Graphic monoids and Hegelian "taco"

0:00 Testing. These notes are recorded on the morning of the 4th of July 1989 at Bangor, before going down to breakfast, before a bit more fast talk. Talking, talking, 1, 2, 3, 1, 2, 3, 1, 2, 3, testing, testing, testing. The name is a special pleasure to welcome people from all over the world, and particularly those who have been here, what I said yesterday, the visit of seven people from Tbilisi to this International Category Theory meeting. I think the applause... Yeah, I'm sorry, I'm not saying you're going to laugh. That's getting that through, and you certainly know this will be a regular occurrence, and indeed a start of visits both ways. I also have some other notes that do come in. Ms. Brenna Merisman has said that she is willing, or pleased, to have two passage fuels of the Cahiers for the proceedings of this conference. And this means the papers will be refereed to both the normal journal papers, which were announced as part of this conference. But this, there'll be 200, maximum of 200 pages will be allowed. So we'll have to deal with the papers as they come in, profitably, but what we did in fact was have some group sessions to decide what was going on, in which chairman reported back in the afternoon, and I think the general feeling was that this in fact went very well, and maybe could be a useful feature there in my conference. So I think that's it.

2:30 There are only a few answers I have to make, but it's only a few minutes. Do you want to say something about the trip tomorrow? Yeah, right. Jim was saying something about it yesterday, but let me say it again. The trip tomorrow is about cleaning off the lungs. It's a modern concept that's evolved and developed over the past decade. It's focused upon even very, very simple examples. To produce very elaborate theories. In some cases, these theories are not merely elaborate, but even potentially adventurous. So, I suppose you can focus upon one particular pre-morphism category, which is actually a home domain. So, how many such entomorphisms are there? Only a second's not. It shows that there's an entity, that's even less than a second, and two constants. Two constants, I suppose, are all supposed to call those D0 and D1. Well, actually, what else is there?

5:00 If these are constants, they satisfy this equation. Or I should say, these four equations, even if they don't. Now, in order to try to apply the semantics and structure of mathematics to this particular example, as I've already said, we consider it has to move away from the present example. Now, what can we expect to be the abstract general derived from this, that is to say, the abstract theory, which is true of this example. Let's look at the other kind of general, which is called the possible realization of that abstract general. The American Maths Society publication is called Contemporary Maths, volume 92, which contains the proceedings of the Voltaire Conference on Holger and the 87th Conference on Category Theory and Computer Science. I have published a paper where I consider certain generalizations, certain very simple generalizations. I consider their monoid forms that they consist entirely of identities and constants. So if C is called a constant, it is only if rho of X, CX equals C. By contrast, we know that the unit element satisfies rho of X, one times X equals X.

7:30 You're given a monoid that consists entirely of constants and the identity of them. Particularly, they're all different ones. All of them are different. But, in fact, they're a much stronger identity. Two variables. The difficulty is only one variable. Neither of these conditions applies. Let's say A is either a constant or the unit. We have AXA to AX. We're all X one of these two times. But if we assume that the monad is so special that it only has those two points, then we can also say, well, hey, so this is an identity in two variables, which is true in any such monad, in particular, in this one, as opposed to these, of course, that have no identity. All of the things that go with a particular example, in the abstract general of this theory, back to the category of all possible representations or realizations of this abstract theory will first be this example, or this more general kind of example, subtext. Now, what possible uses does such a trivial thing have? Well, some of you have heard of that for a moment. Well, let's go back to this example. So, I propose to use the word application.

10:00 What's variously called three-sheet, right action, well, counter-variance of right action. Let's say a small category, application of small categories, application of C, C, appreciate on C, the right action of C, and then concentrate the best values on this from C. These three terms, the more you get multiplicity of names, I can't think of at the moment. Well, yes. C-sense. C-sense, right. Write a C-sense. Yeah. Any other questions, Peter? No, I just remarked on behalf of Hella. That's right. Because, because, that's what they are. You can think of C as a theory. An application is a... The broader context is in which the theory operates. So what's an application of my original example, three-element example, of the water?

12:30 Well, the moment's thought shows that this is the same thing as a reflexive direction graph. These things of course form a topos with the Euneida, Cayley, Dedekind object. Only one object that is homonoid. Again, a reversely multiplying application that's called I. I want you to think of some kind of abstract integral. So, in an application, an arbitrary application, so let's put it like this. If X is an arbitrary application of alpha 1, function category S is alpha 1 alpha, The morphism of I is by Unitas Lemma just an I-shaped figure in X, but that's a general terminology suggested, which in this particular case is just called an edge. And by Unitas Lemma, since I has three elements but two are constants, when you split it in potents, constants all become global elements.

