Algebraic set theory / FW Lawvere: Topos theory, algebraic geometry, categories

0:00 For many purposes, it's sufficient to have a universe in an object that is a monomorphism from its own solar object, however, it turns out that it's no more difficult to require that you have an object such that every other object in the category has a monomorphism into that object simply because Certainly, if you have one of these, then you have one of those. And if you have one of those, what you can do is you can cut down the whole category to those objects which have such a monomorphism, and the result will again satisfy the first two axioms, C, S, and P. And so we might as well take the easier-to-work-with condition that every object has a monomorphism, A, B, U. Now, some of the models have not been struck. We'll actually be of this form and then we'll cut down again in order to satisfy this condition. So let's call a universal object U such that every object in the category has a monomorphism to U. For axioms, I now let me state a couple of facts about these. And then I look at those maps, which are small. These are the maps in small maps. These are the maps in S over C.

2:30 Then I have a system of small maps. Every slice of a heightening category is a heightening category. And now what I'm saying is this system of small maps satisfies the axioms S for a system of small maps on C. Moreover, this also satisfies P and satisfies U. That is, if I have a universe in C, then I also will have, if I have a universal object in C, I'll also have a universal object in the slice category, namely, U cross X will be a universal object for the slice category, and so CX with this structure is again... Did I give this a name? I guess I called the system there a class category, so C is to have a name for it. It's again a class category. What you can verify here is the axiom P for the slice category. And this one I would like to give you as a homework problem as well, and I'll give you a hint for P. So we have a map here F for the slice category. And we want to think of this as the fiber-wise power object of this map.

5:00 So that is, to construct it as pairs, x and x, below f inverse x would be one idea of how to do this. Now, unfortunately, we don't always have this map f inverse x. If we had a math, F inverse, which takes every X to its fiber as a sub-object of C, this would be exactly the characteristic math of the small relation, which is the graph of... How do we have such a math when F is small? If F is small, we're done. However, what we can do is, if F is not small, we can observe that we do have a, in general, we have a letter F. So we can construct this by saying x alpha f lower strings of alpha, the direct image of a small object, is below the singleton, where the direct image map takes us here from the x. You can compose with F, and then you can take the image factorization, posit, that's of course the direct image, F lower streak of A, and now if this one is small, then the map here to 1 is a small map, and now by the quotient axiom, this one is also small, and so this object is small, so we get a small object.

7:30 So we do have a direct image operation like this, and of course you can do it with parameters, you can either do A as an argument to construct this map. And then, using that map, we can construct a power object in this way that can interpret this formula now. So, C, interpretation of the formula is, of course, this. D, we can construct this way. We're pulling back x. And now we just have to interpret this condition. So we have the x here, which is up, which is turned around because we think it's in the left. And now we'll hit B as we pull back. Now we have all the units we need, and then we project out that first factor in order to interpret the sum. This is a construction that I've just done. I've just stated a couple of propositions. Some of them are sketched in the notes. Some of the sketches are given in detail in a fuller set of notes that's available from the Albrecht Set Theory website. It's a paper by four authors, South and Woods, Simpson and Schreiter. This will give you complete proofs of anything that I don't have time to move on. The first is that if we have any small map, then we can take pi from this.

10:00 This is a small map. It's small. I would preserve small maps. And it follows in particular from this, I have small objects, B and A, that I can again build with exponential, and this will also be small. So, this is small. For this and some considerations regarding P , you can show that P is a sub-object classifier for small and it follows now that the subcategory of small objects, which I'll write this way, is an element because we have the exponents for small objects and we have a sub-object classifier for small objects. What is this? Well, this is small relations, one from U, the second projection. This is small sub-objects, but these are exactly the objects of sets.

12:30 So, P of U is an object, but C, and I guess what I'll start my first lecture with next time then, is the question, what is the set theory, the elementary set theory? My question, in the sense of the explanation, before I possibly interpret elementary set theory into the universe in a category of classes, what is the set theory this category of sets has to do on in relation to just to have practice and to consider, in my view, this, the identity of this.

20:00 That implies, of course, you'll have the same records as many of you think have, depending on the, some kind of, minimum or minimum.

