Category-theoretic approach to logic

0:00 See you later. Okay, good afternoon. I just would like to first to give two more references, two more than I have in the book. And part of this book is very important and clarity of telemetric etiquette or telemetric topics. Actually, what's special about this book, it has no examples, no application in the internet or logic, but that's, it makes it for someone to already somehow agree with it, who may use it on telemetric topics theory, and that can be really very interesting. It turns out conceptually. I'll give you the right pass. Okay, for now, just to remind very, very quickly a couple of definitions, probably there are some new people. So by categories we understand the collection of objects, together with the arrows between objects which are thought of sort of structure preserving transformation between those objects so we have kind of world of things which change we might have of course the strong photo transformation of given object to itself

2:30 and also they transform to speak to each other this is intuitive idea and then we introduced something about arrows we just learned how by very simple properties. We could speak about what is set theory when objects are simply sets and maps or arrows are simply functions. What is the injection, subjection? And by injection, here we define this to be correspondently monic, apic, and isomorphism, right? And defined purely algebraically very simply then I introduce what was called the universal property meaning that there is certain diagram which commutes and commuting diagram just means that we be whatever path we go through arrows we have the same at the same result which commutes with some unique arrow that that's called universal property and then we defined this important notion of initial object product, co-product, and always when we have some construction of concept, we might have a dual construction simply by reversal of all the arrows. And finally we spoke about exponentials, which is in objects representing all arrows from between two given objects A and B. Like in sets, we might have set of all function from set A to set B. It would be also another third set, right? So set of all functions. So we also define these objects through its Solskjaer external algebraic properties, okay? All right, and the one, just one more thing. Given a category, in other words, just a natural category, meaning I don't specify what is a category, a group set or whatever, you might construct easily another category, right? And I will mention two. Actually, I can mention one, but I would like to stress. One is a split category or a common category. and this is basically just a sample A, then we pick up the object in this particular S for example.

5:00 And then we can see the old arrows for example to S. So again, this is function A. Everything is very good. You can see the, say, bold arrows of this form, and then, you can find a new metaphor, right? Sometimes it's like this. To be the object of this new metaphor, it will be slash arrows, right? The arrows in particular would be such arrows between A and B which make this triangle green. And actually we consider an important case when we talk about sub-option. We define that sub-option A to be simply a morning arrow, right? To A. And then we define an arrow between sub-objects, between sub-objects to be against actually arrows that this triangle compute. And by the way, since those are morning, right? And here I have, for example, two arrows, say A and B. Then by definition of what in the morning, what I did, that the age would be left. And so that's why this is partial order. Partial order, by definition, is such an area where between two objects we have no more than one arrow. Of course, it's a theoretical result, but you wouldn't have to identify it, right, but we might, of course, there was in one direction, one direction, but still, we don't know that, too, their composition would be necessary identity in those ways, right, so it would be identify it, so we have to identify it, but the same, the same sound.

7:30 Right, but also, you might have to generalize that, and instead of text, take whatever diagram, whatever piece, or a category. For example, you might take something like that. Okay? And then I find again, in euletic order, of all forms, okay? Of all forms, such that I have all these categories, but this will compute, okay? So I have this top statics of phones, and the arrow between them is such that the whole thing will compute, okay? That's clear. And by the way, the initial object on this category will be drawn out. So I can basically define all this when it goes to construction in terms of, so the format that you want to do is give a telephone or initial out, it's just construction. And second, also simple construction, I can, given a category A, I can consider another category which is called the arrow, so I can take both arrows and consider the next option for the new type voltage, which is a square. And why A square? Because so we take this absence of arrows, right? But then we need to define arrows. What is the arrow? And then, for example, we have an arrow like this. We have another arrow. Everything in the A. And naturally we define an arrow between those options, those are not options. We define such a pair of arrows of aim that the drain will be.

