Discussions incl FW Lawvere & M Wright on Bimodules & other topics (contd.)

0:00 And theorem is still true. So this gives you the following, that you have a criterion for totality under this assumption together with the scope of Cleveland. Now, recently, and then, we can do, in extending this idea, is the following. The following is true, that is my theorem 4. Anyway, A for complete, and suppose you have Z small, such that when you iterate this process out by turns, out by the smash column, you get O of A. If you would like me to comment on how many algorithms you can have, and I think there's only a maximum number or so.

2:30 That means that there isn't a limit. And also, I'm just trying to prove the order of the algorithm. Yeah, there are two kinds of ones. But I think there is, I think there is, I mean, see, for life's sake, I think this is, well, See, you can sum this all up like this, that if you have three algebras, that's an ideal, and you can't conclude that all of those are the same, but if you say you have four, then you can conclude that you can't make the best copy of a figure, because, I mean, you see the thing is, you have no...

5:00 I was sort of half listening because I was trying to get some notes. What was this point that you mentioned you'd run past Bill O'Bear if you had a similar idea about characterised and characterised sets? Oh, well, it's rather similar to something he was saying to me this morning. Well, first of all, the category of sets, again, is a total category. You look at the innate embedding, and it transpires that the evaluation of set-valued, contra-variant set-valued functions at 1 is less accurate to the innate embedding. Sure. Okay. Now, let's see. 1 is an initial logic. So we recognize this thing as using... This is just good old-fashioned inverse limit. Yeah. And... And it has a left adjoint delta, the very diagonal. In fact, you get this much for sets to the a of a small a. You get such a string of adjoints. And that doesn't characterize what you're getting, unfortunately. But in the case of sets, because this is now diagonal, And because the empty set is the terminal object in sets opposite, then evaluation of the empty set is the co-limit function, so it provides you with one more adjoint.

7:30 And then inspection of what results then shows that there's a further left adjoint to the evaluation of the empty set. If I'm not mistaken, it's the functor that sends a set x to the x-fold sum of the functor Han-Blanck-Hemdek. So, for the category of sets, we have this particular configuration of adjoints. The problem is, if you have a total category B, and the string of adjoints extends that way, is B necessarily the category of sets? Well, it would be a conjecture that it was. That's, well, yeah, I'm raised to that as a problem, some years ago, and maybe the Walter referred to that, but, uh, I haven't done any work on that. Well, certainly, yes, the idea of having a, you know, unique characterisation of sets, just in terms of the way that adjuncts extend, uh, that gives you just set and not any other closed category, no, not any other total category. I mean, it's sort of a sub-referential characterization, because the innate embedding is already using sets, but it's still going to be nice to use. Well, yeah, except that you can think of the United in fact, you can think of it all in terms of comma, category, what, yeah, yeah, I guess, this is an old sort of, an old story in philosophy of mathematics, I mean, can you really think at all without implicitly having an ambient classical topos, without implicitly having some, and obviously I have to have some version of operational collection, and sort of set up the language, so, so it won't actually help anything, because that's, I was hoping this would, this would help, I mean, yes. Another interesting thing, incidentally, about the category of sets is that you have no such string of adjoints extending on the right. You see, if you, just for a total category, you look at sets to the B-office, and then you look at the other gadget, the opposite of sets to the B. So, totality speaks of the left adjoints united. Now here we have what I'll just call Z for collionator for a moment and to ask for a right adjuvant to this thing is to ask the B of B total, so called B cototal if this thing has a right adjuvant. Now, if the left adjoint to united exists and the right adjoint to pro-united exists, then they interact nicely with the Isbell conjugation factors.