15:00 In the category of applications, you get a constant of points that is matched with one. So the d0 and d1 reappear with a slight abusive notation because really they're given from gender maps. Splitting them, you get two points. And so first, x delta 0, x d 0, x d 1 are the endpoints of the edge. In general, we can speak of incidence relations between six, as we've undersaid, we can associate between applications, there's a discrete vibration, discrete vibration over C, labeled in C, and C associated to... So this is actually a category. So that means that we have two objects, x1 and x2, or there's two figures in x. We can speak about all the actions that d1 and d2, the set of all such d1 and d2, equals the x of these two figures. So, for example, in a directed graph, you might have a figure like this and another figure like this, but they have an incidence which has to be a point.

17:30 So the speech vibration, or comma category, the category of elements, again there's a list of names, associated to an application, contains in it the category whose objects are figures, Now, so the thing is that in some cases, for some C's, we have a simple picture of these figures. I want to put a kind of distinction that in a certain general abstract sense, the discrete vibration always gives us a kind of picture. We can spread out a picture of this action and that gives us some idea about the incidence. It is possible because of the particular nature of C. We usually draw graphs as we draw a graph. Points are points, edges are arrows, and the instance relations are evident just by looking at them. They are determined by calculations involving the actions, in this case of T0 and T1, which in general are labeled by the elements of C. What is this possibility of picturing? I don't know exactly, but at least I claim that that's where the general point of this sort, so I propose to call graphics, by definition, a category in which, category C, in which every, and the work is in one way,

20:00 C satisfies the identity of AXA AX. Is it CAA? Yes, thank you. Now this may be slightly too general, but I'm not quite sure. But in any case, we do need to consider categories, even if we start off with monoids. We're inevitably led to categories in a particular application, even in C where a monoid, C slash X, isn't. So, C-slash-S, X, bank-slash-bank, this construction always gets a category, but it will continue to satisfy this condition, because after all, in the discrete vibration, the labeling function is stable, so each n-morphism monoid is a sub-monoid of the monoids that are below, by the label. The same identity, the same property will be propagated to every discrete migration. But another construction is just splitting intercodes. We know that the category of applications of C and the category of applications of C bar, where C bar is the category you get by splitting intercodes, are always equivalent. And in fact this is very helpful in picturing how the figures really are already in the case of graphs. And that's for minimalistic sufficiency, that's fine, but on the other hand, to really picture it, it's somehow more natural to include this category as the, uh, it contains the 1 as well as the, uh, as the, uh, the data.

22:30 Of course, splitting intertotems also preserves this, this identity. So, the graphics, subcategory graphics. It contains all the graphic monoids, that is, the monoids that satisfy this identity, but moreover, it's closed under these two important operating categories that are connected with applications. Given a particular application, you look at the discrete vibration, or given a particular category, you split the intercodes in it. Now, first, there's a left adjoint to the inclusion. In other words, every category can be reflected as a graphic, but since it's also obvious that if I take C to the power of A for any category A, that will be a graphic if C was. This inflection is sort of like a sheet that implies that the reflection preserves product. Why am I calling these graphics? Well, so the idea was that the one example would be started with, under the property that these applications were graphs. That's the origin of the name. Actually, there's another one that's supposed to be suggested. But, uh, in the sense that from the point of view of SID, these two categories are reflected in these different graphics. But I'm sorry, any of you can be, you know, can be, as a reflection, you can mention the graph. Yeah, that just doesn't make sense, the right and the wrong. He's like, how do you make the quality of the graphics laws? Oh, well, it's the fact, simply the fact that the one example that we started with,

25:00 as a profit of its application, are graphs, directed graphs. But then, but another... That motivation is correct in the sense that every graphic has a picture. Every graphic has a simple picture. That's the kind of amazing thing about RRK. Let me just say a couple more words about the graphic monoids. So, for example, three graphic monoids on an alphabet of n letters has a integer part of n factorial times e, no, i.e. N factorial times the sum a to 0 of the n of 1 over k factorial. And this is, in other words, this is also known as lists without repetitions. N with the outside bound sheet. N with the outside bound sheet. Lists without repetitions. See, so the idea is if you have a list without repetitions, such as a, b, c, There are a number of different types of equations, such as B-U-V. You multiply those into three graphic monoid, simply by canceling the second occurrence of anything. The idea of the identity is that the second occurrence of anything can be cancelled. So you just cancel that second B and you get A-C-U-V. No, A-B. A-B.