10:00 That's why that's clear. Okay, of course, it's very, very basic construction as well. Okay, now I would like to continue. you. Okay, the next step of generalization would be consider a category of categories, okay? Not necessarily of all categories, whatever this would mean, but the idea is just take categories themselves as objects and now we should again define what would be arrows in a way and those arrows are called functors and now we just should look what would it mean that functors preserve structure and this is a little bit more complicated why because we already when we say that object is a We assume that there is something inside it, right? We don't speak as before, absolutely, absolutely. There is some object and arrow and then we just making some simple algebra around. And here we already look inside this object, which is a category of its own, okay? So, a very general definition of what is functor is usually defined as a map, which sends objects to objects, of course not one-to-one, not necessarily one-to-one, which sends identity errors to identities, and basically that's it. If we don't have more special properties, which I'm now going to discuss, and also one important thing, it also, it maps all errors between A and B to to arrows between their images, okay? And we might also consider a case when direction of the arrow, the sense is preserved, or otherwise when it is reversed, okay? And in the first case, the pattern is called covariant,

12:30 and in the second case, contravariant. And for contravariant, people often just write this A opposite to B, which would mean covariant, a functor from category which is opposite to A, which means which is dual to A, okay? Category obtained by reversal of all variables. So, this just means that this functor is counter variant so people think that okay it's nothing more than that okay then we might think about more special properties of functor and one very important is uh to be faithful to be faithful and to be faithful means to be injected on arrows but not on all arrows but on arrows between given objects. The idea is that it need not to be a function, generally speaking, need not have some special properties about objects. It might be set theoretically speaking, whatever function between two sets, if you might think for simplicity about objects of two categories as forming sets okay so it could be whatever function but what is really important what going on about arrows between two given objects and their images okay and it's faithful when it's injective it's not how say collides arrows between given objects okay let's go faithful and it's called full is its surjective on arrows, so we don't get any new arrows in our image category. Those are important properties. Okay, that is the pointer, and before I discuss some sort of categories of categories, yeah, probably examples, An example which is probably not interesting but still important is forgetful functor, meaning we take some mathematical objects, sort of as a structure over a given set, like a group, and then we just map it to underlying sets, respecting all these conditions.

15:00 Okay. Now, we have categories, we have functors between categories, so we have a category. To have a category, we might, of course, satisfy all those axons of categories, meaning we might have this identity functor for every category, composition of functor which is associated, and that's it. but really also important notion is a natural transformation it's like next step of abstraction and that this is a arrow between functors okay we might We might consider two categories, A and B, and think about arrows, we have some, say, two counters, and then we might think about transformation of one counter to another. That's called a natural transformation. We might think about it, that's why I'm trying to give this example, we might of course define what is error category, what is more useful about errors, but it's a little bit more complicated than just this. Exactly because our objects are later, I guess, in terms of structure, which is pretty simple. So it's not enough to say this, and we have to make something more explicit. But basically, I think this will be, okay, we have, here I take just some object of f to a, okay, and f is narrow in category a to a prime. And here we have two functions, AF I just considered. I denote an image of A in B. So this is an arrow in B and this is arrow in B. And now, again, of course, we demanded this square purpose.

17:30 Those arrows here are components of nature's affirmation. And McLean, in his book, he taught that a real mathematical reason to introduce categories was natural transformation, because everything that I told before, from a mathematical point of view is rather just a particular way to speak about the usual thing right but something like natural transformation it's already it's not impossible in a simple case but it's rather difficult to spell out in said theoretical language and a mathematical example is a vector space, which Martin Utila, you take vector space, you take its dual, you have an isomorphic space, okay? And here, yeah, of course, if we have functor which is going in both directions and gives identity functor, it would be isomorphism, okay? But then, the point is that isomorphism could be natural or not natural, right? If vector space is final dimension, you take its dual, it would be isomorph, but not naturally isomorph, in the sense that isomorphism would depend on choice of basis, right, and would break when you change the basis. take a double dual right then they are naturally either more meaning you you might change basis in both okay and again next level of abstraction we can fix two categories and think about category of all factors from A to B and And that's called Fanta category. Okay.

