Discussions incl FW Lawvere & M Wright on Bimodules & other topics

0:00 Well, somebody was trying to make it clear to me last night, and I'm just hearing about it, it seemed to me to be very beautiful, but I guess I need, but am I right in thinking that there are two quite different ways of doing quantiles, or that historically there are kind of two lines of approach, because it seems, I mean, in one version of a quantile that I was hearing about, you had a... The structure, you had item potency, but non-commutativity. And in the other, the other version that I heard, it was the other way around. And I couldn't reconcile these things. Now, is it that there were two separate programs going on? I'm afraid Mulvey is a bit of an obscurantist as well. Mulvey is... Chris Mulvey, physics, who was together with Rousseau, first launched this formal generalization. Why did they choose the name Quantal? Was it that they thought there was a connection with quantum? It doesn't seem to have worked out. Without having actually worked out a connection. Yes, good. In that case, my first kind of feeling for the thing was right. There doesn't seem to be a... Oh, sorry. What shall I do? Can you just explain to me, you know, what the generalization was there? Because I'm still, I'm still, I still don't in fact know either what a quantile is or exactly what linear logics are. Yeah. Well, the exact example is, oh, I guess there might have been one or two more that, yeah, I didn't, so it's not quite a complete, there should be ten, I think, altogether. The original is missing there. Oh, now that's the original there, I think. Yeah, that is the original.

2:30 Four pages. This has four. There's one. Yes. That's complete. That's complete. I've discounted that. Extra last page. Wait a minute, I'm turning it. Just checking. No, that goes with those. There was no extra last page there. That's the last. Oh no, that wasn't it. No, that's good. That's complete. Shall I just put the complete ones up here? I think that'll turn up. Oh, well, here... no, no, no, that was the original. Never mind. Well, see, they... in fact, we're still learning from all the ancestors. They worked on this jointly for a time, but then they had some... but it's all based on this idea of a suggestion, about a suggestion of a connection without the... which, you know, in turn derives from the suggestion that C-star algebras had something to do with it. The whole series of unjustified things is quite far removed from it. Hmm. Which is a further pun. But, I mean, I have a feeling for why that should be, but can you just give me an example, Paul? One foul can't be K-algebra.

5:00 Mm-hmm. All right. But A is associative. Mm-hmm. Now, suppose that you have, sorry, you consider now all the... I want to make a category that has two objects, which I'll call K and A. However, the morphisms are bimodules. The bimodule, let's say, more generally, if you have two rings or rigs or B categories, a bimodule means basically that you have a right action of A and a left action of B. Objects which are separately associated as actions should be in moreover commuted, the usual idea of bimodule, but now you see bimodules can be composed, so if you have y and x, then xy, actually these are categories with several objects, rings are categories with one object, if you have several objects, then x itself is a family, it would be indexed by. The objects of A and the objects of B. And when I say action, it means you have a map that changes A, and it acts on the x-part that's in A and brings it over to A-prime, the bifunctor. But in the case of algebra, I mean, with one object, these A and Bs are all just stars.

7:30 But anyway, so the basic idea of composition, x, y, is that you, it's what's ordinarily called the tensor product over B. Yeah, I was going to say, this is like a tensor product. It really is a tensor product. You take the tensor product, let's say, as k modules, linear spaces, then you squash it in the middle by the action. There's an odd reversal. I call it xy because I write composition as xy. But on the other hand, in terms of elements, that gets reversed. So y is acted on on the right by b, that's right, whereas x is acted on... On the left, same b. So you impose this. When you say tensor product over b, it means that you impose this kind of relation into the ordinary linear vector space tensor product over it. So this is the definition of composition. So that's an associative, and you have a by category whose objects are the b categories or, for example, algebras whose arrows are the by modules. And between bimodule, you have an obvious notion of a map that preserves the actions of the bimodule. Preserves his left and right actions. Yeah, yeah. It's a bi-category, meaning a two-category with a wobble in the associativity. Because this tensor is only associated up to canonical isomorphism, that's the difference between bi-category and two-category, simply that... By category, and that's the possibility that the composition of the horizontal arrows is only associated up to isomorphism. Okay, so now I want to apply this in this particular case where I have A as a k-algebra, and, oh, an example of a bimodule is a bi-ideal. I was going to say, I mean, these are kind of, yeah. It could be just an ideal, you see. So now, so there are basically four different kinds of bimodules.

