Proving Yoneda lemma

0:00 So, I guess it's countervariant. What is it? It's turning the arrows round. So, that makes it countervariant. That's right, country-variant. These are the usual ones that are usually called co-variant, sort of. Yeah, so it turns the arrows around when you apply it. Okay, so... But if I do it twice, I get exactly back. So then that gives me the image that is the double jewel, and the map is now going the correct way that it's been flipped again. Yeah. And it turns out to be the exact same thing we were doing. Yeah. Okay, so I think... I'm really pleased, I understand this example, which is, I think, the Ur example of a natural transformation. Incidentally, I feel I want to get straight in my mind what the difference between this dual functor and what the connection is between the dual functor and this kind of operation here in the opposite category because reversing arrows there's a contravariant functor a sort of identity functor I guess from the category to from C to C op right that just just is the identity function right it's the identity on objects well I mean it's the flat bone right yeah Yeah, but it's the same. The arrow is the same. It's just moving in the opposite direction. It depends on... Now, remember, because, okay, this is kind of subtle. What have I done with that? Did I put that... What did I do with the... God, I've stuck it down somewhere. Oh, well, never mind. See, the point is that... Here's my category, a.k.a. And C op just consists of exactly the same objects, and exactly the same arrows, except that domains and codomains have been switched. Remember, the point about arrows is they don't have to be functions.

2:30 Yeah, sure. Okay. So, let's see. If I take, what do I call it? I don't know what I'll call it. I'll call it F. There's a function from C to C, A is A, for any object A, and F of A goes by G to B is B goes by G to A, right, in the other category. And all this means, I mean, it just, I mean, I don't understand, it looks like it's the same damn thing. It looks just insane. Well, I mean, because it, given, so once we get an interpretation to the category, and to the arrows in the category, then these opposites are likely not going to be anything they're going to be the opposite so for example in the category of numbers where the arrow represents the left than or equal sign then in the opposite category it's going to represent the greater than or equal sign which is certainly not the same but the point is this is important when you're talking about categories you're not talking about interpretations the category is the thing you're looking at so if a category is just a collection of objects a collection of morphisms or arrows that satisfy composition and so on. And because, of course, here, the composition's got to be in the other direction. Yeah. Right. I guess. I'm not so sure. I think. Is that clear? If I'm disappointed, huh? But if you've got A goes to B, and B goes to C... A goes to B by F, and B goes to C by G, and I look at this in the op, under the op, let's call it the op-pongal. Sure. Okay. So that's got to turn out to be C goes to,

5:00 by G to B goes by F to A, right? Yeah. So, if this were talking about G following F, here we're talking about F following G. Yeah. Okay, so what happens is the composition gets, you switch, you turn it around. Yeah. Okay, so that says the composition map here goes to its reversal in the op category. And the domain and co-domain functions are changed, but the arrows are the same. That's what I was... that's what I was thinking. So a camera was just... It's just like it. Well, it's more than just an ordered pair between the two, because there can be many between the same things. They've got an opposite identity, but... Yeah, somehow a tagged ordered pair, and then you just take functions from... It's like the functions are switched rather than the... Notice, incidentally, that we're now doing a no-no because we're all hung up on what a category is as a structure and you're not supposed to do that. But I mean, I suspect that all the people that rail against that sort of thing do exactly this sort of thing when they're talking about these issues. The only hesitancy I'm expressing is in calling these two things that are clearly distinct, the A goes to B and the B goes to A things, calling them the same, because part of the whole point of category theory is to get out of the habit of calling things the same, when in fact what we mean is they're related by some nameable transformation. Yeah, but that's, ah, that's the key point. That's the key point. Because all this works for ordinary folks that are doing ordinary algebra. But when you do category theory, you're not allowed to confuse identity with some kind of isomorphism or something like that. So I guess what I'm saying is that really G star is the same as G, right, but domain of G star equals codomain, right, is codomain of G and codomain G star equals domain G.

7:30 And composition, so that one. And G circle F, let's say in C, equals F circle G in CR. So I've got different composition functions. I've got different domain and codomain functions. but the objects and the arrows are identically the same. That's only one way of putting it because you could, I guess you could keep those the same. So God knows what the best way of doing this is and presumably Presumably there isn't one best way but some interpretations work better and some interpretations don't they can do. But it's interesting I mean, because I pin John Bell down on this. It's interesting that all this stuff about, you know, leaving things vague except up to isomorphism doesn't work when you talk about arrows. Arrows have got to be what they are. What's the word they use for it? They're discreet. Because there's a... What that means is the identity operation on them is really identity, and not some vague-out equivalence relation. So in the statement of composition, or in the statement of accessibility, F composed with G composed with A in the two different bracketings really are the same object. yeah right so these things are i mean he seemed to be bemused by this and thinking there's something there's some fundamental flaw this way but he couldn't see any way of avoiding it and he quoted law beer as saying well of course these things are always discrete when you're dealing with categories to act. Well, these things only hold presumably in a category, but if you go up to a two category, then you can have the two, and the elevation of this is just exactly where you introduce things that go between arrows.

10:00 Then you can have associativity not on the nose, but only up to associator. So 2 is supposed to be sort of like a variable 2, so it could be, it doesn't have to be TF, it could be something like in a topless or something like that. Um, so, well, no, so you start, so you start by defining a, you start by defining categories with the objects and arrows, and then, um, by appending a numerical prefix, you start ascending to a hierarchy, so you have two categories that are, that have objects and arrows, and two arrows, two morphisms that go between the arrows, and then you can have three category, which has all of that, plus three morphisms, two morphisms, and so on and so forth, and in general an M category. Okay, now I don't know why, but Lawviger is very down on these things. I heard him give, he gave John Bias a bollocking for refusing, why I don't know. but actually I was going to say we want to get on unless we want to work through some other examples of these maybe we have to do those on our own of natural transformations I mean I think I think the other ones are quite complicated I think this one is a natural transformation there is a op well there is one that takes a group to its op which is a natural transformation, and that's very similar to that, you know, where you just take the op group just has its... It multiplies in the opposite direction. Yeah, in another way, yeah. But that's taking a group qua category. No, no, in a category of groups. The function that takes each group to its op, there's a natural transformation between... So imagine, in fact, the inverse function, the function that takes any element and takes it to its own.