20:00 Okay. Yeah, actually Fanta, when we introduce Fanta, it gives us a way to speak, to express this category theoretical notion more, how to say, category theoretically. For example, we might speak yesterday about subcategory, meaning category which is closed under morphism, right? But now, given the notion of functor, we define this subcategory as monic morphism into this category. So we somehow move in the same spirit as we replace, just talk about subsets, saying we just take some of elements, not all of them, right? But then in category theory we introduced this idea of sub-object as monic morphism, right? Here we apply the same to the very concept of category and speak about category in this sort of category theoretical way. Okay. Yeah, here I just, that's already explained. just one more thing about coma categories an example which is very important about shifts because it's also a way to to define two ecological spaces, but also local corner market, meaning two spaces are local corner market, because there is an open, such that, such that there's a restriction of the map to this open corner market. Okay, and then you have to find a slice category in choosing some particular space, which is called space-based, and that means you get what is a sheet.

22:30 So, base, ecological space, and the surrounding of other spaces, which are located, or located, or located, or located. and she can be considered it's a kind of and you'll see that it's also a Yeah, just for terminology, people after this work of Grothendig, now they call pre-sheave just whatever contravariant functor two sets and then if you also define some some kind of topological structure on it on this functor it's already considered as a sheave okay okay now I just going to introduce some advantages what we can get from the talk about functors and one important thing is a jointness enjoy The idea is this. We have, again, two categories, right? And we have a pair of fronters, but going in the opposite directions. And then, we might have this situation. that we have a one-to-one correspondence, actually there is way more, I use this more clearly, that we have one-to-one correspondence between elements of A which are like sources for this And those who are targets from the other part, right?

25:00 And symmetry, the same thing in the second part. And here we have this pair of factors. They are all adjoined to each other. F is all left adjoined to G, and G is all right adjoined to F. And that's, again, something which is mathematically very interesting. For example, you know, that is sort of trivial example, if we have terminal categories, somehow called, again, we have, we think about category of categories, okay, and now we, instead of speaking about terminal object in a category we're just speaking about terminal category and then we have we have this only arrow from any category any unique functor from any category A to this terminal category right and what would be say it's left and joint if it exists okay it's not necessarily if we take some particular two categories of course and we have a question a functor from a to b it's it's a question whether a joint left and right and don't exist or not okay but since we we speak absolutely we might suppose that it exists, and then see what it implies. And here it's evident that we would have this opposite left-in-joint which would send always this terminal category or all the element of terminal category to initial element of category it must have initial object okay just because there is only one arrow up there so we can't have more than one down there okay and the less trivial example is this we can see the diagonal functor okay which means we take whatever category A and then we form a sort of product category just by taking objects

27:30 of this product category pairs of object of first category right and taking arrows as pair of arrows okay we construct like this and then we think about adjoints and And so we can see that, actually right, the joint would be product in first category. Okay? Just to see this, consider just an element of this product category, which is just a pair, maybe. Okay? And consider these two factors, working like there and that. and so this diagram is in this product. Okay? Okay? So, if you consider what happens with this disease when it goes there and back, okay? And the important thing here, and here we just take sort of one element, image of element set from our initial technical detail. And the important thing that here we must, since we have this injunction, right, we must have one arrow here, which is exactly the image of our arrow H. Okay? So we have something like universal construction here, which is In fact, universal construction, when we just replace this X and A and B, we just have B and Q is our projection, and so we have a product in A, okay? So in this category, we just take some element, not necessarily from diamond or whatever, and this right adjoined factor sends it back into A to the product of A and B.

30:00 Actually, I would like to discuss about this example, and then if we consider left adjoint, that would be a co-product, that would be a sum. You can easily check. But kind of philosophically interesting, and what I'm going to discuss further. There is kind of trick here, I think. And trick is this, that look, how we reconstructed this category of pairs, with two pairs. Which means that somehow we had in mind the idea of making a Cartesian product, right? But this we made in a kind of external... I just told you, okay, we take pairs of elements, of objects, right? We didn't make any category of this universal construction. But if we try to... And then somehow we got back this construction of product, because product is exactly just a Cartesian pair, right? So we sort of smuggled the idea externally, right? And then through injunction we got it back. actually people when they try be sort of more category theoretical they would say the following they would say okay we would think about sort of enveloping category or I was explain next time enveloping topos okay and do all that to construct this AA in a wider topos okay and then in this all the rest and somehow we we get this product and this is interesting this is an analogy with geometry when we trying to define I already spoke about this analogy with the idea of say manifold right when we're trying to define better manifold we first we think of you think about that in some in

