Session: Lextensivity Rig Geometry / Internal Categories / Quantales (& others)

0:00 Topos theorists have gotten away from the origin of topos theory in Rotendieck's generalized spaces and categories of spaces by space, a lobe being a generalized space, or a topos being a generalized space and discuss all those things ad nauseum, but ignore to actually seriously consider

2:30 The unique features of such toposes really need to be finally investigated so that topos theory can be applied, after all, to the algebraic geometry where it was born. Now that's just one component of a program for simplifying the foundations of algebraic geometry. Such a program, believe it or not, was launched by Grotendieck himself. I was privileged to, before I was actually stationed in Buffalo, visit it for a couple of days because Grotendieck was giving a colloquium talk in Buffalo where he in effect said, So that the foundations of algebraic geometry should be redone, in particular using topos theory more explicitly than had been done, and so forth. But now, unfortunately, this program, again, the algebraic geometers themselves did not carry out this program, sketched by Grogan-Deep.

5:00 Maybe they didn't fully understand it. On the other hand, it should have been our... In that case, it should have been our responsibility to carry this out, but it still hasn't been done, so we should actually do it. Anyway, this axiomatic cohesion, it should be thought of as at least one ingredient, one background ingredient, perhaps, for such a re-foundation fifty years later of the foundations of algebraic geometry, according to Partly according to ideas of Rodenbeek himself. Now, of course, we consider that the smooth generalization, synthetic differential geometry, the C-infinity category, which we already noticed in principle a long time ago, was such a natural generalization, although of course it has many features that are distinct from algebraic geometry in the narrow sense. So the axiomatic theory of cohesion, for example, tended to geometrical examples more than originally thought of. So I wanted to give some examples which are really more of a semi-combinatorial nature than smoothness, but still actually can be, as I said here, approached via sites of definition. In ways very, very analogous to Rotenbeek's algebraic geometry, this is something special even within the special realm of coding. These naturally arising sites often fail to have even very basic as extensivity.

7:30 And so how we, how we, I was reading the obituary of James Eales. I just wanted to say that the axiomatic cohesion could be again considered as 50 years later. The setting for global analysis, because all the ingredients of global analysis, homotopy, extensive and intensive quantities, differential forms, infinitesimal differential equations, infinite dimensional differential equations, all that stuff is automatically present in any one of these topos. It's just lying foul. It hasn't been exploited. So, setting for global analysis. I want to recall this first principle. Which again is a sum of many of these examples of axiomatic cohesion that topos enjoy. The fact that within the context of the topos, with the idea that typical infinite dimensional spaces arise by taking function spaces of ordinary macroshocky spaces like three dimensional space and the line.

10:00 And so on. But Euler already, in principle, had a much deeper idea, namely that the macroscopic spaces themselves can be obtained by forming function spaces of infinitesimal objects. And that's actually a theorem, an easy theorem if you think of it, that I put here, a ray of geometry over k rigs, and I'll be talking about it more later, you can think k is a field, for example. It's important that it doesn't have to be. It could be any rig, for example, two, that the line that is, and hence all the microscopic spaces, arise as retracts of function spaces, of infinitesimal spaces. And since these infinitesimal spaces are basically Euler's idea that... Real numbers are ratios of infinitesimals, ratios are processes, ratios between A and B are morphisms that take A's into B's in a good way, so ratios of infinitesimals are real numbers and that applies to, so that one can arrive at the infinite dimensional spaces of global analysis in two steps of function spaces, starting from the infinitesimals only. Which would sort of encode a position between Heraclitus and Berger and Russell, namely that you can't actually move even if you're in one place. That's not a philosophical content of these internals. And in my previous paper, which is intact, you see that there's a more general, looser interpretation of this sort of... And finally, the infinitesimal thing, namely objects that have the same number of components as they have points, so that every component has a unique point. Of course, again, the infinitesimals are much more special than that, but at least, even in monotorial topos, it's like reflection graphs, where these generalized infinitesimals...