12:30 No, no, explain. I don't understand that. So we've got group, okay. And group is groups under homomorphism. Okay. So now what's group R? That's what you're saying, is it? No, I'm I'm just saying that there's a natural transformation that looks very similar to this, which is... No, it's not from group off. No, it's a functor from group to group that takes any group to its opposite group. Meaning? Well, the opposite group of G, that is... G, that's... G with its off or something like that. It's just inverse. Tell me what it takes. as a relation, you just take its inverse. Oh, sorry, yeah, okay. You take its inverse as a relation. Yeah, okay, okay. So, um, yeah. So it does that, and then there's a natural transformation that goes between that and the identity on groups. Okay. So, which is just the inverse. Let's see, how does that work? Here's G1, going by some, by some metamorphism phi to G2, and here's, what is this functor called, op, G1, and here we want just G2, right? Is this the way the natural transformation goes, from the identity to the op? Well, maybe we've got both things in both cases. We've got off G1 and off G2. Oh, right. And G1 and G2. Well, I guess we've got that up here. All right. So this is in the original category. So we're going from the... We're going from the... So we're taking the identity operation to the... Yeah, so it's off the way to... Don't forget the diagram for natural transformations is sort of this prism shape one arrow in the top, and then you have a square in the bottom, where one side, one face of the... So you've got these two functions, so this one's the identity, and this one's op, and

15:00 again, the identity goes to there, and op goes out what you're doing, and there, and so if you have this, then the identity functions obviously just by there, and then you have to do an AWP function, I don't quite remember, but then the inverse function that takes any elements as inverse in the group is a natural transformation of this, so here you get this, this square commutes. So what's the AWP function? AWP function. what's the value what's phi r that's identity i'm sorry i'm sorry it's just fine yeah i think it's just no it's not it's not a terribly subtle example i just looked up another example it does work it's it's still the question is if phi works does phi The trisomorphism is the same map. So I of A of B. So you have to show this group homomorphism, right? Yeah. So it clearly has to take identity to identity again, because identity is just the same here as in here, and this takes identity to identity. bullet this. So the question is whether this equals that. I think it does because the flip goes twice. So it's... Is that going to work? No, what are you doing? The question is whether this. This will work. These are just the same underlying set. And the question is whether between the two is whether it's a group homomorphism yeah whether it's whether the same the same function phi yeah is still there's a group homomorphism with respect to phi up the final sorry sorry is that a star yeah so you've got to find out whether that equals but that's equal sorry that yeah yeah i think it works because the flip happens twice so that's

17:30 So it happens if this is . And she pulls . Yeah, and now she pulls . So let me do two flips. Yeah, you're right. I see. I mean, you almost have to show the function. So it works, but that's the way it does. So that works. When you say inverse, that means you send each object to its inverse? You send each group element to its inverse? Yes, exactly. that's going to, that again is going to switch the order of modification. Yeah. Okay. Because, because A, B inverse is B in reverse, A in reverse. So that's, yeah. Okay. That's actually one of the most puzzling things in teaching elementary group theory, because this is about how to handle permutations and which way to think of permutations as operating. You know, So when you, do you think you're replacing the elements when I write A, B goes to B, A, or A, B, C goes to B, C, A? Is that right? Mm-hmm. Okay. So you're saying replace A by B, B by A, C by A? B by C and C by A. Yeah. Yeah. and so what is that what does that mean when you do it twice you see it when you're when you're labeling you know you've got a triangle you want to talk about the symmetries of the triangles you got ABC yeah so now all of the a B and C attached to the triangle as you move it or the attached to the and depending on which way you're looking at it, you've got either G or G-OB. I mean, the two ways of looking at it give you those two groups. I don't want to go through all that crap. But I mean, that's what is going on here. Well, now, I've come to the conclusion reading that I think we've got to look at the Yonita Lemma as given in what's his name?

20:00 but I think now is the time when we have to go back now we've got to go to we've got to go to McLean and do it by McLean because he is a section on what he calls universal elements. And that is the central idea here. And we've got to master that before we understand Yonita. But let's look at Yonita. You've got McLean with you. What? I've got a copy of McLean, yeah. But let's look at Yonita in the way it's presented here. And see if we can just glean what it's saying. See, what I find really interesting here is that all the complications come in these things because of this large, small distinction. So I can see why these guys, they keep bobbing their heads up against this business. And I can see why they're pissed off at the way things are done, right? but do they have a serious alternative i mean i mean i you know it's it's not that how do you say it it's not that there isn't a genuine gripe it's the question is the gripe with mother nature or is it with the way we're doing looking at mother nature i mean you might complain that why is physics like it is and you know it ought to be newtonian or whatever and you you know just the sort of puerile resentment that things are as complicated as they are by the nature of them or you or you might think you know that that's just the way things are you've got to live with now that i'm i've all along felt that in my bones that that's But I don't know how to, I'm too ignorant on this stuff to really be able to pin the argument down. But that reminds me of another thing, which I forgot to send you guys.

22:30 I was so wrapped up with the Max in business yesterday. but I've got a thing that Nosen Janofsky sent me which is a set of transparencies for a talk given by this guy, this Canadian and it's a talk about foundations in category theory I mean the talk is maybe 3 or 4 years old it's about foundations, issues and category theory, and it's, we're juxtaposing Kreisel and Lovere as the two protagonists in this argument, and it gives, well, you'll enjoy it, because it's a nice set of transparencies with pictures and little things that balloon up, and the guy really knows his business formatting these uh these things but um it's interesting because i didn't i had i wasn't i'm not familiar with this interchange between uh mainly the claim in chryso i guess but my but lobby was in and uh this literature that goes back to the early 70s and it seems to me that dead right i mean i i at first when i looked at it i had the horrible thought my god did i read this stuff and then repress the memory of it and then just think that i thought it up all on my own because chrysal's arguments are exactly the same as mine i mean i think chrysal is right but the question i mean but then the question but that doesn't that doesn't vitiate what the category theorists are saying it's just he's saying that they do that they're not doing foundations and he has a lot of interesting remarks to make about that I mean for example this is a characteristic sign that you're involved in foundations is that you're moving away from the common vocabulary and into things that are more that are required to be more prolix but

25:00 are fundamentally simpler and that's exactly what happens isn't it when you talk when you talk about set theoretical foundations all you mean is that you're giving definitions in set theory of notions like ordered pair, function, relation, structure, and so on. Well, where does it, my mind is wondering, I'm sure it's this maxim thing hanging over my head that's, I trust it's not just senile decay, but I mean, what I'm, what I'm convinced about is that somehow we have to understand how this business of large and small factors into all this stuff. Because in McLean, he just talks where Bossert talks about classes and sets, McLean talks about sets. He just talks about sets. But it's this large, small distinction that's critical. Okay, so let's see if we can figure out what... Let's write down the Uneta Lemma and see if we can figure out exactly what's going on. I guess the first thing to understand is this business about the representation of the uh represent representable functor and yeah so let's let's think about that i mean when you say something is representable you're saying it what's representing it and how is it being represented so that's what's not clear to mean so let's look at that example so we've got we've got a category C and a fixed object C in it look okay okay so this is the harm function right so we got HOM sub C of C is just the set of all, what is it, it's a set of all A going by F

27:30 such that a, a, f, belong to c, right? So it's just the set of all morphisms into c. It's coming out of C, is it? It's kind of like that. Remember, he doesn't actually talk about that home function. That's all he talks about it for. Well, it's C going to... You have home C and then of A. OK, so... And then you have subsets of that. Oh, I see. Okay, so, okay, so what it should be is, it's, it, this functor is susceptible, it's the union of, uh, home, C, A, such that A ranges over C, right? That's what it is. This is the union of all these. But I'm saying HOME sub C of A is the set of all maps into A. Yeah. Should we just forget about HOME and do it the way he does it? Wow, that was confusing. You can do HOMS that one. What's that subscript C on the top line? Is that the category or is it? So I see, this is a, this is a. The argument is A. So the subscript C just means that C is the particular object that we're using as a.