10:00 We have two objects. So a bimodule here is simply any k-module with no, so it's very general, any k-module. Here it's required to be an a-a bimodule, so the same a acts on both the left and the right. Whereas here, this is really just a, and you assume everything is a k linear space, so x is just the right a-module, so it's understood. Whereas this is a left-A module. This is just any k-linear space. This is a special case. So now this is a perfectly, I mean, this is a, this is an associative, well up to isomorphism. Or you could limit yourself to ideals, right ideal, left ideal, bi-ideal, and just sub-vector space of A. It amounts to being a sub-vector space of A, and it's a special case. Yes, like you do in... And when you're thinking about the ideals in terms of the underlying vector space, isn't it? This is a right-A ideal, homomorphisms, which are a special case. So this is a left-A ideal, two-sided ideal. Well, they're the special case when the ideal reduces to the identity, yes? No, no, no. Sorry, I hadn't understood in that case. And when you say homomorphisms are a special case... Oh, actually, no, no, right, right. So if you have a homomorphism in a ring, say, then that defines... I call it x sub phi. It's really just A, except that you need a right action. So the right action is just a multiplication that A has by virtue of the main ring, but the left action is by a phi, right?

12:30 So B acting on A is defined to be P multiplied by A. So you use the homomorphism to define a left action, and we have the right action because it's already... An example of a bimodule. Right, I see, I understand that. That's for the sense in which it reduces... This is what they were saying. This damn terminological multiplicity. Bimodules, profunctors, it's all the same thing. Right, okay, I see. That actually helps a lot. That's what I'm saying. So in particular, you always have this phenomenon that an actual functor... I said rings to start with, but they could be linear categories, or more generally V categories, where V as a base, V could be vector spaces over K as a special example, but all three of these, I mean, this one concept in three names applies in great generality. But you always have, right, so a particular homomorphism of rings is really just a linear functor. Functor means preserved multiplication and being in a linear, V being linear spaces means that it's linear, you see, so it's linear in multiplicity, hence the ring homomorphism. But more generally, if you have many objects, it's called a linear functor or an additive functor. Whenever you have an actual functor, an actual functor gives rise to an example of a propunctor. It's kind of a special kind given by a solid arrow. Now, this xb, I thought I was going in the same direction, p upper star, it's again a, but now we need a left action of a, whereas the right action of b, you put it on the other side, so a dot b is defined to be a multiplied by phi of b.

15:00 So the actual homomorphism or functor gives rise to bimodules or propunctors or generalized functors in both directions. And these are actually adjoint to each other, because adjointness makes sense because it's a two-categories, a bi-category. Yeah, yeah. Well, this just is a jointness in bi-categories, this property. No, the point is that, okay, this is all the V-categories, and all the bimodules between them, and all the homomorphisms are bimodules. In any bi-category, it makes sense to speak of adjoints. What are the adjoints in this case? Modulo, what they call Cauchy completeness that I introduced in my metric space paper, modulo that, the adjoint pairs of these bimodules are precisely those that arise from actual homomorphisms. The special case of a root ball. I was going to ask you about metric spaces, as a guiding example to understanding this. I think that will probably help me understand it more clearly. So if A is a metric space, then what is a bimodule? General bimodule. Well, it means that, so you have these points of A and B, right? So, given any two points A and B, you have X of A and B, which is a real number. Think of it as the cost of getting from B to A by the process X. And it satisfies the fact that it's an action, that you have an action. It means that, sorry, that if you take, what did I say here? A goes on the right. A of A prime A. Tensors, but you see tensor means adding for that case. Take the actual distance from, if you take two points in the space, this is the distance between them. And now you could consider trying to get from, to get to A from B.

17:30 It would be non-symmetric because... Well, yes. Yeah, because it might, yeah. So, so the cost of getting from B to A plus the cost of getting from A prime to A. Sorry, the distance remains the same, plus the cost by means of the process x, you know, your particular connections in the travel agent world, whatever the given moment x, right? Yeah, I was going to say, I'm quite certain you're going to make a joke about that. You can use it, it's not a joke. So the actual x cost of getting from B to A, and then within this world A over there, Just taking the distance. Well, that should be greater than or equal to the cost of getting from B to A-prime. Yes, yes, getting to A-prime from B. So since tensor product in this world just means addition of numbers in the metric space, that's the action. Yeah, in the case of metric space. The fact that the space acts on the bimodule is just this, and a similar thing on the other side. ...compared with the cost of the particular process X should also satisfy the same kind of inequality, because the act is greater than is an error. But the point is now, right, suppose you had an adjoint pair of these things. Well, then that would in fact be an actual distance-decreasing map. In other words, a V functor in this case is just a distance-decreasing map. In such a way that distance is decreased. And so from that you can derive the special kind of x's. And you can define x to be, well, what's x of phi?