12:30 Boiled down to just graphs that have only loops and no errors that aren't loops. Function spaces of those do contain the natural site for generating the topos. In some weaker sense, Euler's principle applies even there. Go back to the main train of thought. Why do I mention extensivity? Why did we come up with extensivity in the first place? To work with serious things like adhesion categories and bar exact categories and the like, extensivity seems a very trivial sort of property. Of course, it's not trivial in the sense that it's rare, but it seems so easy. But one of the main points was that it seems to be things that schemes have in common. This, in fact, is an algebraic category, a multi-sorted theory, but you can ask, when is that a topos? Well, it's a topos if and only if the original thing was extensive, but this is one way which is related to what we're planning on, in which the extensivity arises. On the other hand, if this site itself were... Or the opposite of the category of finitely presented algebras for an algebraic theory, which is the sort of typical way that Grosbeek constructs these geometrical models. But algebraic geometry starts with algebra to construct geometry. So you start with finitely presented algebra.

15:00 Well, the opposite of the finitely presented algebras will sometimes be the extensive category. Not usually. So I will call the category of algebra as coextensive if that's the case, and if the category moreover has finite limits, we introduce this word lextensive, putting together lex and extensive. Some people find this sort of thing ugly, but I don't especially mind, so I'll use it to pretend to use it. And then I noticed that this is really a strengthening of distributivity. The mere existence of products together with extensivity implies distributivity of products over co-products. And so the first question is, what is this topos classifying? If we start with the site of finite presented algebras, then we know that the full preaching topos is simply the classifying topos for arbitrary algebras of that sort. So this is the subtopos in the extensive case. What is it the classifying topos for? Well, in the case of Riggs, Riggs is coextensive, so we can do this to it. Extensivity manifests itself in a very concrete way, namely via partitions of unity in terms of disjoint impotence elements of a rig such that the product of any two of them is zero and yet the sum of all is one, then the whole rig is uniquely the Cartesian problem of smaller rigs such that the i-th becomes one and the i-th factor, so that's how the extensivity is seen inside the algebras. So if we want examples, concrete examples of extensivity, we can build on the example of Riggs and look at various elaborations of Riggs, so you can say that this is the theme of the rest of the talk, what are some of the possible elaborations of Riggs which still maintain this feature of extensivity.

17:30 So, well, for example, we can take a given rig and look at k-rigs, just like k-algebras with k as a field, k as any rig. Well, okay, still the same thing is true. The free k-rigs are a special case, of course, of the monoid k-rigs, starting with any given community monoid, and I'll be referring to examples like that. We could consider k-rigs acted on by a given group. That will still be an algebraic category that is so extensive, or a monoid or a Lie algebra and so forth. But there are some examples of these, not quite as obvious, that I will call core varieties. So that's the core varieties from the title I'll get to in a moment. You can ask for simple objects, and of course the simple rings are simple rigs, namely the fields. What are the simple rigs that aren't fields? Well, there's exactly one, namely the two-element rig in which 1 plus 1 equals 1. So that in itself would be a reason to look at two rigs with a special interest because of the simplicity of the scalars. And known linear algebra, the community of algebra, and algebraic geometry. I'd like to point out that the example that launched ring theory was not a ring, but a two-rig. Namely, the consideration of ideals as generalized elements along the taking of principle ideals.

20:00 Even if you start with a ring, you don't get a ring that way, you get a two-rig, which doesn't have that. The sum of ideals, which of course is a semi-lattice sort of thing, and therefore satisfies two equals one. And a third way in which the two rigs came up was in Channing Willow's work in Como 90, Looking at the rig, which are just isomorphism classes of objects, as Grotendieck did, of course, in the linear situation, here we're in the Cartesian situation, but then reducing that rig in two ways, namely making it a ring, like Grotendieck did, by tensoring with z. And getting, therefore, quantity, abstract quantities associated to space as they call other characteristics, but also at the same time tensoring with two and calling the abstract quantities there derived from each object the dimension. And the reason for the two, the reason why the addition dimension rate should be, in other words, upper semi-lattice. Because that's how, intuitively, dimensions behave. The dimension of the disjoint union of two spaces is the maximum of the dimension of the two, so it's an in-quote operation. By the way, the usual dimension numbers can be read off logarithmically from the objects here, to avoid some confusion. Two rigs are an important case, alongside rings, in a way. In the computer science literature or classical algebra, the rigs are called commutative semi-rings with unit.

