. If you have a pair of helium categories with an exact, you get a natural transformation. And you can play the same thing with regular chemicals. Comptorily better for each one, almost. So, junction, existential quantum. The next step is to add some disjunctions. The lightest, the fact is, we're not going to be happy very long with whatever these things are. Maybe that means a distributive lapse. That's making it positive. A little calculus of relations. And it's one of the first constructions anybody ever thought of. I remember getting a paper in the 1950s. How to start with a candidate category and move into the category of matrices. And that's what you do here. There's an easy, positive completion of preload, right? And when you reinterpret this wonderful fact that a capital...

1:20:00 Sorry, this capital is complemented in the form of a generator. Now, a generator in a strong sense, which means that they can cover an absolutely preserving journey, then the resulting set here is named by a map from Hilbert into a... And so, again, you can just say what you can and say, start with a category in which every statement is one.

1:22:30 Then, of course, you're doing classical logic. And so, practically, you're welcome to do that. Because if you do have, and so the representations you'll get here, all the filters, are collectively faithful. I wish, I mean, some of you have heard me, and used to agree with me, starting at least 30 years ago, why are we stuck with the word intuition? We are. I don't think there's anything we can do about it. But it's a real handicap. If you look at what Brouwer had in mind, we don't have that in mind at all. ... which some people consider a contradiction in terms of... ... which is what Heidegger did. It really has made life more difficult for us. I thought of things calling it positive logic or something like that. Hear, hear, hear, hear. Positive logic. But that in particular. And it may not be too late. You know, if we all agreed to it, I think it would work. I don't want to be equally bad. I'm not going to essentially end up taking the position. But that comes with construction.

1:25:00 Something exists without being able to do it. There are times when I feel like being one. Let me point out that one of the possible applications, one that a lot of people are working on, is to take things like the existence proof and use that for what sometimes is called automatic programming. You know, supply a machine with a intuitionistic proof. If something exists and the machine extracts a program from it, and it's true for everything I can construct, then I'll say it's true. The flip side. And essentially you're saying nothing exists other than things I can construct. ...holds himself an arrogant constructivist. And it seems to me that there are people who sort of want to claim to begin with that they're humble constructivists, but there's a tremendous tendency to slip into arrogant constructivism without being aware of it. Humble could be translated as mathematician and arrogant as philosopher. Whatever constructivism is discussed as a philosophy, it's more of the arrogant nature. There are those who really seem to believe things aren't true until they have proofs, which as far as I'm concerned is the ultimate, and that you can go to arrogant constructivism.

1:27:30 I'm not a Platonist. I understand the problems with Platonists, but to go to the other extreme is as bad as hell. So the name I've been using for the categories that embody the first order, standard first order logic, intuition is the first order logic, is the nice old word LOCUS, and so forth. It's LCCC just locally in the sense that given a math from A to B, In the category sets, so this is going to take a pre-reader, pre-logos, can be basically represented in a gavel. Punters, these set-ladding punters do still have, can't do some language here.

1:30:00 Let me say that a category is projected and it's one of the nicer, that's too big, we'll cut down, like so. And the maps of this category are maps between the focal categories that make this point. Now let me call this boldface C for the moment. All right, so it is too big. It's a class. So cut down to, at the moment, I mean, you can get more precise, but for instance, cut down to the vocal categories that are no larger than A in the cardinal, and representations into the... In fact, I think I want C, not C-O. So, and what's this? You send an object A, it's got to go to a functor, a piece of that, of a vocal category as defined by F.

1:32:30 It comes equipped with a puncture from A into it, so maybe I'll call this S for the moment, and this is gamma of S of A, where S is part of this structure, in fact, so I'll say gamma of, as defined by F, of S of A, and this is covariant gamma. Then it becomes, amazingly enough, just a matter of identity checking to see that this is a representation of Lowry. This preserves all of this structure. The objects of C are collectively fatal. The functions of the form EF and then followed by SC. This family, as S runs through C, the objects of C is fatal. And this is not enough. You need to know that if you pick any F, that's from there on. If you look at the collection of maps and then go on to whatever you call that in the category sets.

1:35:00 This is a faithful representation of that. Faithful on F. ...linguistic problems and you may as well assume A is countable, and I could just stick to countable. And the naturality of this thing that was handed to one by nature. How do you sell that to somebody who knows about cryptography? So, now we switch categories or something. For that name in this context 30 years ago, let me use the name I used there. If you have something here, there exists a map from C1 to C1, T of this, it's called this F, T of that plus, I'm not sure, what opens it, it becomes a matter of identity, this thing is a preserved representation of, preserves everything up to universal quantification, it's remarkably easy, of course, I guess you didn't need any of all of this, but that says that I can get into a category at all.

1:37:30 How can I dominate this by various things? And the first thing that might occur to one is that every category is dominated by a tree. So let's think countable categories. See an arbitrary category. Consider that this will be a non-arulia tree. And consider its objects to be frequencies of mass going up. Sometimes I like the trees going to the right. You look at all the nodes of this tree. Are named by finite composing strings of maths coming together and it simply says I don't care what you think you can dominate any category.