10:00 I'm not sure which way around they go right now. But if you're thinking of these as down-closed subsets of an ordered set and these as unclosed, then this is really the thing which sends the down-closed set to the up-closed set of its upper bounds and so on and so forth. There ought to be quite a nice lattice theory of characterization on that, too. Well, but the point is that if you have this, and if you have this, and if you have this, then you can compute imps in terms of subs. You can just go around this triangle. But the rather surprising thing is that totality does not imply cototality. The easiest example that I know of is the category of groups. Groups is total, but it's not cototal. So what's the point of all this? Well, just that properties that the left adjoint T and A data has tend to encode the kind of category that you have here. I mean, just as, I wrote a paper on lattice theories some time ago, which is the lattice X down-close subset, which is the A structure. So which is just as, just as the unator structure, isn't it? Yeah, the unator structure. But if the left adjoint is unator, which gives you supremum, itself has a left adjoint, that's precisely complete distributivity, modulo the axiom of choice. Okay. Now, in fact, I mean Frey has shown that this is left exact if and only if he is a government of topos. In the presence of totality, Cartesian closeness of B is equivalent to this thing preserving finite products. So exactness properties are sort of reflected in the stuff. So I mean, to have more and more adjoints is to, you know, get a sign of a greater and greater degree of exactness of the categories. So it's rather surprising when you take B to be sex here, that while you have a string of five on the left, you have precisely two on the right. Because I was just saying a minute ago that, you see, if you have a total category, then it is Cartesian closed if and only if this preserves finite products.

12:30 So, if you were to have a string any further at once on the right would be equal sex. That would force preservation of some, which would give you strictly complementary sub-objects. That's right. It would say that sets opposite Cartesian quarks, which... Yeah, yeah, yeah. So it's rather interesting anti-symmetry, as it were, which is inherent in the category of sets. I mean, the co-limit theory of sets and the limit theory is really quite different. And yet when one thinks of them just simply as a topos in terms of the exactness properties, it is complete and co-complete in a very straightforward way, or at least when you just think about it as a topos. That's right. Presumably some of this stuff can be internalized. This is very interesting indeed. Sorry, I didn't hear your name. Richard Wood. Mike Wright. Well, I'm not. I'm sort of, I'm here under plenty of false pretenses really. I was involved in organising this conference at Cambridge last week that Phil was talking at. He was invited up to give a series of lectures to the history and philosophy of science department on category theory and foundations of mathematics and there was a workshop on foundations which we organised after that. I'm not a mathematician at all, my background is in logic, some logic in certain fields, but really my main interest is in philosophy of mathematics, particularly foundational problems, and it's now strictly an amateur interest, I'm not at any sort of institution, I just happen to know people like Colin McClarty, and I was invited up here to... I think in order to realize the depth of my ignorance about category theory, but I've certainly got an awful lot out of this week, and particularly even more out of last week at Cambridge with Bill. It's very, very approachable, but this is very interesting. The state of my knowledge about the category of sets is literally, I guess, about 1972, I mean, the obvious attorney on top of it.

15:00 But I had to, you know, this is very interesting, this fact that you don't, you know, you don't have, you have this asymmetry between the completeness and the co-completeness, particularly when you try and internalize the, yeah, particularly internalize, because here, what, yeah, sorry, sorry, no, I was going to say something about the diagonal functor, but no, go on. If you're a logician, you might be interested in this paper that I wrote on distributivity. We're very interested, yes. Have you got a copy? I gave the last one away to Martin Hyland. Oh, is that the one that he was just looking at? Well, perhaps I could get a copy of it. I spoke on those matters last summer at Sussex at the category theory meeting, but it contains a slight improvement in that it's got Rainey's old theorem characterizing. All of these are completely distributed lattices as quotients of complete rings of sets, but you get a very slick proof out of that adjointness, but from a logical point of view, I think the following is interesting, so I say that a lattice is constructively completely distributive if Well, as I just said, when it's essentially total, so that the supremum function itself has a left adjoint in addition to its right adjoint, the downseg. Now, the thing is that it's fairly easy to show constructively that, let's see now, if a lattice is completely distributive in the usual sense, it's easy to see that it's constructively completely distributive because this property, At first blush gives you the usual equation for complete distributivity, but where the things that you, the subsets that you soup over are constrained to be down-closed, and you just sort of chase around the adduction and you see that's what it means, and so it's, you get this implication quite freely. Now, in the presence of the axiom of choice, you get the reverse implication.