22:30 That's better. You have a shorter ring, shorter name. It is inappropriate to mean that the multiplication has binding means. In fact, any multiplication you can think of can be faithfully embedded into a two-rig. Given a commutative monoid M, any commutative monoid M, you can simply take the monoid rig of that thing over scalars of 2 equals 1, and that has the very simple interpretation that it's just all finite subsets of the given monoid under union with addition and the usual setwise product as product. Why is this extensivity important? For the program of algebraic geometry, it's because, of course, the algebraic geometry is to construct categories of spaces, which of course we automatically think topos is, sorry, from algebraic categories in such a way that every space has an algebra of functions and every algebra has a spectral space. This is a contravariant adjointness. Experience shows, really, that one shouldn't expect dualities to be equivalences, but at least you can hope that they are adjoints, or even adjoints in which one composite is close to the identity, even if, in the simple case, could both be the identity. So, specifically, we take finely presented algebras of some algebraic category. Function algebras themselves are left-exact functors on that category, and on the other side we take a subtopos of the classifying topos of those kinds of algebras, and the full classifying topos, of course, is all the contravariant functors, and then we take the con-extension and get the spectrum in one direction and the function algebra in the other direction, and so forth.

25:00 The essential need for having a subtopos, not the full thing, that is, in other words, the need for introducing some grotesque topology, is the fact that neither the affine schemes nor the category of all pre-sheaves have the geometrically correct co-limits. If you take the intuition that projective space is gotten by gluing together two affine spaces along another affine space, This gluten process, of course, should be a co-limit. Well, of course it's not a co-limit of atom spaces. We know that's trivial by Liouville's principle. But on the other hand, if we take the co-limit among all functors, that's completely wrong as well. So we need a subtropos for that sort of reason. In other words, a non-trivial global topology. Just in order that... Basic, geometrical co-limits like, for example, projective space will come out to be the co-limits within that topos. Now, okay, extensivity, again, it's a very, in some sense, a very shallow thing. It's a kind of, it's not a first step, it's a three-fourth of a step towards solving this contradiction, this profound contradiction between the two wrong co-limits. If the algebra they were talking about wanted to do something like algebraic geometry over any of them, of course, if you start from the end of, you always have a nice sight, but starting from outside, you have extensibility of the algebra, we can assume that this subtopos is contained in GMC, in other words, the product preserving function. This is basically the idea of finite disjoint coverings. So, finite disjoint coverings should behave well with respect to the algebra, because not only is the function algebra the disjoint sum, the product of the function algebra, but in reality we go in the other way as well, the spectrum of a Cartesian product of an algebra will be the disjoint sum of the spaces. This is a very shallow remark at a certain step, and the...

27:30 It's made possible if the algebra is extensive, if it only has a degree of C. We think of this as the Gaeta-Grotendi construction because in the lectures that Grotendi gave in Buffalo, notes taken by Gaeta, this principle, this fact about the extensivity, not in those words, but it's brought out as As being the sort of first step toward resolving contradictions between geometry and algebra, and the way it manifests itself in the study of these topos by means of the sort of sites is that the finite distance coverage is sort of trivial and so therefore you can concentrate on single map coverage. For deeper studies like the Zariski, the Petal, etc., etc., those are studied by means of single maps because the finite disjoint couplings are considered to be taken care of. But I wanted to pose this as a kind of question of general category theory, namely, all right, so we see that the spectrum and functor connecting the extensivity assumption. It does preserve finite products in both directions and not just in one direction. The two exactness properties that are in common are the pretopos on the one hand and the opposite of the extensive algebra category,

30:00 included by definition by extensivity, but not much else because the algebra category can never be bar exact. This should be clarified. This is the sort of thing that category theorists should be able to do. What's the spec-hunter, say, view that's going from quantum algebra to the pretopos? What does it really preserve? What probabilities do the two categories have in common? And as I say here, I mean, obviously pretoposes are very good. In terms of exactness properties. But one shouldn't conclude that the opposite of the algebraic category is bad. It has different good properties, very different good properties, and as an example I mentioned the whole idea of modules as abelian group objects over the constants, which is used all the time in Bach's thesis and even before that and after that. Because, you see, the pretopos can't support any modules. If you define modules that way in a pretopos, you'll find that there aren't any, essentially. This was already cited, in effect, in Ross Street's talk. Basically, that principle that there are no code algebras, no cohomonoids, and so on. So, the opposite of the category of the 5% of K algebras does support this whole category of modules. This is something very good. That's just an example of a consequence of some other kind of exactness property, I claim, but I don't know what the formulation of that other exactness property is. Oh yeah, sorry, it's an inclusion factor between algebraic categories. It's a full inclusion, so both categories are algebraic categories, and moreover, it's induced by a surjective map of algebraic theories.