30:00 Oh, right. OK, in that case, then your definition's right. OK, I thought that was the category. then it's, yeah, that's right, if you put back in A there. Okay, so the whole C is a functor. So it's a functor, so what are we going to, let's call it what he calls it, okay, so what does he call it? So it's curly C brackets, and then C comma A is, um, that's the set of all arrows from C to A. The functor, you take an element of the category, and then that element defines the functor, and the functor takes any element of the category to the set of arrows that go between C and that element of the category. Yeah, exactly. ...to set. And it's given by C of C bracket. Brackets A equals, yeah, no, no, no, without, sorry, brackets A. C of C, I'll do this. Let's go up. A of A equals C of C of C A. We've got Holmes, yeah. Okay, then set the values between. Yeah, let's start with C. Okay, so the first thing to notice is that if A, if C is large, then what then it doesn't go into set does it because there would be a class of these things well there may be at least yeah well so we can't talk about it as going into set but he does yeah yeah yeah If it's not locally small, then we're not guaranteed that the home sets are in fact SATs. That doesn't work. Oh, sorry. So it could be locally small, but still not small. Yeah. By having a proper class of objects. Did the SATs reach objects maps to another great pair of objects? Look at what he says here. Given the category C, at least objects C and C, define a functor.

32:30 No, it's right, but that doesn't work. It has to fit some of the co-culture. So the category of sets, by first putting 5Ca equals that. Okay. Okay. Now, wait a minute. So what is... What is this sign of CA that means? That's the sector of the, well, quiet. That's the class of Irish from A to C to A. Okay, so let's suppose it's locally small. That's what he means. That's what you need here. Given a locally small category, a fixed object. Mind you, actually, he says in his definition of a category, It consists of the following a class of objects, and for each pair of objects, a set, with elements of an arrow. So he's only interested in... So by default, when he talks about categories, he's only interested in all these... All these small ones. So is the definition that you've learned in your notes? Well, that's weird, isn't it? Or is it? Well, I mean, I think it probably is, well, it's not a full generality, but can you think of a category that doesn't look at this one? Well, that's because we can only think of concrete categories, yes. I can think of a proper class of morphisms, that's very odd. I can barely think of a proper class of anything. Well, groups. At ten o'clock on a Friday morning. in the category of categories, I suppose, all these bizarre things happen. Yeah, right. What does he say, actually, on this? He has a little... He has a little... So, where was it? In Wikipedia? Uh, yes. Because it occurred to me that the full generosity might not specify that anything was a sack. Hmm, hmm. But it does. And the Wikipedia does? No, Wikipedia says... that I've found there, and it's usually pretty good for mathematical definitions,

35:00 says that in full generality a category just has a class of objects, a kind of class of morphisms between a pair of objects. But if we restrict the latter to be a set, so for every pair of objects there's a set of morphisms between them, then we call that a category locally small. restrict the collection of objects to be a Santa. It's just small. So he's dealing only with locally small categories. Well, McLean isn't. I venture to guess. I'm not sure if it will make a difference, because you'll have to restrict these certain things. Well, what does he say after the business about, okay, so the problem is, you want to say that, so already the category of categories rears its ugly head, and so what is that category? It's got, what are its objects? Its objects are categories. Its morphisms are functors. Does that just, does that explain it all together? Let's see, what is he saying? natural transformations we know where is this which what do you think for the definition of category that's on page four and then the stuff on no no no I'm trying to get the place where he you need this okay here it is we're okay now I'm on page ten okay so he's saying what he gives the definition of natural transformation and then says it is an obvious matter to notice that when FG and H are functors and you got natural

37:30 transformations that they compose. Okay. Compositional, clearly, so Stephen possesses a unit at each functor f. This is just the natural transformation 1 sub f whose a component is 1 sub f of a. 1 sub f of a. Now, again, in Kara's argument, would deduce the existence of categories whose objects are the functors from a to b okay this is a different this is not the category of categories right no it's just it takes the functions take the functions from a to b as morphisms and the natural transformations between them no no the the objects of the natural transformations and the natural transformations or the morphisms. Okay. But since A and B have merely classes of all students, what do you mean? Not by merely, he means, so we can only general. Yeah, we can only say. We can only specify that we can't say that there, in addition, classes also sets. He does say that. There is no general way to prove the existence of a set of natural transformations between two functions. exclamation point so this is only a restriction for him because he's saying that all categories that are worth the make are the locally small categories is that right so if we allow non-locally small categories this objection wouldn't arrive no no but he has eliminated these other things is he is he saying here that if we tried to construct a category of this kind then oh look he couldn't show that it was locally small and so you couldn't show that it was a category because his definition of category enforces local smallness but we would say well we don't care they can have so if sorry if the underlying categories have a um if the underlying categories a and b maybe we only need to worry about b i don't know but the underlying categories a and b have class have each has a class of objects forget about what the i mean he says it's a it's important well it makes sense it's the one you're going away from it's important it's one where you're starting if a has set size okay so it's so it's b to the a

40:00 that's what we're talking about yeah we're talking about b to the a right yeah that's the one that's problematic and if this is class size yeah we've got problems yeah yeah okay but the problem is specifically the problem we've got is that b2ba would not be locally small yeah and so on his definition wouldn't be a category at all but on our definition on on a definition general and what they were in a more general definition would be just a Yeah. I don't know why he's so worried about that. Because there may be... So, I think somewhere here he says that if you assume there's such a category, you get an immediate contradiction to the thing about the set of all sets. And it's not here. He says that earlier, he talks about Russell's sets earlier. Well, he talks about that on the very first page, and then somewhere in here, he talks about that if you had something, you could, I can't remember exactly what it is, if you had something, unfortunately, if you had such and such a thing, you'd get an immediate contradiction to this theorem, part of this theorem. Yeah, here we go. Where? Page six. So this is, I think he's trying to do the category for a functor from a category consistent on math, et cetera, et cetera, et cetera. So it's like, given two functors, this can put, yeah, so the identity functor of a category A is clearly a careless argument. What page is this? So page six. A careless argument could thus lead to the conclusion that categories and functors constitute a new category. that this can easily be reduced to a contradiction using proposition 1.1.1. Yeah. Oh, well, no, because he then says the point is that in the axioms for a category it is required to have a set of morphisms between any two objects. Okay, so the contradiction you would get is it wouldn't be a set of morphisms, but if you haven't put that in to begin with you don't get a contradiction. So you can see why these guys are pissed off about all this.

42:30 But, you know, unfortunately nature is that complicated, that's the trouble. It would be so much easier if Robbie's law applied all the way down for black body radiation. Unfortunately it doesn't. Well, that's Rayleigh, isn't it? Isn't that the way you pronounce that? Is that how you've heard it? I think I've probably heard it pronounced both of those. When you get your undergraduate education in a physics department, pronunciation is... I know, I know. Well, that doesn't lead to a contradiction with it. It does, if you insist, because it's clear that in the canon... Oh, what if you then take it? Oh, yes, I see. So then you're having to assume that these things form a set. That forms a set, and that gets your contradiction right. Okay, I don't think that's all there is to it, actually. But, okay, so just why not find time with this or something? So this is a different kind of problem. this is similar but different kind of problem, I was confusing it in my head, I thought I remembered this passage and thought it was applying to this stuff on natural transformations, but it's so can we see why this has to be, why this can't be locally small is that obvious? if there's a proper class of objects Well, there's no guarantee that it would be locally small. It's just that if you make it local, if you make A small, then there is a guarantee that you'll get a category. So that you'll get a locally small category, sir. So if A is not small, it's possible that B to the A is not locally small. But if A is small, it's guaranteed that B to the A is locally small. last and you could just do you know everything, one arrow coming in. One more. One more. One more, that's really. Well. It's going into itself, it's a whole bunch of blast points, you know. It's one arrow going into themselves. But, yeah, and then he said the category is going to be small itself.