32:30 enveloping space, like Euclidean space, for example, we might think like Möbius ribbon, something like that, but then we are looking for a way to describe it intrinsically. It's something similar going on here. We just somehow, naively, we have some external framework where we define simple things like a pair, right? But then we obtain it back in our category. Okay? What's going on here? Yeah. But anyway, anyway, it's sort of impressive. We have actually triple of functors, right? Diagonal functor and two adjunct functors, which gives us one direction, like just a diagonal, just double, right? But the other way gives us some product. So it's not in a trick in a sense, it doesn't worry it does work right but it's really interesting important to think what does it does it mean what's going on here okay and yeah I just mentioned one theorem which actually much connected to that example but I'm not going now to to prove that right joint preserve limits and left joints once preserve core limits. Yeah, again, when I say that functor preserves somehow structure, actually it doesn't follow. We might have a functor, which is functor according to definition, all right? And even a faithful functor, which doesn't preserve some additional structure like limits and call limits. but then has this if it has left a joint is the other way to say this then it preserves limits and if it has right joint it preserves four limits alright now I would like

35:00 to discuss Lemma theorem, which might relate to this question, what is external, internal, what are relations between category theory and set theory, and which is probably more philosophical nature than mathematical, which is called Yonada-Lemma. And I think it must be also interesting logically, although I don't think that there are some logical, probably I don't know, but I didn't see any logical treatment of Yono Dilemma. but I think it's it's it's very interesting conceptually and I I think could be interesting logic logically actually you know the lemma is generally considered as generalization on things like a kelly theorem and stone theorem And I just remind Kelly's theorem, it says that any group, whatever, is a subgroup of a certain group of permutation. And the proof, again, is interesting. The proof is very simple. We just take a group, say at least it's elements. and then we take whatever element from this list, we multiply every element and we have the same element but in some other order And this we may consider as a permutation. And then it might be shown that this group of permutations is actually an item. And Stone's theorem is the result for pulling the algorithm. Except in the case of Stone's theorem, infinite and finite case really differ significantly. So, finite case is also very simple, but infinite case, you know, involves this construction with ideal and more interesting. And you'll see why Yona Dilemma is a result of the same sort, although I'm not sure it's sort of a formal generalization.

37:30 So, here's the dilemma. We get some category, whatever, C, okay, and fix certain object there, okay, and then we consider all arrows from A to whatever other objects, okay, and this gives us a set. that's of course limitation because we didn't suppose it's speaking about category but but here we suppose that another category object would make a class or whatever but all arrows between two to give an object form a set called home set okay this gives us a set and this would mean that we have actually functor, and that would be contravariant functor from C2 sets, okay? And then we look to functor category, considering contravariant functor from C2 sets. And by the way, I already told that it's called a sheet actually there is geometrical interpretation on all this story but i'm not going to speak about this now okay uh so we have a uh funter category from whatever our given category to uh two sets and here we have we fix this wall of holes okay right and so we have this constructed founders and whatever other founders, and then we have a nature of transformation. And now the lemma says, look, the lemma says that as far as we, as we fix two founders, which means we just take some whatever other parameter from C to set, okay, one to write one. Otherwise, let's try that. And consider all natural transformations from F to G,

40:00 then actually those transformations would be the same thing that arrows from A to X to X. This is why it's imprint, because, look, what's going on here? We sort of go to rather high level of abstraction, right? We start with a category, then we consider functors, then we consider natural transformation, is natural transformation. But what is coming back, right, that those natural transformation are basically the same that eros in our category we started from. It's a little bit like, well, it's like Kerry's theorem. You know, we also, you know, even historically there was basically first group considered by Galois, groups of permutations, right? But then we might generalize, introduce introduce abstract concept of group, right? But then we have this theorem that says that actually we didn't go far. We are still there. And that's a little bit the same kind of result. And for the proof, it's actually a rather strange proof because there is nothing technical about this proof. And rather the idea that people of this proof is just is this is is go to such a high level to abstraction then then it becomes not just evident in some I don't know intuitive sense but but just you can do otherwise necessary in a sense and then it's the case if you just think what does it mean the statement you you see that it's sort of tautology this lemma it's kind of tautology but highly non-trivial you know because so we must just check that this square commutes right here we have we have elements of the natural transformation we're working on our