20:00 It says that the cost of getting to a from b is nothing but the actual distance from a to phi of b, because phi of b is a specified point. The general bimodule, there is no specified point. You just have for every possibility what it would cost. But in this particular case, for example, in metric spaces, you do have a... Given B, there's this definite A, and the cost is the actual distance. Yeah, because here's the distance function. Whereas the reverse thing, it's the same sort of formula with the order of reverse. So the thing is, if you have an adjoint pair of bimodules, essentially, if A is Cauchy complete, Cauchy sequences have limits. Then, in fact, an adjoint pair does come from an actual map. Now, one more abstract case would be ordinary relations, where, again, if you have two relations which are adjoint, well, the adjointness says you have two adjunctions. One adjunction says that it's a well-defined map. The other one says it's everywhere defined, so it's an actual map. So the adjointness, adjoint pair of relations, or bimodules, is essentially the characterization of Which bimodules actually arise from actual maths? This is the philosophy which turned around backwards gave rise to the definition of Cauchy completeness. What I just said is literally true if A is equal to its Cauchy completion already. In general, an adjoint pair of bimodules gives you a function from the points of B into the completion of A. Because an adjoint pair can be proved to be essentially a Cauchy, a frivolous class of Cauchy. In the case of relations between sets, everything is already complete, so... So this is a little bit about what these, the point I'm getting at is now, okay, now, bimodules are very general. You could restrict to ideals, things that are actually sub-vector spaces of k, k cross a, so k tensor a rather, rather than just, so those are called ideals.

22:30 So there's a non-full sub-bi category of this two-object category that consists only of the ideals. It's still associative. So, where is the non-associativity? The so-called fontiles, they have confusing examples of fontiles, the whole thing. They take this perfectly nice structure, which is a bi-category with two objects, and they try to make it into one thing. So, they say, well, it's all the any sub, this is sort of now the most general thing. This is the most restrictive, and these are two different two-sided, a sub-vector space might be a two-sided A ideal, might be a one-sided in either sense, but if you, you find that if you compose these things, it all works out, in other words, if you compose a right ideal with a two-sided ideal, you get another right ideal, if you compose any sub-linear space with a right ideal, you get a right ideal, or compose with a left, you get a left, and so on, so it's a, this all works out, you see. Yes, but you don't have to worry about this identification because you start with subsets and you define a subset in the same spirit as this, everything that can occur as a product, the ordinary product of ideals, this composition is the ordinary product of ideals as ideals, but there's four different kinds of ideals, they sort themselves out. This is all very clear, nothing mysterious, nothing non-associative, nothing, it's only one-sided units and all that jazz. But to make it more confusing now, what you can do is, you can say, well, there is this particular homomorphism, since after all, A is a K algebra, there's a canonical inclusion, because if you take the unit element of A, and you can multiply it by any scalar, and algebra always involves that, which is a homomorphism, so you see what you can do to bugger things up. So this is an example of a right...

25:00 So, any of these other three kinds of things, or let's say these three kinds of things, you can interpret them back here by composing with this. When they write xy, for example, what they really mean is x-phi-y. The phi put in the appropriate place because there are four different cases. Things that can't be composed by the ordinary bookkeeping, you can force them into composability by inserting the phi either on the left or the right, but now it's non-associative, so you've created this big mystery, which is called quantiles. Right, so that's really just what quantiles are. They're this thing but taken as a... It's taken as a single vector space and viewed as having a non-associative classification. Some products will have two fees in them when they should have had one if the associative law was true, things like that. The thing that really bothers me, I told them this, and they said, well, we can't do that. In other words, we can't explain this because we've already established our sect, you see.