45:00 If B is also small, I guess that's obvious, because B to the A will then be less set. And so the function is coming in. b were set and a were group this would be this would go tits up wouldn't it b were set so the groups are what forget the groups into them um well i thought the objects would have to be class size in order for this to really mess up oh the objects because if they were structures they'd have Yeah, it would be class-sized structures, yes. Because if you had set-sized structures, you're normally going to get set number, set-sized home classes. Yeah, OK. Yeah, so group to set. We're going to get a finite number from any group to any set. A set number, a set, yeah. Sorry, we're going to get a finite number from any group to any group. okay so all right well bearing all this in mind then which I I can't even remember what my difficulties were let alone what the solution do it just you just want one good point since you're talking about something I don't know if I said to you John Byers says he doesn't like the we've got to talk about the category of all categories, because really the collection of all categories is a two-category, because you've got the categories as objects, functions between them as morphisms, and where are the natural transformations? Well, there are two morphisms going between functions, and this progression fits perfectly in the definition of the two-category. In fact, it's the archetypal definition of the two-category. But I don't think about two categories, but I presume they're categories, aren't they? No. No, no, no. They've got extra stuff. They've got objects, morphisms, and then double morphisms, which are morphisms between morphisms. Oh, I see. Right. But I need two categories. will be associated with the category, just by taking the tumor to double morphisms.

47:30 Yeah, if you just strip out all of the tumor morphisms, then you get a category, yeah. Just like if you strip out all of the morphisms from a category, you just get an underlying set. Yeah, yeah, sure, sure. Or, or, box. But I mean, he's, you know, he's not saying that somehow you would be wrong to call the category of cancer as a category, it's just that it's more useful to call it. Tells more you have more about it. he thinks there is a category of categories um i guess he well i mean i'm i'm prepared to i'm prepared to concede that there's a category of all small categories okay that seems obvious this is no more problematic than the category of all groups right i'm going to have to guess that john byers doesn't bother himself too much with these foundational worries about stats and classes? Well, I mean, what happens though in this time? Because I don't know enough about this, but has anyone proved a nice result about the category of all categories? What are things known about? Is it? Well, a lot of people think they have. Useful to talk about. The suggestion is that because there's no big so-and-so's theorem on the category of categories, that suggests that there ain't no such thing. And there are a lot of guys that have got wheezes about how to do this but nobody actually knows do we see it sink into a silent reverie each of us lost in his own thoughts No, no, no, I'm not. If we move on to your latest limits. I was just wondering whether this had a use or whether the debate was just a sort of academic one. Well, A, we're in a pretty poor position to be despising purely academic debates. True. I just wondered whether they needed them or whether they could just... I'm pretty sure it isn't just... In any case, I don't think it is purely... I think it is, I think it does hit on practice. Right. I would guess that, I mean, there are certainly ways of turning talk about how you create categories or how you could define a certain category and just put all of that talk into category or talk itself, in which case it would be useful that you would be able to

50:00 have operations whose co-domains would be nominally the category of all categories. So you might want to say for any kind of this thing we can dig up a category out of it. And so the operation that does that is a mapping that takes things from over here and produces a category. So it's a mapping from the category of all those kinds of things to the category of all categories. So you'd at least want nominally to be able as if there were a category of all categories well you can do that i mean you know let's restrict that well you can do i mean i wonder if the groden deep trick doesn't get or doesn't allow you to do something exactly do exactly that kind of thing so that so that the category of categories isn't some particular object at some level in the universe it's a kind of it might be the function which at a given universe produces all categories over that universe including those in which it's including large ones over that universe I don't know well that seems to be far away from the issue immediately at hand which is the name of this now you can see I think if you look it up here in McLean I think you'll discover My God, I didn't realize it. This is from chapter and verse. This is my purchase for Wittgenstein's remarks on the foundations of mathematics. What is it doing inside my copy of... Just like putting some kind of obscene... You know, a bill for, you know, an invoice for an obscene publication inside the Bible, you know. Jesus. Here we go. There it is. You can look, you guys can look on there. It looks like it's the same thing, but... I've got a couple of lists of the notes in there. Um, okay, so... So how... So what does it say? That's exactly what he says.

52:30 Is it exactly what he says? Let's write it down and see what that is. He does. He says specifically what the hijaction is. But that's... Notice that instead of a three page... Notice that McLean has a doodle So does Wikipedia, and Wikipedia says the proof, it's almost exactly where the proof is indicated by the following commutative diagram. Range 1 on Wikipedia has a double square. I don't know, that's a square within a square. But that must be what's going on. Okay, so we've got A. We couldn't even find that kind of system notation like this. No, no, exactly. the proof that indicates it both. Okay, A is an arbitrary category, whatever that means. Okay. And S, the category of sets. Okay. And F, Wrapping A to S. A functor. Functor. Okay. Yeah. Let's see. Sorry, no. Yeah. Okay, we've got an arbitrary functor from that. Okay. Okay, so A lies in A, and whatever this is, so I guess we'll, how are we going to write this as you write home in a line. Is that the way you could do it?

55:00 Well, the way Bolsa does it is just to use the symbol for the category instead of home. So he brings that A that you've got as a subscript, that's a full thing, it just deletes the whole. Which is quite elegant or something. Well, it's not clear to me that the harm thing may actually be clearer what's going on. So A of, what was it? A blank. yeah okay so if I fill in the blank I get the set of all mass morphisms from arrows from a to the B if I fill in the name of B so this is mapping a right the set is that right yeah and now so what is this on an arrow that is So, if you've got C arrow D by F, then A of A and C goes by F to D is what? So that has got to be a map from... It's a map from AC to AD. So we've got A goes to C, and we've got these maps, a whole bunch of those, and also a bunch of them going to D, and we've got this one map here, F. And And so if I, so I want, yeah, okay, so if I want to take this one, G, then I just compose it with F. Okay, so it's blank composed, so it's F following blank, right? So it's just F composed with blank, right? Yeah. Yeah. Is that right? Yeah, so that's evaluated at G, is that?

57:30 Circle G. Okay. Okay. Okay, so the content of the limit is that there exists a bijective correspondence Now what is he, what does he call it? he calls it theta, some F and A, because this A has been fixed out, fixed out, that's been actually picked out, right? Yeah. That particular object. Now it's going to map what? The natural transformations from this functor, this representable functor, Uh, port A of A and blank, and F to F. Okay, now wait a minute. And we want to say that this is going to put this in the one-to-one correspondence with, wait for it, F of A. Yeah.