22:30 In fact, there's another way that this is often called the notion of split epimorphism, P, although P comes equipped with a given section. In terms of split epimorphism, P is given, and again, this is kind of the third way of particular totals. As C, I'm just going to take the arrow category as the property that punctures from it are the same thing as morphem, and so there are two. There are actually three non-trivial concretes there, but forgetting about E, there's a concrete which attributes the name of P, so it's a semi-symbol of P. If you say the morphism in a category is the concrete of two, then this internal picture is the internal picture of. Okay, so there are two concretes there.

25:00 So here we actually have two punctures, and so multiplying two times three, it's six. So there are actually six punctures that are introduced here. Again, you could find two more if you could see as well. Six punctures. Now, the two inverse unit punctures are just the obvious thing. I mean, this is roughly what sounds like a tautology, but if you have a diagram like this in the category U, well, then he isn't math, is he? There's a total of four slightly non-trivial functions, and actually, those methods actually compute as a tensile product or a direct number, which is like a computer that will generate the data.

27:30 And this, this is a, I think this is an example of fundamental math, because if we were taking an arbitrary map, and then we were producing a set of things, these other things are the basic tools in analyzing that, ways of analyzing a given map, or pictures of a given map. We have the pictures and the two standard, the ad-hoc combination of 2x minus y. So, so these are, these are important things to know about, because we use them all the time. They also have this Y is D, column D. Namely, I call this column D, column D, the Dirichlet, the Dirichlet. Sometimes that's the concept of objective number theory, namely that all of the basic relation of abstractions associated to categories like these grids, grids.

30:00 One hint is, if you make up a really bizarre notation, so N to the power of minus S. That means, for example, it's something that's got n points, point and set, point and set with n elements. It's uniquely a linear combination, or uniquely a sum of representations, set coefficients, basically. So if you represent those objects in this kind of an expansion, and ask yourself, how do they multiply? Take an ordinary Cartesian product, two objects, with each one expanded. How do you expand these? So then, going along with that, I ask the question, which certain object here would you call the re-monitored function, the strictly categorized? When I came up, I remember what the counter-categories were. Like me, in describing academic categories, one of the first things is you've got to isolate this two. Which category is two?

32:30 He noted that it has exactly three endomorphisms, three endomorphisms, but he said, therefore they're characterized as two, and then I got to thinking, well, no, this one also has two. So, the first time I noticed it was as a concrete thing as some reasonable conjecture. And then, three years later, I had some outrageous conjecture about the problem of equality and equality is a large and small. I had some outrageous conjecture. Peter Johnstone came up with a counter-example to my conjecture. His counter-example was any other than the single, virtually, topo. And now if you look inside it, as I say, there is one object, the Riemann zeta function. And of course, everyone knows the Riemann zeta function is actually the Euler zeta function. And the thing is that the Euler product form that is true, continuous exercise, proves the Euler product form, proves the formula that we mean in the objective sense, there is some kind of isomorphism. If one of the objects is the zeta function, maybe the closest objective approximation is the right-hand side.

35:00 The notation 1 over 1 minus x really does the shuriken for that. The word Algenburg, the three monomers. This is the way that we compress really infinitely structured algebraic relations because we just have 1 plus x times 1. On the one hand, this is a well-known equation that the free monolith satisfies, but on the other hand, if you solve this with the formula, you have a different set. So obviously the objective of your formula is that they have an objective of differentiation, and if they involve this expression, then that's the thing to take. In some sense, a rather strange calculation. On the one hand, there's a free product, a very extreme kind of thing, the Cartesian product, say, extreme within the category of monoids, which of course, if you look at the commutative monoids, the products and co-products are the same thing, and the category of non-commutative monoids, the Cartesian product of free monoids, seems like a rather strange, because if you take a finite product of free monoids, then because, but then, so therefore...

37:30 The top has a multilink. All these multilinks run together to get the tree of union. And the zeta function is a tree localized with multilinks. I'll explain that in some sense in a fairly different way. So actually you get not just an isomorphism of sets or of objects and so forth, but you get an isomorphism of monoids. I forgot one ingredient. You have to take a limit, a very simple option, generated by some crime, if you take a bigger, actually there's an injection map going up, precisely because they're monolized, so they have a zero on them, so you normally think a bigger product projects down, but you take a direct limit from that sketch, so the idea you see is that when you see this infinite product, an objective number theory, quasi-infinite...