27:30 It's a sect. Mowbray is especially... Well, what is the plug for this? I mean, I see, you know, I can see that, okay, so you think of this. So, you haven't done your bookkeeping, you've messed up your domains and co-domains as a result. Try to avoid the elementary bookkeeping. So who are the set theorists? What do you mean in what context? Well, I mean, functions are co-domain in the universe. Oh, yeah, you mean like... Fights have a, in type theory, have a definite co-domain called the type, which is a sort of vague domain. Just to try to avoid the bookkeeping, the fact that everything has both a domain and a co-domain. So this is once again the same thing. Yes. I hadn't seen the analogy in the case of set theory, but... That's what I've held in my lectures in the first lecture. Well, yes, I understood the bookkeeping stuff in the first lecture. It was very clear. I just hadn't seen the application to set theory. I'm saying that people prefer the message that comes out of this attempt to avoid bookkeeping to doing the actual bookkeeping itself, which is simple. Yeah, because actually this bookkeeping is simple. I mean, all you have to do is to keep track of four different ideals, essentially, or rather the action. Well, there are four actions, but two of them, as you say, are just... So what do you call this again? The... where you've got the... An A-A bimodal. An A-A bimodal. No, but the actual... Bioideal, rather. Yeah, yeah. It's very curious. But then having... Of course, the alleged connection with quantum mechanics... I was going to say, what is the alleged connection with quantum mechanics? People like Alan Kahn and his followers talk about non-commutative topology. So the idea is if you have a commutative ring... Everybody knows what amount of space associated with that is the Zariski open subsets, the set of Zariski open regions, the lattice, which can now describe the space. It also had points, but it's lattice now. So again, by kind of formal generalization, they think, well, quantum mechanics involves non-commutative rings,

30:00 Objective backgrounds of subjective quantities that we're measuring, that should be in some way a system of regions that forms a lattice, but not a hiding house, not a distributed one. Yeah, an all-distributed one. And what should we take? Well, let's take, since the risky open sets correspond somehow to ideals in a community case, we should take the lattice of ideals of the algebra, of the non-communicative algebra, and call that non-communicative topology. Now, this is already somewhat dubious, you see. It's a formal generalization without an actual physical story to go with it. And who do you say pursues this? There's a guy called Kummer, who's, well, one of his papers I read of non-community. The original thing was due to Kahneman and Birthoff. Oh, sure. I mean, the whole business is about quantum logic, which I have got. I must admit, I probably wasted a great deal of my time on quantum logic. So they said it's the lattice of subspace and Hilbert space, but now the point of view is more the algebra of operators, the lattice of ideals, or some kind of ideals in the C-star algebra. So that's the algebra A. And the quantile people, they said, well, let's consider... They actually consider one-sided ideals. Now, by means of this canonical theme, you can turn a left ideal into a right ideal, a strange way, because it means blowing it up quite a bit, and so just on these, let's say, the right ideals, which are the arrows like this, you can define this non-associative. Yeah, right, right. Just on those alone, to be more precise. Now, suppose we have two, which are both right ideals.

32:30 What is the right ideal? ...a thing that's no ideal at all, an arbitrary sub-vector space, but it gives another right ideal. So this is the basic example of the quantile multiplication, when you see it has the p-upper star in it, right ideal set. But it's not, as I understand it from talking to Paul Taylor last night, or listening to him, it isn't. The linear subspace is a separable Hilbert space, which...

35:00 ...was the framework for the so-called quantum logic, but don't actually form a quantile. Hilbert space, I mean, the Hilbert space as a topological space does form a quantile. Have I got that right? I think that's what we said, but yeah, it would just be trivial, it would be locale, it would be distributive. I mean, that's trivial, simply because it is a topological space. But the linear subspaces don't even form a quantile. So the payoff... But you see, the idea is to use the algebraic operators instead of the Hilbert space itself. Oh, in the way that you do in C-star algebra. I see. The C-star algebra is some of the operators on this. This, of course, is important. If you don't take all of them, then certain of them are appropriate to the kind of physical system. So when I say linear subspace, I mean linear subspace of A, of the algebra. Not at some space in which there are relationships. Well, you know, this is extremely interesting, but now armed with these warnings about the bookkeeping and the bad philosophy, I'd still like to go away and read about these things. Because I'm very interested in kind of how bad ideas originate. I think there's a lot of... This will at least give you a guide to understanding. As an example of the theorems about quantiles, what might be... I'd like, as I say, I would like to know more about it. It's again, it's this constant straining to think without having, without keeping track of domain and codomain. It's the problem which I think David Finkelstein has. I don't know when, I missed in fact when you talked to him at the meeting. But I think he has, is hung up particularly with this problem, but... And that's why he's so obsessed with semi-groups and with the idea that you could define a kind of partial category or a quantum category. Is it the case, and I don't know if I'm telling you this, that also Daniel Scott was...