1:00:00 now what's this oh that this is my okay so there's that such that what between natural transformations between natural transformations from this puncture, this representable puncture, to F, and elements of what? of the set F of A What the fuck is happening? Does anybody see what this means? So let's go slowly through it. So wait a minute. There's a little bit theta of F A constitute a natural transformation in the variable A when A is a small when strict when category is small the bi-directions also constitute a natural transformation in the variable F what does that mean? that's extra stuff that we might understand if we understand the first thing so there's Right, so it says, observe each of the notions, universal arrow, universal elements, representable functions, subsumes the other two. That's a universal arrow, I don't actually know what's going to say relevant, is it? But this is what he says, the argument for proposition 1 rests in the observation that every natural transformation from a, a, comma, dash, into f is completely determined by the image.

1:02:30 Every natural transformation wants to change the rotation, but that between that and that is completely determined by the image, and I don't get what this bit says. Um, the image under, sorry, this, A, the image under that of the identity element that goes from A to N. Okay, so wait a minute, what's the set up? So we've got... So what's... So we've got, let's see, we've got... Oh right, so to figure out what a natural transformation has to do, you only have to know what it does. To the identity? To the identity. To, no, to figure out what a natural transformation of this particular sort... Yeah, in this case, sorry, yeah, yeah. So can we draw the usual prism diagram and see what the cast of players is? So you have A here. And so suppose you have two. Well, this is a bit confusing. You've got three things. We've got a set down here. Yeah. You've got three things up here. So you've got your initial A. and then you've got B and C some bunch of arrows there's a bunch of arrows here and then the mapping F, well it's just going to take if that is F, B and C I don't know anything more about that but A and the other takes so wait a minute, this thing is but you've got the things going the wrong way. These have got to be the sources of the alpha arrows, or these things, not these things. Oh, you mean this should be? Yeah. Oh, I see, yeah.

1:05:00 Yeah. Between A and B. A, here, between A and C. And I'm right here, I think. At B, sorry. Yeah. Yeah, let's see. yeah okay and here's alpha B I suppose you have some iron here is alpha a alpha Farrell Yeah, whatever that is. Okay, that's already making a bit more sense. So what it says is, do you have a piece, Judy, because I still repeat this song in the darkest, blackest bitch, dark night? Okay, so this set is these ones, lovingly recreate, and this is these ones, yeah, okay, so that's the set. let's set that okay so this is just this is just composition by phi on the right I didn't mean thinking that number was yeah it's this so the way to understand this is not by talking about connection by understanding what exactly so many of us looking at the point is trying certainly so what we want to understand is again i get a natural transformation is determined just by the uh... How does that sentence finish? No, no, but actually, may I make a suggestion here, because I think this is important. I think what is going on here, partly, is people's refusing to talk in constructive terms, right? And what you actually want to say is not there exists a unique bijective correspondence, but we can effectively define A, right?

1:07:30 We can explicitly define A. what's being said. But supposedly McLean gives a more constructive... No, he does, it's just I don't understand what this, what his subscript notation is. Ah, right. If we knew that, if we knew what this thing was, a underscore. Um, subscript. Because it can't be the natural transformations. You know, it can't be what we used to call that, what's the natural transformations? That value at A to A, because that's what we want. Well, it's a value at B. Yeah, but that's exactly what it can't be. It's a notation part. So I don't know what it is, man. Oh, I see, yeah, yeah. See what I mean? Well, there is an alphas of A. Yeah, but there's no good here. It's not good. But he's saying that any natural transformation between this functor and this functor, sir. Yeah, that's this setup. Yeah. Is determined by... Just by its value. At the identity. At the identity. That's astonishing. It's astonishing, but at least it's intelligible. I see what you mean by that. A is a function over the whole category. that takes, yeah, the result of this image, the result of this function. So if we add... Yeah, so that's... So if we... That makes sense. No, it is a sign that we can make sense. And I can prove it. So, we've got one of these outfits for every possible... A particular one for A, yes. Okay, and now you want to go over to A here as well. Yes, yeah. Yeah. So you want to go over to A there, so you want to. Why have you got A at AA? Um, because we're looking at these sets here. So this is. So is that a phi or an A?

1:10:00 That's a, that's a, that's a curly A. Oh, okay. Um, and. And both of these are, all of these things are curly A's. Is this right? Am I doing this right? I don't know. The point was that it takes anyone... This guy here is the set of all arrows from A to A. So identity is one of them. Yeah, identity is one of them, and what's the... What does a value offer? Alpha A. Yes, what is Alpha A? What does Alpha A do to the identity? that determines what it does every other value so how does that correspond to what bias that I have to be in? I get it, Jesus, I don't know but he confesses that he never understood what the original limo was so I mean the fact that we're the fact that we've got the fact that we're confused we can take heart for the fact that John Byers confessed confusion it's just that there are so many different levels of it it's like that I was getting very confused with that about the double deal last week because there were functions and functions and functions you've got to remember the difference between taking the value of a function at a function and composing two functions and that's what was confusing me in that example yeah so that's that's why i was tempted to use the lambda notation yeah which i think you know all the stuff is sort of especially when you're doing these homsets and stuff i mean it's calling it it's crying out for it yeah back when i was going to that algebra course a couple of years ago, the guy just sort of knew what the diagrams ought to look like and knew what the formulae ought to look like and had such a well-developed instinct for this, but he didn't even bother to make a distinction in his notation a lot of the time between composition and higher order application. So a lot of the things that... So you might define a delta or something or a particular object

1:12:30 as actually a function from some idealized thing into the topological space you care about. But then the functions that we define take those things and turn them into something else. So they're, in that definition, functions that act on functions. Well, his notation wasn't really... Well, the point is that... ...a distinction. that every function, I mean, if I have that, if I think of this, then lambda, well, I guess lambda phi, that is a function Yeah, yeah. That will take that could be applied to psi and give us a value, right? Well, we know it gives us that, right? So that function composition is itself... Is a particular case a function of application, namely the application of one or the other of these lambda lines? But then, of course, we've got the function that takes phi and turns it into this thing, and that's itself a function as well. A function called apply this thing on the left. so do you think you understand what this how how this works well that's that's why it would have a name attached to it yeah if it's surprising and true then it's worthy of a name if it's just true mere truth doesn't carry much weight in these circumstances it's going to be surprising and true we should try to just define when it says it's just determined by this but how does it determine that so suppose I have a natural where is his diagram what's his proof This proof is just this doodle. The proof is indicated by the following community diagram. Of course, he's got all these different... So, suppose you have some B. You have to define for each B what alpha B is.