40:00 There's a unique conker in many categories in one. On the other hand, an object in a category is a conker from one. And so if we take this smaller object, we get there. So by taking all the conic tensions, we don't get six conkers. They get far fewer because of the massive collisions of the adjuncts. So there's the trivial thing, the identity. In the picture, the typical picture, the internal picture, there's a dot, which are the pitch points, and then everything else that goes into that. And that would be considered to be either a castle, which is where all the pheasants would rush into the castle and travel.

42:30 You have this simple expansion. Every object is a direct sum of connected ones. Connected ones are nothing but a number in a way because, at least for abstract sets, it's more complicated. For abstract sets, it doesn't matter which point you choose. It's essentially just how many components are there, and then for each component, how many components does it have? And on the other hand, there's a sum here. Which is both left and right-hand joints, with the same function left and right-hand joints. We should obviously call it the set of components, therefore components. On the other hand, the right-hand joints, sort of the Cantorian idea of extracting points, means six points and they're the same number. So that the set of components and the set of fixed points are actually in bijection, even if the picture has a function to it. You have this, all the admonitions are the same, you've got to hear it, you've got to be aware of it, and say that this is a triangle of space, and it's called, in context, I think, it's not positive.

45:00 What does that mean? It's here. But notice that it's, in a way, the fact that you have one so-called, and another so-called, it's a full subcategory, all these admonitions that are one-sided. There are three grounds for us to have an essential inequality in this category, where central, or the center of any category, means the natural endomorphisms of the identity function, and if you have a natural endomorphism of the identity function, well, you could take its fixed points and so forth. This inequality, whether its fixed points agree with it or not, anyway, splitting that inequality can actually create this other category of knowledge. Actually, it's a category that's carrying a sense of input. It represents a certain type of quality, of course, over the days. It somehow generalizes, miscarries you along the way.

47:30 You can imagine that it's, well, how many different sizes do we often have? Five or six? It's probably more than that, but at least every already noted size. There's usually some way that you can stretch it so that it really says something about every topo. Every topo partakes of the intuition, in fact, particularly from the head, the heart, but the mathematical way distinguishes the mathematical property. I claim that there's some topos that really should be called generalized spaces, but they're not called generalized spaces for many reasons. Others should be called categories of spaces because they're simply mathematical. They should be called categories of qualities, but very, very specific. And of course, really, yes, they're really properties and morphisms. How about witty topos? We really should mean utopos as the arbitrary rotative topos, because Roglic himself introduced this whole technique which he relativized. In fact, there are some very crucial examples of rotative topos which should not be defined over understanding them.

50:00 So the qualitative distinction is here. Now notice that I just wrote down some... If we have one topos defined over another, we have a gamma lower star. In different contexts, different works. So really this map represents some kind of contrast. So if we call it global sections, for instance, that's why. It's a view that you get by applying the internal X and then transporting the internal X along the particle series. So global sections, in particular, global sections... The term mobile section comes under. It expresses the idea that objects in E are more variable. We have the one that has the more constants, the ones in E are more variable. So this has to do with the intuition that, again, sometimes quite literally, in today's sense, a topos is considered variable.

52:30 We are more active, more active. So already, active and variable should be able to express in terms of action on the mathematical part why variables require their life. The gross way of their life is to go to the example set of the ground. On the other hand, the third way is especially if we're thinking of algebraic, geometry, or cosmological, these are spaces that have underlying sets. So in that case, what we're saying is really that he is more cohesive, probably intuitively the idea of several domains.

55:00 Cantor's really big move, not a big move to hear about, it was just himself thought. The great discovery was that you can actually extract the same abstract things that have no cohesion. So, there are these intuitive ideas, there are these words that we use, what are the actual mathematical properties of the topos of the Ewell-Morris diagram. There are special cases of variables. There is the trivial action. And in the case of, there are the sweet states immune to that special condition, which adds a subcategory to the case.

57:30 So, it's not a great geometry though. It's really an expansion down the drain. The geometry is clearly a special form for totalist theory. I'm sure that's why. But again, well, this really does have to be looked at more closely. Special properties. And so, those of you that in 1973 observed another poem, Federico Gaetano, Part 2, what's special about, what comes up with, what you translate into our modern world experimentation by saying that it has to do with extensive categories. By here I mean categories of algebraic space, monastic, patito, cosine.