27:30 So a general right action of a graphic monoid, an application of a graphic monoid, if you think of the elements as states, then if A acts, that may change the state. Then anything else X could act, it would also change. If A acts again, it won't change it anymore. So there's a kind of, you get the vague idea that in an application, there's some kind of hierarchical structure. Things that are more detailed but are reflected into things that are relatively fixed. They're not completely fixed, they're fixed by the thing that you want to read. The three things, any graphic, so these things were considered in 1947 by Schutzenberger, apparently without any application in mind, at least according to his recollection, just with the idea of trying to find, they knew that lattice theory was important, they also knew that the quantum mechanics of non-communicativity was important, but they were trying to find the concept of non-communicative lattice, or really non-communicative semi-lattice. And indeed that's the kind of application which I had in mind as well. The main result of Schiffinger's paper there is simply the finites. The remark that finitely generated implies finite variety of algebra.

30:00 Another paper by Kimura used these monomers to give a structure theorem for slightly larger classes. The general model has to be built up out of pieces that are either like this or dual, and also a powerful structure theorem for these things themselves. You see, if you force it to be commutative, you get a semi-lattice. And so just forcing it to be commutative, for example, if you apply that to the free one, you just get the free semi-lattice, and the free semi-lattice is two in the end. And you just remember the set of letters that's turned in it. You get the order. It's sort of the support of the words without repetition. This is the wound-orphism of monoids. And in general, for a graphic monoid, you get a symbolized. This is a kind you can call a support point. Support of a product at the union or the intersection. So the partial structure theorem says that, take an arbitrary graphic monoid. The fibers of this mass, the fibers of this support function, are always the same tributal time that I started with. Everything is either constant. Well, I mean, the fibers, of course, are semigroups, not normal. So the fibers are just semigroups in which the multiplication law is xy equals x. So that's a, that's a partial structure. I'm not complete with it. It doesn't tell you everything about the multiplying things and the different kinds. But also, Kimura, I think, did not consider applications, that is, either in either sense, that is, actions or possible things that these strange guys might mean.

32:30 So my claim is that these things can be fixed. So, for example, We take delta 1 cross delta 1. Delta 1 had 3 elements, so this has 9 elements. And on the other hand, the idea that delta 1 divided as applications was I, the generic arrow. So we take the product of the generic arrow itself to get a generic square. Here this is the unit element in the middle, and then 1, 2, 3, 4, 5, 6, 7, 8, plus 1 is 9. So the nine elements are the square. This was a great revelation to me because I remember in school I learned about groups, that there's a group of motions of the square, which is okay, but I never knew exactly what a square was. Of course, a square, you could think of it as this nice continuous metric space. But obviously, in this context, it's being treated or it's equivalent to some kind of combinatorial object. But what is that combinatorial object? Well, I say it's just this monoid. It didn't go into operations. In fact, it's a much stronger condition than the graphic. Because... So in some sense, one should teach about graphic monoid before groups. Because they set the setting, in terms of bookkeeping, the bookkeeping setting in which the interesting groups can move around. So the idea is that this whole square could be moved over to this edge. So this edge is more than just a subset. In some sense, it has two aspects. There's the subset, but there's also the whole operation that reflects everything. Likewise, everything could be reflected in this part. So you have basically four independent operations which generate this thing. Although the corners, of course, aren't. You know, if you call this a and b, then it happens that ab equals ba, and that is the corner. So the whole thing can be squashed into the corner, but first squashed into b, and then squashed into a, or the other way around.

35:00 So this is the, in some sense, the combinatorial structure, the idea of the square, and the famous group is nothing but the group of monomorphisms of this monomath. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Before. Now here's another example. Now the directed graphs, of course we all know, these are the natural bookkeeping setting for doing more interesting things like composing, i.e. categories. But what about two categories? In two categories there are two very interesting compositions going on, horizontal and vertical composition. But actually we can't get started with those until we know the bookkeeping setting for that. Of course, the bookkeeping setting for that, we can immediately make up a name for that, but that's not done already. There are two graphs. Now, the two graphs should also be applications of the monoid. What monoid is it? Well, the picture of the monoid has as its picture precisely the generic example of an element of the two categories. The generic element of the two categories. As you know, it's something like this. Right now, we have two categories, two functors, and one natural transformation, and you take this abstract picture, which has five elements, this one has nine elements, I, now, the nature of the United, where's my I? The United object I, the nature of the United lemma, is that when you have an arbitrary figure, it's the unit element of the mono-english.