1:15:00 So, for every B and for every map from A to B, you've got to define which member of the set it is. so that shows the bijection, that defines the bijection, and after that you've got to show that it's a bijection what's its value of this given that we know this given we know what the value of this identity is well, you see, okay let me just speak in highly poetic terms, it's not surprising that knowing that of the of this fun of this transformation on the identity map here because a action after all is playing a very special role in this whole business namely because we're talking about the home the this home based on a yeah so I mean so I mean if there was some other thing in there I mean then it would be surprising, but the fact that it's A is surprising that you only have to know its value at one function, and at one element. I mean, admittedly, the one element being A isn't so surprising, but, you know, I mean, basically, often we're determining things. This may be another way of just saying if we just knew what alpha, if we just knew the map alpha A, that may be what it's saying. Well, I mean, that's, but it's actually saying something stronger. We only knew one value of the map. Alpha A. No, that's one value of the map. Alpha A. What I'm saying, if we only knew the value, if we only knew how to calculate Alpha A at A. At the identity on A. At the identity on A. So, look, you've got this. Wait a minute. You've got to know all these values that each one of these are. arrows taped down in its respective set and all you know is the value that this is a here is these are a's not five's right yeah okay so you've got down here so a gets matched down to here so this is a a so this is some big set of stuff as well but one of them is distinguished which, namely, the identity, so we'll just draw it like that.

1:17:30 And that goes down to F A. So just knowing, well, there we go, alpha A and that, you know, just its value then is enough to determine all of the values of these in their respective sets. Yeah, I mean, I believe it, because I believe it's going to, things are going to filter through this. Yeah, well, this is a kind of choke point where all these maps are going to have to filter through somehow. I mean, I'm not claiming I see the proof, but I'm claiming that it doesn't seem quite so implausible to me. So what the... But I mean, let's just... I'm the happy guy that doesn't see what the problem is. So what the Lammer is saying, in some sense, of the natural transformation is very rigid in that once you've got this one bit of data now and now, then everything else is locked into place as well. That's right. There isn't further flexibility that you can... There isn't further flexibility that needs more information in order to... Okay, but where do we get... Where does Bias' remark come in? I think once we understand... But, I mean, there's got to be something to do with these functors from... that somehow, functors from categories to set have nice properties. Good. Okay, well, that's... but they're represented by maybe the point is that the map of alpha at the identity can be construed as a little structure of some peculiar sort sitting at A right? and that's the structure that this it's saying that this is that could that be it? oh yeah this is right

1:20:00 do you want to know what the value of this one is at some particular value, add some particular one of these to this thing lies in here so that thing's in here so before Stuart buggers off let's just remind ourselves what what we thought that the pot of gold at the end of this rainbow was initially. So the question is, are there essentially any categories apart from concrete ones? A concrete category being something defined by means of one of these structures, you know, okay, and by a type of structure, okay. and now we thought or rather I thought when I saw this thing that Stewart had excavated that Unita's limit in effect says no there aren't any they're all concrete categories because every non-concrete category can be in some sense instantiated as a concrete category and they're just equivalent in the appropriate sense so that so that would actually solve a lot of mysteries it's not going to get rid of the big large small problem because that's going to be persistent at least it's going to say the sense in which I mean because that remark those remarks of McLean about you needing to get right generality maximum generality, because Bulbaki was heading toward a general, he wanted to do all this stuff about universal elements and so on, but he wanted to do it via some humongous general definition of what we would now call concrete category, a general definition of a mathematical structure. And McLean says, well, that was the problem, you didn't want to do it in that generality yeah well I think there's we got a lot there's a lot of food for thought here usually carry on depending on how long you carry on you might even still be here when I'm finished talking with James or I come

1:22:30 back and join how long are you going to be with James I don't know our meetings when it's the three of us tend to run on for about an hour well it's just the two and Alexander is gone because of the baby okay, do you know the answer to the following mystery, is the baby's name Isabel or Isabel and if it's Isabel this Richard can address is that a well known Scottish variant of Isabel Isabel is I don't know whether he's called So Isabel is a Scottish version of Isabel well it's plausible then that it is Ishmael that makes sense Alexander's always claiming Scottish connection so are you his accent sort of betrays something I didn't even know that the baby had yet been born so I'm not going to return to you didn't even know that it had been born but I'll find out more But all right, well, I think you've solved the problem. I think it is Ishmael. When I finish talking with James, I'll send you a... Yeah, well, let me... Okay, yeah, yeah. Because I think we might... We might get lots of to get through this. Yes, I'm there. Yeah, so if we can, it would be productive to get through it. Well, this is the diagram. Okay, okay. All right, so you be the expositor, listen. Well, I don't know. Well, it seemed to me that you'd convince yourself that you could set up the problem, which I don't even see how to do yet. Okay, we'll keep that. I think that would be better it's better just to keep the let's just remind ourselves. I mean, I've got I've got trouble remembering what the claim is. So the claim, well, let's just Let's see, we'll just, just my job is for quickness and that question. I have to redo our natural transmission going down the way, don't we? You don't have to put the dots there. No, let's try it.

1:25:00 and that goes by and that goes by so the claim is that that we've got a universal construction A is a natural transformation alpha, yeah also alpha is a natural transformation so alpha between well, I don't know, why do you do that From this arm puncture to F, then, well, yeah, I guess what this claim is is that if A1, A2 are natural transformations for A2, then, and a, and that's a bad idea, alpha and alpha dash, alpha identity from a to a equals alpha dash identity A to A A equals alpha dash equals alpha equals alpha I think that's saying too much actually Richard I mean what does it actually say it says yeah I know but Well, what we really say is that there's a bijection from... So we can construct, we can define a bijection. Define bijection from the natural transformations, what does he call them? Natural transformations. Nat from A. A of that HOM functor to F. Yeah. Yeah, there's BITACs of WIF. I put that in. what does that mean? That just means, sorry, what does that mean? What does that mean? What does that statement mean? What does it mean to say it's isomorphic with F of A?

1:27:30 Well, just that it's a bijection. Does it mean it's just a bijection? Yeah. Sorry. No, no, no. It's got to say something else. what it's got to say is that to every okay now wait a minute it's got to say that to every natural transformation of that kind what? F of A now is something in the category of sense right so there's an element of F of A Let me just look at this now. I think it's just meant to be a dijection. I mean, McLean's statement of it goes on to state the dijection, which is this. takes this so some natural transformation from that yeah, from between that and that that gets mapped to alpha sorry, you can't have an A as the first argument Oh, OK, no, all right, OK. Oh, I see, it's... It's got to be mapped to what? It's got to be mapped to something in F of A. Oh, it's not as much as we thought. It doesn't say as much as we thought it looked like. It's that composed with... Well, that's just... Well, let me figure out what that is. that of that. That's a bijection. That's what the theorem just says out right. Let's try and prove that and then we can think about what it might actually be saying.