32:30 There are lots of other reflective subcategories that are even algebraic categories. For example, a variety is thought of as analogous with closed sets. Then there's analogous with open sets. You can introduce inverses, monoids inside, or sorry, groups inside monoids. There's not a variety. It's like open, you see. But there are even subalgebraic categories. Which are not even, neither closed varieties nor open, like Wilgen algebras, are not a variety inside distributed lattices, which is what I'm interested in. Variety, of course, inside other things. Why? Find some examples of extensive algebraic categories, more general than just k-rigs for some k, or even g-group. There are several others such as inside extension. We start with the co-extensive category, algebraic category, and look at a variety inside it. It may not be co-extensive again. The sum of them will be. A simple-minded, i.e., categorist way of making that be true is that the inclusion should preserve limits of both kinds, not just one kind, because, of course, going back to the definition of extensive, you know, when you say that an extensive category, a map into a sum, uniquely splits the domain as a sum, well, the factors that you get...

35:00 The sum ends that you get are pullbacks on the objective. So a function that preserves pullbacks and sums will preserve extensivity. So the insight in the co-extensive category for which the inclusion preserves co-limits will again be co-extensive. It usually fails, but there's a standard categorized way to ensure it. Namely to say that this nocturne should have a right adjoint. So this is what I call the inside a given one rate category. The inclusion has it further and so such a variety will be, such a category will be more extensive if the original one was. Now the question of which algebraic functors have right adjoints was studied by Gavin Ray. Of course there are a lot of full inclusions. Those are, it means that, let's say we think of the single-sided theory, some equation involving just unary operations with the property that if you look at the elements of any algebra that satisfy those equations, it'll be a subalgebra. That's the core, the core of an algebraic sum. It's the core which satisfies this additional identity. So those equations become identities in the... In the subcategory. But if you take anything in the big category, take the elements that satisfy those equations, that should, again, be an algebra. That's it. So there's my old joke there, you see, that a core is the first four letters of the word co-reflection. That's a kind of algebraic equation.

37:30 You can extract a core at the once of any algebra onto the subcategory of algebras. And the subcategory of algebras, well, of course... So the Frobenius analysis of algebras or schemes and so forth over finite fields is an example of that because, as you know, you take certain powers and you can take bigger and bigger powers. This is an example of Q, which means, more generally, you take bigger and bigger sub-monoids at the center of the category, and then you look at the fixed points, and this gives you an analysis of the arbitrary algebra or arbitrary space, because, let's say, in that particular case, unlike the more general case of the unary equations, this all makes sense, just in terms of the topos, you don't even have to know.

40:00 And of course this is a kind of cumulative count of fixed points rather than an incremental one, so it's natural that the zeta functions arise when you're trying to count those things. So that's a kind of very special further example of this in their use. The other need not involve homes, it's just the fact that the system that's to satisfy the equation should be closed under the operations, that's much more general. So my actual examples, the actual content of mine is simply some examples of this phenomenon within the world of two rigs. I've set it up now for that, that you should actually believe this is something important. But there are some very amusing examples, really. It means Briggs in which addition is idempotent. Why not look at things also idempotent? Hence the algebraic category. But it's not a core variety, because if you have a variety which is idempotent, then x plus 1 will typically not be idempotent anymore. Even more, you know, we had a problem with x plus 1. So let's assume two equations. X squared is X, and X plus 1 equals 1. And then you can verify grids generally. There is a subcategory of those grids.

42:30 This inclusion is a variety by definition, and it forces these equations to be true. This is the standard thing. But in this case, there is also a rite of joint, because we can, sorry, this equation itself obviously implies that 2 equals 1. This is actually contained in, here it's really a variety. It's a simple calculation. Take any two elements x and y, both of which, well first of all, we're assuming that one's satisfied, you see, by starting out. Take any two really, you can look at the two elements x and y that satisfy this, well then the product and sum will both still satisfy, and so you get, just by extracting those elements, you get a core. Now, this is actually a well-known system. This is distributed lattice. Multiplication, again using both, multiplication is the inf and fumum operation with respect to the ordering given by addition. Every two rings, of course, you have, you know, you say that an element y plus y equals z, right?