1:00:00 Again, another story that I'm talking about. So now, all extensions, I'm sure that they'll affect something that we've experienced with the Austrians and Americans. There's a lot of, by now, a lot of papers that use this term. It's actually a combination, and this is even more abominable, and extensive. They've come to the idea of large-study X twice. As it's been explained, lack of Australian time for final limits. But one need not say that this is related to each other. It turns out that both products have a special property, which is the way to answer to the following thing. It's two different questions, but they're almost the same. But suppose you're on the side of C, category C, to find a product. And if you look at the category, you can never really think of C as an algebraic theory.

1:02:30 So you take the algebraic, the product, simply the product, and you're done. You can ask when is an algebraic category. That's a certain condition on how the products are in the same order for certain problems. If you have a category C with the products, you can ask to embed C into some topos in a way that's reserved. In that form, you basically know that the answer is disjoint and universal. The word extensive is not just in order to have a shortening for this, but it expresses sometimes a third aspect, and it's directly about the category rather than about the species or the alphabets. It is that if you, so it's a category with co-products, and if you slice it by the co-products. Obviously, related by a function, possibly move A and B together, it comes out with an A-plus, thus having a co-product, a six-hundred, and so the extensive acting is that this is, and if you want to, you can throw in, so extensive is because of the role of extensive qualities in thermodynamics, so to speak, on one end, and the discharge is, at least in some cases, perfectly consistent quality.

1:05:00 Objective numbers. Objective numbers today that we mostly know about, just like the most logical, is things that are contravariant and both multiplicative and additive, like the random continuous functions or frame of subsets. Covariant. Extensive. Forming, contorting, abstraction, putting the signs of every object A, measures, these measures are covariant. The usual theory in throwing the hammock is that you recognize an extensive quality by the fact that if you divide it into two parts, it adds up. It's linear. It's usually shown against the exercise that you have.

1:07:30 It turns out that the inverse described is actually an extensive category. It has some pullbacks. It doesn't have most pullbacks. And it does have pullbacks along product injection. And so, in fact, that's really the case. Anything over a plus b, you have the coproduct injections, the coproduct injections are horrible, and so you get c sub a to the minus c sub b, and then it turns out that the c is actually the sum, and there was any math, any math with two domains as a sum, uniquely splits as a sum of two maps where the two domains are intrinsically different. So this is sort of the most primitive form of the idea of the additivity of the thing over the face and expressing its size. So, all right, so in particular, if you have many, many extents of that, the Pumper's X product is really the means of the face. So they just satisfy this exponential law. All contemporary companies with effects, status studies, they use themselves, which we call GE for built-in deep-guided, grid-riding, parts of set-asides no-extensive, so that we assume to preserve some, which kind of preserves the parts that Connacht considers limitless.

1:10:00 And so, as you can see, this is the work of the no-extensive category, which preserves Connacht limitless co-products, but both sides are where things blow up. Well, don't guess, because things say algebraic geometry is orbit spaces and space and, say, co-ordination spaces, or you want to glue together, glue together, and then the typical sites are bad, in a way. There's this notion of pre-topos. Pre-topos could be defined as something which is simultaneously exact in the sense of far, and also extensive, splitting the two aspects.

1:12:30 Quotients and sums, and so on. So, of course, the topos, pretopos, are nice because they're duvetic topos, and they're practically equivalent to coherence topos. Canonically, it's quite small as well, but it can be constructed from theoretical point of view. Topos and pretopos are obviously thin. They're small, but they have all the right impacts. This is very bad because you can't understand very well exactly which pretopos it is that I'm talking about. Because you started with different sort of data, so the different sort of data, what it has in common, the whole fight, why aren't the concrete examples, why did those arise, they arose precisely in a sense to struggle with this problem. How can I understand, how can I have very good focus, like in focus. And yet understand those in terms of the site, which is definitely not a feature of those, but in some sense, kind of extensive categories. 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 categories are also good categories in a good and quite different way. It doesn't match up with the typical examples. And then there's a category, which is this. You take K as a rig, and then you look at A sub K, which is a category of presentable K-rigs.