1:30:00 So it takes any natural transformation between the two things and matters if there is something here. and its value, the value it takes as... At a given natural transformation, alpha. Yeah, so that's, yeah. Alpha. So suppose we have this alpha. Is, so inside here we've got the identity element at A. Okay, so that's an arrow per meter A. So that's I, I, and 1A, which goes from A to A. So in there, so what it takes this transformation to, natural transformation to, then here is, yeah, it's alpha, alpha of that. So every, so the bijection. Alpha of A of that. Yes, yes, exactly. Alpha of A of. thing. And that's a bijection. No, that's not surprising that you can go that way, right? Well, in fact, you get that immediately because you can define the function. I guess the surprise is that, what? Is that this injector or is it a surjector? So, I was trying to prove, so what's being defined in a sort of slick way that makes everything more straightforward than we thought originally? Well there's not how You've got to be able to define it by ejection. I guess it's quite a high level one, so let's call it that, right? Yeah. So the value at a natural transformation, this and this, of phi, is that element of this. So that's an element of this, definitely. You've got that, first of all. that natural transformation takes the identity of some affirmation into some number no it doesn't it takes the identity map

1:32:30 to something riding on the arrow there no it takes because this is taking it as an element as an element of this thing it's a member of this Yes. And objects in this. Okay, so it takes that map to whatever alpha A takes. Yeah, it takes it to, yeah, exactly. So, so we know that this isn't a function from this to this. Now I've got to show us a bi-jection. Sir, suppose you have two alphas. You see if two alphas go to different things. So if you have that, then alpha... What are subscripts? Alpha and alpha. Or alpha and beta. alpha b and alpha c i think alpha sub b and alpha sub c no no you've got you know it's got to be you've got to have two different natural transformations just separate completely oh okay okay alpha prime over alpha and beach okay okay so you've got alpha and beta so that That means alpha at A of the identity equals beta at A of the identity. No. Why must those two always be the same, though? No, that must mean that, uh... Okay, now from that we've got to deduce that alpha sub A is equal to beta sub A. No, we have to deduce alpha C equals beta C for every object C in the category. Yeah, exactly. So alpha C at some... No, alpha C just equals B to C, right? Yeah. So alpha C from... Alpha C just equals B to C. Yeah, but to prove that's a map, those are two maps, so only way to show that they're the same, you have to show that they take the same value,

1:35:00 the same argument. Okay, so why... So what sort of argument do they take? Alpha C is a mapping from A, C, into F, C, and Alpha, and B to C, is a mapping from A, C, yeah, we're both mapping, yeah. OK. Yeah. So, well hang on, let's use me, because that's what we've got here, and that's what Niflaine's diagram suggests that we use. Right. Well, it doesn't matter. So you want to say alpha B equals B to B for all B. Yeah, that's even all B. And up in the top line, too. it's funny how i'm already beginning to absorb the natural transformation thing so it doesn't seem complicated yeah i think i think first you first look at it yeah look at the alpha yeah exactly and now it's okay but okay so let's suppose we have these two here, so alpha b and b to b on our map. Right. I just don't see any of the phantoms. But what is the diagram? So what's an element of this? It's an arrow from a into b. So I'm alpha b of f from a into b is the same element of f of b that, no, yes, yes, is the same element of beta b of f from a to b. So suppose you have some a to b in here. In fact, we We have one here. Yeah, okay, so we've got that. Sorry, it's being here. We've got to say what value this takes. So we've got to use two things. We've got to use the fact that everything's coming out of A, and we've got to use the fact that these things, these are natural transformations. with the same thing on the left-hand side. No, no, it isn't the same thing on the left-hand side, is it?

1:37:30 Yeah, it is. It is the same thing. Because what you want... No. Shit. What's going on? The natural transformation is that this has to commute. I guess that's what's going to give us a... I don't see how many of these sensors are starting here. Oh, my God. Let's go to the lobby here. Oh, hi. Did you get my email? Okay. Okay. Well, okay, but I think you'll have to just redo it yourself, because, I mean, it's not just immediately transferable as I've written it, because I've got notes to you and so on in it. so if you notice in the in the bottom part I've got little notes in caps saying is this what you mean okay well okay you got them I was afraid that I was in such a rush when I sent it I was afraid I turned off the the Frank was afraid I turned off my mold break before it had actually

1:40:00 gone but okay so that's all right if you need any if you need to talk about anything just bring me okay all right fine great then then that'll solve your problems just get it in the mail okay i'll talk to you later richard and i are working on on the your native swimmer okay talk to you later fine who's that arena she was she she wanted some help in getting with her english on her application. Oh, yeah. Right, this is better. First of all, it's the inverse map. Yeah. So instead of showing that this is injected, it's injected directly. You know, you've given this... What map am I talking about? Yeah, but this inverse map. He gives the inverse map to phi. Yeah, so we've given this. Sorry. We've got a map that takes things from the natural transformations two functons, from this functon to this functon, into the second one. And what is that? Now, with this one. That's fine, then alpha is that. Okay. And now he wants to define psi of... Of some A, which is where A lies within F of A. Yeah. Okay, so he gives that. Okay, and it's going to be a natural transformation between this and this. So it's going to define its value at some b, yeah? So it's a natural transformation. So at b, which is here, I guess, yeah, its value at b is going to be f of. I'm sorry, B after A. Yeah, let's just write that. So you want to define, for anything in here, a natural transformation from this to this. The natural transformation takes, for every element of the category, A. So it means some B in A. it takes

1:42:30 the elements of this for the elements of this so it takes an element of that namely F so in this thing A is in F A B is in category A and F maps A to B and then this is the function And so, wait a minute, so how does this work? So where should this whole thing end up, then? It should end up in this, it should be a natural transformation. This thing here, oh, you mean all this, should end up. So that f of little f of a. It should end up in that of a. sorry, f of little f of a is a natural transformation or is it the result is it the application of the natural transformation to some element in other words I'm asking where does phi of a applying to f phi of a sub b apply to f where does it end up here in f of a again because it takes an element of f of a into a natural transformation from this to this. And what's a natural transformation? Well, it's a big math that takes any element of category a to a function, this thing here, that takes numbers of this. Oh, no, sorry. Yeah, it's meant to end up in f of b, sorry. Yeah. Yeah, so this whole thing should be in f of b. So, I guess the thing to check is just that they are inverses of each other. So... I can't remember about the original one. The best one. You've just got two functions. So, I guess it's looking to write them, I'd say. All right, you're good. So, let's take phi. But the problem is phi doesn't occur in the bottom one. No, it shouldn't. So you want to do phi of... No, I shouldn't. I shouldn't do psi. You want to do phi of psi of A.

1:45:00 Okay. And psi of phi of A, presumably. Yeah. That's right, John. Right, okay, so... OK, so what does that do? So it takes it to this natural chance. Sorry, phi of anything. Oh, well, you've got to see what that psi of A does. So it's going to, phi of this thing is going to be this thing at A. So it's going to be psi A at A of 1A. Yeah. That, which is this, equals F, first of all, of this F was your identity, F of identity identity at A, evaluated at A, which is going to be F of the identity of A, it's just going to be identity. Yes, of course, because it's fun for us. Yes, it's got to be, this is going to be the identity of Little A. Yeah, so this is a map from this thing into itself. Now you take it then by its function to here, so you've got its image. Under Little A, yes. Yeah. And then so there's got to be a map. Yeah, so hang on. I think it's one F A. That would be bad news. It should be one A. All A. Because that would give you A. That's what that's supposed to be. It's supposed to be an embarrassing sense. Well, what's once a little A? It's the identity. From what to what? On what map? On what set? Well, what level are we working on here? That's exactly the problem, there's always a problem there.