45:00 It's a semi-lattice. So these are just distributive lattices. The subject is algebraic geometry over distributive lattices, what all the subtopologies are. The thing which I'm emphasizing here, namely, there are distributive lattices which are in the sense that you can't, C is P. This is still a subcanonical, so the feature that if you take the basic idea of local, this classifies local distributive.

47:30 Simplicial sets. A lot of people talk about simplicial sets. This is the classifying topos for distributive lattices that happen to be totally ordered, totally ordered and non-trivial. So, as I said, I think it should be possible to have a complete understanding of, as you mentioned also, sets to the power of finite sets, which is the classifying topos for eucalyptus. As I said, lumen algeris is not a variety inside distributive lattices, but of course it is a subalgebraic category, and also subclassifying topos is a subtopos as well, so the role of the lumen algebra classifier, of course, it starts off to decide from distributive lattice. It should be possible to have a good survey of all the subtoposis, and in particular those elusive operators like the co-hiding boundary of a subtopos should be calculable in this context. This is a rich, particular, not only talk about generalities, you see, there's particular things that need to be calculated there. Examples, by itself, doesn't give a core variety. Even inside two rigs. But on the other hand, the x plus 1 equals 1 by itself does. And so we have these things, finite disjoint coverings, and in more detail, you can see that the sort of generic one should be thought of as an interval in a way.

50:00 This is equivalent to the any problem. It's easier to think of. Between 0 and 1, so they form an interval. But it's an interval with a non-trivial multiplication because it's the monolingual of a variable, a squared variable, getting smaller, of course. You can see that another issue of this equation is that one of these divides another means it's smaller the other way. Now, in principle, when you take the formal sums, this relation means that, in fact, all you do is join zero. To the moment you have the multiplicative modality that tests zero, that's the, that's the, so, you know, in the interval you can drastic reduction modulo this divisibility implies ordering business so that you get only the subsets which contain no divisibility.

52:30 There's a finite set of pairs under addition of exponents with the property that within that set you never have irreducibility. In other words, that's the sort of canonical form and word problem for those particular polynomials. So, as I said, if I read this, as one could say, entropically, which is just a fancy word for minus the law, So that multiplication becomes addition and sup becomes int and so forth, then it looks very much like the recently popular so-called tropical geometry. So I think that actually this predicator topos, this example of generalized algebraic geometry would fit more or less exactly to what people study under tropical geometry. As the final example, I want to take the opposite of x squared and root of x, which again has an equation. It says x plus x squared. Again, you can verify if you start with any two-rig, the elements satisfy this one, the sub-rig, and so it's again a core variety. You're going to have an algebraic geometry in this sense over that. So I end by pointing out that there's a big supply of... There are a few of this sort, mainly the dimension rigs of Chanuel in the case of that the diagonal has a complement, those that don't have repeated roots, the space has that property, so you have a separate role of extensive category, you have the Burnside rig, but then you make Chanuel's two reductions, there's our characteristic rig, but there's also the dimension rig, and it will always satisfy this.

55:00 In any semi-lattice, there's a sense of category. Take a square. If the object is separable, then the complement of that is equal to the square. And so you get this, and those are the cases that he can, a variety of, this core variety of rigs, and hence have their own abstract geometry in turn, namely that of the algebraic geometry of the topos arising from. For a single sort of theory, for a single sort of theory, yeah. Yeah, yeah. And the question was?

57:30 Well, is that in character with which equations they are? No, I don't know, I don't know. As I said, a simple case would be where the equations themselves involve operations for Charles and Lomovorffisms. But in the more general case, it's sort of hard. I mean, I could just, these are very simple examples, but I just calculate by brute force. I don't know any other way to see that the elements satisfying the equation are closed under the operations. Yeah? Do you have a classifying topos description of four varieties? Is there a condition on the classifying topos that you're aware of? It said there should be one though, yeah, you're right, there should be one, so that sort of, yeah, is it just, wait a minute, the subtopos inclusion preserves sums, is that possible? Yeah, maybe that's it, that the subtopos inclusion preserves sums, in other words, well, I think actually Mackay has some work about this, you know, the case when a single morphism suffices to describe the covering, yeah. That's what I was saying. Once you pass down to that level, the rest is only single morphism coverage. If what I just said is not true, something very like it is, as you generalize. First of all, our apologies to Bill for this strange voice that at some point interrupted the talk. It seems that he will not disturb us in the future, so it was just one.

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