42:30 object A and here it's object B. And the idea is just to take identity, just to take identity to prove that this error actually exists for any given error like A and B, right? And So the proof basically is take an identity and you can't say that it has no image and you can't say that it doesn't have an inverse image just because all those functions exist like hypotheses. So at least for identity it must work but this is enough. That's basically the proof. But I thought for the same reason. I mean, Stone Theorem says that Boolean algebra abstractly conceived may be realized as algebra of subsets of a set, right? Yeah, and here again, we say that whatever error in EPS category... I see the idea. Yeah, the idea. Yeah, I'm not sure that it really generalizes in a formal sense. It's rather just the same idea. Yeah, and why I think it might be logically interesting, this Yona Dilemma, because in a sense it allows to reduce higher order construction back to sort of first order construction, right?

45:00 No, I speak informally, of course, what's going on here. But just natural, basically, natural transformations between two functors, one of which is constructed from a given error in a given category, and the other is whatever functor, two sets. Also, it's important that as a target category, we have here a category of sets. Otherwise, it's not a theorem about abstract categories, okay? You have a dilemma. So, those natural transformations are just in actual nature also bejection, isomorphic, to our arrows of categories we started from. And that would mean that, in a way, informally speaking, we might probably replace this higher-order construction, like natural transformation, backed by something simple or probably the other way around. but in some way it looks like, you know, complexity, which is growing, of course, right, when we come to this high-order construction, but in a way this growth of complexity is limited. it. No? Yeah. And actually, a kind of representation theorem, representation result, which is corollary to Yonade Lemma, is called Yonade Embedding. And it says that for any category C, okay, that's probably closer to Stone's theorem, actually, as representation theorem. We don't have exact representation here, right? We don't have isomarchies, but we have what's called Iona de Embedding. We have a functor from any category, right, to functor category or category of pre-sheaves, we might say this. That is category of contravariant functors from this category, two sets. and moreover this functor has good properties is full and faithful and in a sense that's that's maximum what we can expect from functor which is not simply simply equivalence

47:30 yeah just just arrow in this filter category is the same as our arrow we started from. Okay, and now actually I want to touch upon, I'm not going to introduce any serious system of foundation, but would like a little bit touch upon and probably discuss this foundational issue. What is foundation of category theory and whether category theory could be, sort of as foundation for the rest of mathematics, probably for some part of mathematics. And actually such system, I think it doesn't exist. There is a paper, old paper of Bill Lavier, and it dates to 66, called Categories as Foundation for Mathematics. And what he is doing here, he is trying, he makes sort of formalist move, he is trying to describe categories or some kind of formula which is constructed by simple rules, but then, but that's sort of standard move, right? But then he just immediately comes to this category of categories, and basically the idea is this, I call it Pythagorean, because there is, the idea is this, for example, we just postulate there is something like terminal category one, right? And then we might say there is another category, like two pair, which has exactly two points, and points would be, of course, arrows from morphism, factors from one to two, but then we don't know what are morphisms, what are objects. Okay, but now we can say objects are one element, right? Objects are points and arrows are two elements, meaning arrows are morphism from two to whatever category, okay? And say to say what is composition, what is commutative triangle, we might say that those are three elements. and actually I love you he also needs four else perfect Pythagorean like one

50:00 two three four and some ten basic morphism between them from which he through injunction he derives he derives other thing yeah but but I think it still doesn't exist any really coherent system which would be kind of purely category theoretically category theoretical yeah and actually this language of functors allows probably give better sense to to this diagram which I already showed yesterday right we might think categories as category of we might think about those errors and we might even suppose that it should be a pullback for for some you know other object but then of course it's not clear in in which in which envelope and category things are going yeah myself i think that probably an interesting interesting direction of research would be not push this sort of pythagorean thing it's really not clear what we start from it's all the circle sort of circle but that question of course is it vicious circle or not uh but i think a interesting thing would be probably relativized in a sense i really devised in a sense this idea of foundation in the way I already mentioned we can really think that we start from some category which is rich enough to construct what we need our sort of working category so we start from categories which we don't know where from and then we construct things in it but the idea is that actually we can always relativize our discourse to some new category to some new category we basically can think we might probably