17:30 Just knowing that you have the complete distributivity equation for down-closed subsets, you get it for arbitrary subsets by, it's just a little game of interchanging some quantifiers. The axiom choice will allow you to do that. But you use epimolar factorization to... Well, it's one of those things where you have a... For all there exists, and you change that to there exists for all. Yeah, yeah. So, yes, sure, it's quantified, it's commutativity, really, of existential and universal quantification. Well, doesn't that just amount to epimonifactorization? I'm not sure if it does. Well, I should have analyzed it, though. I didn't try it. But the nice thing is that the improvements are there. That's in this paper, is it? I really should like to look at that. Could you send me a copy if I give you my address? Thanks, that's very good of you. I'd be very interested in reading that. There was one about choice up here that I meant to ask, you know what you were saying was so interesting I didn't want to stop your flow. But you were saying that there was something in this construction that only worked modulo the axiom of choice. Oh, that was... And I didn't understand that. I couldn't see why that could... I was saying, sorry, I was saying that... That was before you started talking about this. Yeah, but I think I had... Was I... I think... Was I mentioning that to Walter? No, no, no, this was when you were just talking to me. Sorry, I... It's exactly this. The modulo, the x, and the choice, these are the same thing. Ah, right, I see. It was just that. It was about down... Yeah, it was about down... It was about... Down-and-up sets. I'm sorry, what's the terminology? I can never remember, but... By the way, the proof of choice, given the equivalence of these, is dead easy. And it just rests on the following thing, that you can, for any partially ordered set x, you can show that dx is ccd. So if you're given this equivalence, then any set of down-closed subsets of an ordered set

20:00 Thank you for watching. And it's well known that the action of choice is equivalent to power sets being completely distributed. Sure, yeah. So you've got to find a system that's all completely... Yeah, that's very nice, that's very convincing and nicely tightly tied up. Yeah, yes, indeed. Yes, indeed, I mean, that's obviously one of the... And that was after all the way that the initial independence proves. I mean, even before, sort of, Kernan people, when they thought about permutation models, back in the 50s, Moshtoskyan people, and I think even Bernays. They were trying to prove the consistency of choice before the co-work, but it was precisely in terms of analyzing the way that the power set construction worked over permutation models, and all that stuff in the algebraic way of thinking about permutation models. I mean, before they had all this machinery of functor categories, it was really difficult to think about in terms of the subspaces of algebraic spaces. Yes, that's very interesting because this also turns out that in the failure of choice, one of the corollaries of failure of choice is that you have vector spaces with different bases of different dimensionality. And this, that I would imagine can be captured as a lattice theoretic property, or rather that all your vector spaces do have bases of the same dimensionality is also done, doesn't... Is that equivalent to distributivity, complete distributivity of the lattice? I'm just wondering whether there's a connection.

22:30 Certainly that property holds for the action of choice and is equivalent to vector spaces all having bases of the same dimensionality and failure of choice does give you amongst all the other ways of thinking of failure of choice and vector spaces with bases of different dimensionality and that feels as if it ought to be a property that could be characterized in terms of complete distributivity of lattices. And given that one doesn't want to think of vector space structure as underlaying by a sort of lattice. By the way, there's another wrinkle to all this stuff, and that is that because this work on CCD is constructed, it all works in a topos. And so we're well accustomed to thinking of the sub-object classifier of a topos as being an internal locale. In fact, internally it is a completely distributive lattice. But you can't see that because... But it's a property that can't be seen because when dealing with toposes and change of base, we look at geometric morphisms. If the direct image of the geometric morphism applied to a CCD lattice, it doesn't give you a CCD. Sorry, say that again. I want to be sure I got that right. If we go to topos E and look at a geometric morphism gamma, let's say, with inverse image delta, left exact inverse image delta. Okay, so it's an internal locale. It's well known that gamma of omega is now an internal locale in F. In fact, as seen by E, this thing is completely distributive. But gamma of omega is not completely distributive as seen by F. Perhaps one of the reasons why this didn't surface so early. It will be a completely distributed lattice if this morphism of topoid, this geometric morphism of topoid, is what's called an essential one. So if delta preserves not only finite limits but all limits and so tends to have a further left edge on it, then we get the complete distributivity as carried. And of course that is the situation with powers. I mean if you have E and E to the I,