1:47:30 So, OK, what's the F? There's a filter from this to this. So what does it? Subjects to objects, and maps to maps. What does it do to this map? A to some map. The identity map from A to A. Well, it takes A. So it's f of 1a is the map from that to that. It's the identity map from that to that. So is that what you said, 1 subscript f, eh? Yeah. Sorry, you're right. So 1 subscript f of a, which is, of course, a. Because we know, because this is the problem that we're, as another mysterious category where you can't say anything about what these are we know what the identity maps are upset yeah well we know that well in particular we know the identity arrow is a map yes yeah yes exactly sure so okay so we've got that direction So, what's 5a? That's a subside of this thing of 1a. of 1 a. So that... So, Psi... It's just going to be, I wanted to have taken an element of, just, oh, no, hang on, hang on, It's because A is a natural transformation, I think, under a set. Yeah. Yeah, alpha. So if alpha is a natural transformation from this to this, then you take this into here.

1:50:00 so this this is an element of that and what does phi sorry, wait a minute I'm just completely bloody lost here okay, so alpha is a natural transformation sorry psi is supposed to be operating from where? psi goes this way phi goes that way so psi is supposed to be starting from element of f of a. Yeah, and taking it to a natural transformation between the homoset of a and a. But phi of alpha, is phi of alpha an element of f of a? Yeah, because that's the really other way. That takes a natural transformation from the homoset into that into f of a. Okay, so it's in f of a. Okay, so. And what element is it? It's this one. It's the one that's alpha of A maps the identity to in here. Okay, so it's an element of two, but you know that its value... Yeah, so okay, so we've got to show... So we just have to evaluate, we just have to calculate psi of alpha sub A at 1A, and that looks like No, we've got to have this. So this is a natural transformation. So we've got to show what it does at some value. OK, so it's that, that, that at some value. But we know that that's just f of, f of alpha. and that's alpha a and one sub a has got to be one sub f of a right it's got alpha a alpha sub a evaluated at one sub a has got to be alpha it's got to be one sub f of a no because that's just It's not a map. It's not an error. Alpha of A is this natural transformation from this into this. Yes, and if I apply it to a particular function, and here it comes out just as a member of this. That's right. That's what I'm saying.

1:52:30 Alpha sub A applied to one sub A is just one sub F of A. One sub F of A is a map. yeah so is once of a yeah but that's because this is a set of maps this is just a set nevertheless ah so these could just could be sets around ducks or fishes so the identity element could be anything on that is what you're saying so show me the identity the element. Oh. Shit. This is a complicated way around it. So I want this. And why is it happening? What is it? There's something funny going on here. Okay. Let's start all over again and talk our ways through it. So what have we got to do? Okay. We've got to show that, sorry. Let's start out with the ingredients. So alpha is a natural transformation. Not a transformation. Okay. So, we take it by phi into active A. Yeah. Which is a sum sac, a bit of alpha. Yeah. And then back into another natural transformation by Psi, because we're very able to submit it now, by Psi. And we have to show that it's equal to alpha. That's what we want to show in the end. We have to show these things over. Yeah. So, to show that we have to show that that this thing evaluates its... Some subscript... Sorry. Wait a minute. Phi... Psi of phi of A is some natural transformation. Yeah. So we want to see what it does at some... Yeah. So we have to take some B in the original category, and then some F, which is a function from A to B. Yeah, basically something here, an arrow from A to B. So it's kind of a value to the x. We have to show that the fact is equal to the fact.

1:55:00 Any choice of these two? Why alpha B? Well, because we've got to show that this equals this. They're both natural transformations. So we've got to show that they're part of your opinion. No, I see. So now the question is how do you calculate the left-hand side? So, this life-hand side is equal to, by definition, F of F of an A. so A is phi alpha here phi alpha is is alpha so this is our problem because this Oh, fuck, that's my... It's because it's not that, it's, this is exactly the problem that we were talking about, it's this, of that, and that function evaluates to that, it's not composition. It's about this gives a function, and then you evaluate this. Right, right. So it's not my own. Yeah, the notation is really piss-poor here, somehow. I mean, huh? Yeah, it's to do with writing things in the line. OK, so it's that. It's confusion between arguments and, you know, ambiguous values and functions. So it's that thing. Okay, yeah, yeah. So it's that of alpha A1A. So what's alpha A1A? It's a number of F of A, and a number F of F is this, is a function from that. 1a is now an element of the thing at the top, that's a, a in it, yeah, yeah, it's an element of this thing, yeah, alpha a, alpha, it's what I'm saying, together, and then f of f of a takes over to f of b here, so, we've got to show this is equal to that, now it's alpha b of that.

1:57:30 It looks like there's something commuting. Yeah. So it's saying, it starts off with 1a. So we know, because this commutes, that alpha a, 1a, f b equals, and that's exactly what we've got, this and this. Okay, yeah, of this thing, equals this map, ah, and this is just going to be, end up being, yeah, okay, that's really good, so that, alpha B of this map of the identity thing, so that is, well that's that's a map from uh the heart of harm let's see that's a map of these hauntings and then that's going to be the identity right on the other side composition such as kick it with the same thing so it's going to be in fact yeah because you take identity here in a you know Now, by f to some function from g, a to b, we saw before that that was the result of applying this map to this is to g equals, let's try it right, does it? Well, it doesn't actually matter, because both of them were. No, no, it doesn't matter, it just has to start playing f, because there has to be f line to 1, which is just f. So here you just get f, so that's it. That's it. Which is what we have to prove. Prove the uniddle. I didn't expect that.

2:00:00 No, I didn't expect it, because it's three pages in Borset's book. But, I mean, maybe he's spelling it out in such gory detail. I don't know. I mean, this is pretty gory. Mind you, that's gory. That's three pages, right? Yeah, considering you have to filter in all the things you said. Yeah. Is that Stuart? it seems to be receiving the same message right now okay now we know it's true what the hell does it say i mean i was prepared to give it a prepared to concede it was true before you actually went through the arc. So what does it have to do with... Yeah, because McLean goes on about determining things, that you determine everything just by. I think we need to read what McLean says about universal element. Yeah, I think you're probably right. But that, you can see what's happened. you determine each and this is the sort of surprising direction right there that you can map each element of this back into you can define a natural transformation from it and also I guess the other way that any natural transformation can be defined in this way any representable things. I mean, Okay, so what are these representable things? So they're sets of markers. They are like they are like Cayley's picture of a group as a group of transformations. So this is that's what somehow we've got the factor I know, I know. That's what Lovere says. So, somehow, I mean, somehow these arbitrary categories become, let me just put it slop-away,

2:02:30 algebras of some kind of, algebras of functions of morphisms of some sort. Yeah. We become algebras of these functions. Function algebras. Okay. Or arrow algebras. okay and so the idea is what is the idea so it's the original the pot of gold that we were expecting about it at the beginning was that every category is just a concrete category of arrow algebra of some species of arrow algebra i guess that's what we're trying to say Yeah, this sort of says it. But I wonder, it might help, because you see, this whole thing is stated in terms of some arbitrary functor F. I wonder how you get biases inside F that it's putting in. Particular F. I guess the identity functor is the one. I think that seems to be the one that most of the natural transformations work with. So it somehow says that any natural transformation between the identity and the home set of an element of a category is just the same size, isn't it? All the information about it is encoded in... No, it's not, no, sorry, it's not identity. Sorry, it's not, because you don't get identity from an average category. It's not, it's something like... Thank you.