37:30 He was professor of logic at Oxford, yes, for many years. That could be, I don't know that. That could have been. Quite possible. I don't know what the chair in logic at Oxford is called. I mean, they all have names for the people who endowed them. Could very easily have been a Wolfson chair. I don't know. I'm not sure how much to read into that. I'm not sure what the chair in Oxford is. I'll be honest with you, I don't know what the name of the logic chair at Oxford is. Angus MacIntyre has it now. But it was, yeah, Dana Scott's chair for several, for about 12 years, 15 years. But, um, now is the way that Girard does the deals with, sets up these quantiles, Girard, I'm sorry, Girard sets up these quantiles different? I have the impression from talking to Taylor last night, listening to him, that in fact he has a rather different way of doing it. I'll give you an example. I mean, you don't actually lose associativity in the case of the way he sits it down. No, no. Right. So all of that construction that you've been showing me there was the moldy construction. Yes, it was the sort of key example, and of course there are other examples. You know, they just start by saying, let's consider right ideals and let's define them. I mean, as ideals, this is really just a set of old products. P is so canonical it's not even mentioned. P is just the... Ordinary inclusion of scalars into variable forms. So, as an example of linear logic... Yes, I'd like to understand more about how these abstract models connect with linear logic. So, again, this is not the only example, but it's kind of a typical.

40:00 So, let's have a Cartesian closed data, a particular distributed data, you know, some of the... Limits and talents all... You know, so this is the typical kind of proof period into which mystic logic... Actual statements are the supports of the... The proof bundles themselves will take actual sums so that the, for disjunction, for existential quantification, you take actual sums so that the principle that there exists a thing is really literally true, you know, that there actually exists a thing. Implication is the exponentiation. In other words, the set of all proofs that A implies B is simply the mass of the proofs of A and the proofs of B. So this is kind of the proof theoretic version of put in variables. That's the big long paper that's in the La Jolla volume, yeah, that's yours. Kelly has a more recent book in the Cambridge.

42:30 If you take U of V tensor W cross Cartesian, then there's a canonical map here. This is the underlying set of the tensor product. Well, tensor product consists of all formal linear combinations of formal tensor products. Well, among those are the actual tensor products themselves. Take a vector of V and a vector of W. So you have this canonical inclusion. A closed functor is not just a functor. It's a functor together with, U of K. K is the unit of the tensor. Take the underlying set of that. That's the key is the distinguished point there, angle of number one. These two things are given. The addition to the puncture makes it a closed puncture. Now, F has a similar property. In fact, F is actually isomorphic too. F of one three vector space with one basis element is isomorphic too. The principle would be given on both sides. This is actually a maximorphism, although not for you, obviously, you're not tensors, you're not mental tensors, but a math is not. Okay, and then closed adjoint means something a little bit more, you know, some kind of equations related. In my thesis, I introduced comma categories. Why did I introduce comma categories? Describe adjointness without set-valued hum functors.

45:00 Adjointness, one way of describing adjointness is... You have two categories, two functors, and two natural transformations, and two equations, so-called triangular identities. So that's perfectly elementary, but you'd also like to describe the sort of homoset bijection The usual way of doing that is to say, well, you have these set-valued hum-punctures, so you have to introduce the category of sets, but the hum-punctures... Yeah, because the set-set is the category in which all the hum-punctures land. I mean, that's the way I was always told to think about it. You know, sets are just the category in which hum-punctures land. But you see, from the point of view of the category of categories, that's a highly non-elementary thing, because the category of sets is a particular object, not the subcategory of the three categories of an actual object. Yeah. And it has all sorts of higher order properties and so on, which really have nothing to do with explaining the kindergarten notion of adjunctness, you know what I'm saying? They have nothing to do with bookkeeping. Well, anyway, so the thing is that in order to describe the... Well, I mean, I know you need far more than... adjoining the states care much more than just bookkeeping, but it does take care of bookkeeping. But you don't need the sets, you see. There's an elementary construction. The comma category is an elementary construction. If you're given the functors already, you just take a sort of double pullback, just using the fact that you have exponentiation by two and pullbacks, and you construct the comma category. So it's completely elementary, algebraic. There's no need to introduce it. So if you're given F and the identity on V, you can form that comma category, which by its very construction has a forgetful functor, and it's coming out of the pullback diagram. On the other hand, given U and the identity on S, which has a forgetful to the same thing, so adjointness just means that you're given a further frontier, which is an isomorphism between these two common categories. So now, this is for categories and frontiers in general. This is the adjointness, an isomorphism between these two common categories, which preserves the U.