1:40:00 Certainly crystallology, it didn't know that any and is dominated by you have your okay then binary tree up here all downward sloping uh branches are going to go to identity maps what you then will pick up here is uh this will be a one that will violate this this is often called It is faithfully representable, not necessarily root, right, so at least to think of it as a common collection of rooted binary trees, in which case you could review that as a Cartesian power of, because it wasn't spatial, but it is actually, and that rooted tree is just a partially ordered set, and a partially ordered set you could always view as a space, so you're looking at a very peculiar space. It's the nodes, it's the nodes of the binary tree.

1:42:30 And the open sets are, well, if I'm drawing a tree this way, are the right ideas. Basic open sets are these things, the principal, subtrees, take a union of those, and those are your open sets. Sheaves on this crazy space, by the way, notice it's a focal space. It's maximal. It doesn't sound at all geometric. So, you notice it's not sober. I don't need to have the general definition of sober. Let me just, I'm going to look at it in a special case.

1:45:00 You note that different spaces can have the same lengths of open sets. So you can have an x1 contained with x2, so it can induce now the opens of x2 falling on back to x1 with equivalence. Understand that all spaces are t0, which means that you can distinguish points by the open sets that contain them. If it's not t0, identify things that are aligned exactly the same on themselves. And also, given any space, there's a maximal T0 space that has the same open set. That's what the simplification is. So you're adding as many points as you can. You're trying to optimize the geometry of it, holding onto the open set. And in this case, what you get, if you don't need to know the general construction, is you have to add a point for every long chain of points. And that turns out to be a copy of Cantor space. So that may be a little bit more formal. Before I'm going to be done, I'm going to want to take this like two-element set of two things plus and minus, maybe plus one and minus one, if you insist, and what this is, is the set of finite words to union the set of infinite words, better notation than this, set of infinite words you call sequences, and so, as well, I've already told you what the open sets are, right, take any right idea on this tree and not only include all binary tree nodes. But all the infinite strings that are eventually in that open set. Which means that if you cut back, this is a copy of Cantor space. And these things are your basic open sets. And that this has the same lattice of open sets. I mean, yeah, that's what it is. I apologize.

1:47:30 And then somewhere you have a plus one and then all minus ones after that. You have a zero here and all plus ones. And, of course, the same thing with the pluses and minuses. That would be two and the other one where you put the standard in the unit of the book. That would be two, zero, zero, zero. The same as zero, two, and somehow get something here.

1:50:00 And here the representation states that something in here is either a finite order of pluses and minuses or infinite order or infinite strength. I don't think I could figure it out with something like 20 minutes after I had it. I had some pictures in mind, and I had some idea of, by the way, this is worse than piano, but it's untamed. Apologies, you can do what I apologize. But it does give me a nice representation here. There's one of those moments when you go, hey, I can solve all these problems by the following definition, find out it works. And then as far as I can tell, everybody else does it very often. All of these things have totally lost the thing that made you guess in that definition. Does this have anything to do with the fact that tree value logic is simple? I don't think so. So everything you're going to do is extract, throw away all the zeros. So let n sub 1, n sub 2 be the indices where such that a sub n i... I'm going to send this to a string that would sound like, I could just say, m1, m2, etc., n i, i goes 1 to infinity. Maybe not to infinity, but this might be the nth string, the finite string. Obviously it wouldn't work.

1:52:30 You twist that, depending not where you are, but where you last were. Where, by convention, n is a 0, z is a 0. What you get is an ordination of pluses and minuses that will be the same all the time. Then it becomes a sort of thing you wouldn't want. And so this gets you from into Hulson space. I'll say she's on Hulson space. And then she's on... You might notice there's exactly one point that goes to the empty string, that's zero. That's the last timeline. It's so simple. Hulson space is going to have at least, in all the other words...

1:55:00 So for instance... No, one half. I discovered there are some people who are worried about the size of the spaces. It's easy enough to see that if you pick a subspace here large enough, you're going to hold onto the total. So for all sorts of subspaces going on in the government, this is no longer open. You're going to hold onto the faithfulness, and that allows you to say, if you're so inclined, that you can get into the space of rationalism. The lesson being that you have a completeness theorem. Oh, so why does a completeness theorem take the free locus or a first order theory and essentially faithfully represent it in, say, sheets on those intervals? And something is true of the free thing if normally you can prove it. What I'm saying is that coincides with being true in this particular lecture. The free locus will have a canonical model. This is one of the ways, you know, time is essentially up, but one of the better ways I know of Witten's category theory is this idea that for first order theories you don't have three models that can be categorized in a sense over any particular category, but if you enlarge your universe to allow different categories and talk about, well, then there's a nice treatment, there's a particular locus for the particular model which specializes to every other. And so when you basically represent that, you're getting the nice, ten minutes to go, nine minutes to go, maybe there are a lot of questions, so I'll stop.

1:57:30 One thing is, for example, I have the maps go down and the objects are on top and bottom. So these are, this is a map, here, this is its domain.

2:00:00 We can check that this is the model structure of our category. And also, this we might call the model structure in which the co-vibrations are point-wise. Vibrations are not. There is also another model structure on this with the same weak equivalences in which the vibrations are point-wise. So the vibrations are vibration here, vibration there. And the co-vibrations are sort of dual to this. So you might, as part of the exercise, check that that model structure exists. Okay, so let's let A be the class of co-vibration, and B...