1:15:00 See, what a rig is, is something that if you join in, you have to develop your conclusion. And the idea is to have a shorter term than commuted, semi-ringed with units. Exactly in this form. And I say this very concisely, because the mistakes change all about rings. Community algebra in a way began with ideals. Why are they called ideals? Because they're ideal bodies. Ideals don't form the ring. And in fact, they form, they form a two-ring. A two-ring system sometimes, obviously, most unlike the ring. This was one, but not as a range of this and this versus, but the ribs are quite interesting because we didn't say anything about the multiplication, the fact that you can take anything you want and make it embedded as part of the multiplicative structure even of a category for any day, including, first of all, the...

1:17:30 It's just a set of finite sub-sets of the pinnacle that is under union multiplication. These are still not really so well investigated because most of the books about the exceptions, as you know, the books that talk about the nucleus, semi-rhythm, and unit usually do not take the approach of abstract algebra. Systematic, you can module, you can get three things, enough three things. One of the standard things in an algebraic category is to look for symbols out there. What are the simple rings? What other simple rings can be used? What are the simple ones? One is two. Every point is either a field value point that you want to generate. For rings, you can see rather clearly what those are. They're very special local rings. One's on the right and the other one's on the left. Anyway, the point here is, since these things are constructed from a module, let's say a center module, If K happens to be a ring, K mod knows other than that that he can use as proof.

1:20:00 These categories, it's very likely that when you know, it's just that there's other examples. K-rigs are defined with respect to that tensor product. I mean, they're defined set-to-rigs. Multiplication. A tensor product can use this thing. A tensor can use this coproduct. That coproduct is passed to the algebra system. And therefore, that's why they're distributed well. They're getting converted K as well. That's the basic reason why they self-medicate off. But my sense is, to detect there's something that's slumbering on. I thought it was hero. Again, it's going to have one minute extra time to write the theory for you. On the other hand, since we don't have subtraction, we can't just say it's important because we may not have a complement. There's a complementary pair of importants.

1:22:30 Since these equations here, mathematics, physics, and so on, that means that sometimes the whole discussion of coverage can always be reduced to a single map. The finite, the finite is the period, but on the first part, which is particularly important, it is always which single maps are going to be counted as epimorphic. The inclusion is restricted, so those kind of maps are epimorphic. And for some purposes that I will discuss next time, there's sort of, to me, a sort of preferred one, the idea not that if we have a map, it has an actual section that it's got to be, but what can happen is if you have what I call a stochastic section.

1:25:00 Your alphabet is low and trivial. You don't have a precise choice of something in it, but you have a definite distribution, its own standard deviation. This idea can be expressed in the spirit algebraic context by saying that spec A is what you would expect as nothing but a negative. This is the opposite. Expect it just as negative. So if this is spec A and this is spec B, say, our favorite means B. The learning process is a way of reducing this delay in functions and making it more variable.

1:27:30 I can tell you this also in the smooth context. It has a sense of coverage. The way that it works here is the same as your product function. It is the one that corresponds to the full size of your memory. It is the same one that applies to all any logic. It applies with its speed. Once you have it to pass the test, it needs to be fulfilled, and so this is the way. Now, is there a standard name for this policy? Is there a standard name for this? I've seen this concept, to have a standard name. If you had any, you'd think, well, that doesn't have to grow. It's almost slightly more general than it changes. In fact, mainly, there are, there's this notion of algebraic punctual, where you interpret one algebraic theory into another, and then there's a substitution along that. It's always at the left-hand corner, at the right-hand corner. The thing is that sometimes there are sub-categories, these grid categories, which are not in themselves of the form. That's of course the standard way of getting stuff.

1:30:00 For example, the g of a sponge and the g of a sheet. Those two grids are not in property. The next is less than the one. It turns out that for a function, I like to call it a core. There's a core of any two-rich, but this two is a two-rich. It's both, and the core exists. Since we included the reserves of everything, it still has the same density, so it's also good. Another one is in some sense the opposite. Max is known as less than or equal to x squared. That's another example. Again, we didn't start with any two-rich, but we have a core, and we respect two cores. And, of course, you can get figures of distributed lattice. The study of distributed lattices and its subtotals is going to be a relief of these things, that it's two ways of learning about intervals, like there are cubes and parts of the cubes that are tetrachromic.