52:30 give up this idea to invest foundation which are nowhere this is a little bit idea we might we might think we always work in a certain topos really i i'll explain tomorrow what are toposes is basically cartesian closed category with some additional structures which are which allow to do logic really there and And probably we just don't need anything like sort of free category, which is nowhere. We might always think in topos and think how we may switch from one topos to another topos. That's probably what would mean foundation somehow differently conceived. okay probably probably I stop I stop now we might discuss because and tomorrow I'm going to to speak finally about applications to logic starting with as I as I very conditionally I distinctly between sort of algebraic application and topos theory so probably I decide I leave to tomorrow and if there are any questions we have time to discuss I said I don't know Yes, but what do you mean by that? No, I just meant that probably it might be interesting for logicians, you know, dilemma. Because apparently it allows to something like, how say, to handle with this problem of order, you know. I don't know what application you want to have. Yeah, probably you would mention. No, if you would take a start on this category, and you want to find that with our units, you need your data to get something. And you need that category and you want to put your status in your data.

55:00 actually one thing standard way to make kind of model theory in categories is think about models as functors to sets, right? As again kind of pre-shifts yeah, and again somehow connect, you know, semantics or syntaxes. But it's interesting that it's really tautology, you think. It's nothing there. You don't have any construction to prove. Normally, not always, of course, but I don't know, some, like, Pythagorean theorem, right? You make some construction, you make a trick and you show. But here it's almost like definition if you just spell out everything. It's like tautology. Tautology but not trivial and actually it's also questions that the relationship between sets and categories. Probably it shows that sets are still indispensable in a way, at least as sets of morphisms, right. But of course there is also ways to make set theory in a topos, right, so if we can do everything, category theory, so we can define what are sets there, and in this sense, the Yona dilemma could be also kind of relativized. If the community has continued taking part in some of the things that we have a category of arrows and results of the conference, also now creating a category of information, is there any reason why we take another step or maybe it's a new decision? Actually, different people think different things about that. It's called N-categories. I've got to speak about that. So N-categories, there are different attitudes, because people like you and here,

57:30 they agree that this is uninteresting. Because they still think that this is close. There's actually two categories right now. It's still important, right? Because the two categories, before I was thinking, you just put in one block function between every category and every point on each side. So, two categories are in this answer. Everybody is. But about this further, every category is very different. Probably one reason to not take that seriously would be exactly this idea that at the beginning we choose to span the hierarchy, which is one of the important motivations for the whole and for the same reason. But there are still interesting results. There are a lot of problems about that kind of course, but again, there is often kind of course theory, as you have seen in London and Denmark. Difficulty is formulated, but not improved. As sometimes people say, all the time comes to me. But just to spell out what you want to say, it's not always easy. You know, John Bias, he works a lot of categories, and he hopes to create a kind of what we call stabilization of the theories, meaning that we get time to do it before we continue to develop structures in this. I don't know the category, but he just, he can't even formulate it. So, I guess I'm going to focus on, so it might be as well that you will see contamination as arrows in a four category. I was in three categories and then it could happen if it has one book, one book, one book, and another so we have a four dimension. So this is all for us. Maybe there will be a few categories which are under the version of ours with this set. It's obviously to relax.

1:00:00 There is a post-op that was a bit But then, for example, you don't have your requirements. No, you still have, you have to work. No, I'm a Russian. You know I'm a Russian, but you have to practice your question. No, this exists, there's no question. But simply you don't have some people like your requirements. And then a little bit up here. It also is not the problem to make objects, you said. But there also is a little bit more to make objects. So there is a question where objects are identified and to form a new job. Actually, that's also why still people keep objects in front of this is exactly N category. You can think about N category as a sort of geometric compare, right? You have objects as orange, then you have arrows as, arrows, ages, but then you have two arrows, two particular S faces, right? It's sort of, to keep this dramatic, but not partially, you will keep larger, right? But on that end, it's formal, I mean. Yeah, I don't think that I get it. No, the question is, what view the difference is in nature, and it's a way of speaking with that. Any other questions? Okay, thank you.