37:30 Which goes over to the particular figure, right? That's how the genetor, Kali, medical genetor works. So therefore, the claim is that this picture is really a moment, even graphically. In which this arrow here is the identity. Because that's the generic two cell. You look at a two category as a two graph of the typical, most general type of figures. Now, in this case, these points are not representable as products of the lines. The capital letters mean domain and co-domain of punctures, sorry, of natural transformations, and script letters mean the domain and co-domain of punctures. So you have this five-element thing. It's not generated by any fewer, however, but it's generated by . Everything is idempotent, but even better than that, it satisfies the two variable identities in this graph. So the applications of this are exactly what you might call two graphs. In other words, once again, the sort of geometrical bookkeeping necessary environment for the discussion of horizontal variable composition. In the paper here, which there are several copies in the seminar room, I have lots more examples of these pictures, so I slowly convince myself that every graphic, and indeed every application of a graphic, has an intuitive display. The display of graphics, this at the moment is not completely rigorous. The idea is, there should be a geometric realization about it. The geometric realization would be the display. But, so it's a potentially mathematical science concept. It's almost heuristic. It's a heuristic observation with every single graph of homoids that you can think of, but can also figure out a unique picture for it.

40:00 So this way of graphics therefore has to have precise and mathematical meaning, but once again, it's supposed to be suggestive. If you are given such an algebraic structure, say the multiplication table of generators and relations, it's finite anyway, then the picture should appear on your screen. Okay, now, why should we want this picture? Well, it's possible classification would be a road map into the library. You see, indeed. In the interrelationships of different sub-disciplines, sub-disciplines of science are not really completely frozen, some people would prefer perhaps that they were, because it reflects the world, hence in particular it reflects the rest of science, so if you take topology for example, topology is not just one row of books in the library, if you take any other book in the library, it might have something to do with topology. It might have something to do. So at the moment I'm idealizing that relationship by passing two subsets, which you can always do to make things seem easier, right? So given anything in the library, there's a subset of all the topology books, or a subset of all the pages in all the topology books, which is relevant to it. So, you see, there's actually a retraction of the whole library. So topology is not just a subset, it's really this impotent operator which retracts the relationship with everything. Now if you take instead algebra, the same thing is true, these two impotent operators don't commute.

42:30 Now first if you talk about the monoids in which everything is impotent, they don't necessarily commute. They're finite actually, but rather large. However, the claim is, or at least let's say the observation is, that it seems that these things really satisfy, should satisfy, the graphic identity. In other words, they are kind of non-communicable semi-levels. Because, as we know, if you intersect algebra and topology, you get two different things. You get algebraic topology, you also get topological algebra. If you intersect chemistry and physics, You get physical chemistry, chemical physics, which again are two completely different shelves in the library, different subjects. So the algebraic nature of the interrelationship of the sub-disciplines might be to a much better approximation than calling it Boolean algebra, but according to librarians I've talked to, a better approximation would be to consider them as a graphic monad, perhaps a linearized version of that. And so, therefore, if you want to find your way into the library, you need a road map. Wouldn't it be nice if your pocket computer could copy that for you? Okay, now, as you might guess, such a powerful accident as the graphing definition means that there are lots and lots of unbelievable theorems which are very easy to prove. Let me just remark two things about graphing mole numbers. One is that every left ideal is a right ideal. Now this requires just a moment's calculation using this identity. Use this identity and screw with every left ideal. Not at all conversely.

45:00 In particular, every principle left ideal is a right ideal, hence a union of principle right ideals. They're relevant to the treatment of this. So, as I say, this requires a moment's calculation. Much more obvious is with every left ideal. Because every element is in it. Every left ideal is in it. Now, this has the effect, you see, that if you look at... If A is a category of an application, a topos of an application of a graphic monoid M, The essential subtoposis, the essential subtoposis, in other words, the cripples and adjoins, which these two are full in faith, which I call unity, that's A, identity of opposites, identity of C, opposites are the two functions that are throughout the identity of C. What did you call that? Max-hyper-presentable? Those categories, those highly-presentable categories. Oh, no, this might be different. For pre-sheet topos, A, the class of all these Cs, unities and identities, are essentially sub-topos. It's always parameterized by the two-sided, idempotent ideals. Now, an inimpotent ideal is quite a non-trivial concept. It means every map in it could be written as a product of two things in it. Okay, so it's very existential. You're not closed under intersection, as our paper shows with mathematics. So they're kind of hard to handle in general. But if every element is inimpotent, you see, no problem.