25:00 In the internal ocean then the geometric morphism here is essential and so so one can... Well sure because you've got no structure I mean you know because you've got such a very simple skeletal structure and you've just got the cardinality. It's one of the reasons why we... What Bill Laudier calls cardinale as opposed to meno. It serves to give sort of the conceptual frame of why products at least... Completely distributed things are again completely distributed. It's easy enough to see equationally anyway. That's very, very interesting. Where are you working, by the way? I work at Bell House in Halifax, Canada. Ah, right. Which was where Bill one time was. I think they're trying to let us know that we should be going, I guess. Can I just jot down your... Yeah, exchange the usual crud. Well, I actually have got a card, but I'm not, as I say, in the academic world any longer, so I'm actually a businessman, which means that I have a card, which saves me having to write down, saves people having to write down my address. That's about the only advantage I can think of it, but I'm also doing all of Bill's travel arrangements for him while he's here. Yeah, yeah, I've got friends in Nova Scotia, I know. Right, I don't know, as I say, I've never actually been to Nova Scotia. I'm afraid John has, but I mean, I know the geography of the place. Thanks very much. Well, that was very kind of you. I mean, I'd be very interested if you would send me that paper, particularly the one on distributive, you know, complete distributivity. Let me just mention, as we go, that there's another paper by Carbone. Mm-hmm. They have a, they show that Cauchy's complete different category.

27:30 That's a nice result too, as it turns out. Can you send me the reference for that as well? So then we know that properties of order ideals do tie up with choice because of all this work on coherent topics. They get sort of delineated off the crawl theory. I mentioned broader ideals that showed it was a proof of the equivalent, that Deligne's theorem was equivalent to the axiom, sorry, that Krull's theorem was equivalent to the axiom of choice. Didn't he prove that? I mean, this is all hearsay, so I'm not, this is hearsay from stuff that I've heard Bill talk about, which I may have got badly confused, but I thought that that was proof back in the early 70s. That was one of the things that Deligne's theorem, Deligne's work on coherent topology, that he actually proved a lot of equivalence results between theories of, I mean, existence of prime ideals and, and this is when they're doing all this work on about the global spectrum of a ring, the existence of, to get algebraic geometry. You had to use the axiom of choice in order to ensure that you had enough primes to get the, and this was the prime ideal theorem, I'm not talking now about, I guess this is a more general theorem about order ideals, but I understood that the prime ideal theorem and the theory of dimensionality and the crawl theory of dimensionality in rings were actually provenly equivalent to the axiom of choice, although the corollary is the axiom. Oh, hang on. Yeah, I just want to go upstairs actually and see the book display. There's a book there about quantals and linear logic that I wanted to get that Paul Taylor was telling me about last night.

30:00 So we go straight back up the hill now, don't we, to eat? Yeah, I think I'm going to go to something else to eat. Yeah, I said, well, the bus leaves at 1.30, doesn't it? Yeah, is there a short cut back up to 9.5? Well, somebody told me there was, and in fact some people went up that way yesterday afternoon. It turned out to be a much quicker way of going than going all the way around on the road. Oh, well, then I do know. I'm just going straight up the road there. There's a pathway up there. Yeah, that's what I mean. I hadn't actually found that yet. Oh, could you just hang on one second in that case? I know which book it is. I'll just simply pick it up and come straight back down. Thanks. Cheers. Do you know where the seminar was? Oh, there it is.