47:30 If you look into this, this is really just saying that the Homs set by ejection It's looked at from a slightly different angle so that it's elementary. Yes, so in other words you don't have to think of it as valued in sets to get it. Right, you don't have to have the category of sets. You just have to, I mean, I call this S, but that's with any Cartesian clause. Or for this, that's any category, V also. The typical object here, you see, is a map that is a pair S, V, and a map from S to S to V. Whereas the typical object here is again an S and a V. We require this to be an S into U of V. That's a typical object of this category. Now the maps are commutative squares on both sides, and you'd say they have an equivalence of categories, it means that the natural bijection between this kind of arrow, thought I was an object in this category, and this kind of arrow, with the same S, i.e. triangle with commutative, and the same V, that's exactly Agilent, that's equivalent to the triangular identities, but put in a way that makes explicit the useful misbijection. There's no reason to mock that. But now we've assumed that these are closed categories, and the closed adjoints comprise one category that has two different descriptions. The thing is that it also is a closed category. Find the tensor product in a completely obvious way, V1 tensor V2, V1 cross S2, find the hom of such pairs or of such pairs equivalent.

50:00 This is an example of linear logic. In other words, this tensor product is a kind of conjunction, which has a Cartesian aspect. In other words, multiplying these S's is taking the conjunction of the proofs of, it's the proofs of conjunction, the ordinary intuitionistic, and the hom as a component, which is the ordinary proofs of intuitionistic implications. So in other words, from this category, by applying the forgetful function to S, you can extract, these become those functions, obviously, preserve the tensor product exactly as a comparison for the homs. You can extract intuitionistic information. But on the other hand, L also has a linear aspect, because there's the other cross from it, there's the V, so it means you can take linear combinations of proofs, you can take things that are ordinary intuitionistic proofs, three times one minus five times the other one, that's in V, so that's the idea, sort of the advantage of linear logic, is that you can take linear combinations and calculate with those as in linear algebra, and then if you're lucky, go back and extract... Again, this is a typical example. It's probably not the most general example of a linear logic, but it shows clearly why it's simultaneously linear and intuitionistic. There's one closed structure, one tensor and one column, which have the two aspects. Yes, I—

52:30 I think these functions, you know, these various functions, these— Of course. So what items, I'm not familiar with that, the, the, the, the, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, what, And so there are four countries, and for those they're given names, but perhaps you would want to consider the adjoint, which would, of course, be the end of the end of the end of the end of the end of the end of the end of the end of the end of the end of the end of the end of the end of the end of the end of the end Yes, this seems to much, well...

55:00 Closed adjoints implies that the common comma category itself is closed adjoints, and that you have closed adjoints between linear and Cartesian and distributive situations from first-year linear algebra, right? Yes, I think that will help me enormously, because I hadn't understood the connection with comma categories at all. And the way that linear logic had been explained to me was, last night, was much less clear than that. It's a very confusing way of interpreting it. Yeah, can I keep this? Sure. So the underlying idea is that one wants to be able to do linear, do implicit logic, but with linear algebra methods. So you take linear combinations. It's like probability theory. You take convex combinations of actual things, or linear combinations, you know, at the same time. You can put convex sets into the vector space. It's also a closed set. It's not exactly linear, but it doesn't satisfy coproduct equals product, but it's much more linear than a Cartesian thing is. It's also closed, you see, and it's closed to junctions. It's a convex set generated by a set and the underlying set of a convex set. You could have the same formalism with probabilities. But the way that Girard actually sets up this thing is not in terms of a category-theoretic background at all, is it? I mean, doesn't he just introduce the natural deduction rule, or doesn't he just introduce a sequence after this? Well, he doesn't do sequence calculus, does he? He always puts the subjective first. Well, no, actually, he's taken up the slogan, he should put the objective first. Well, probably the... He's written an example about a topological space. Oh, I guess, yeah, that would help to make sense of the way that this...

57:30 That would help to make sense of the tensorization, I guess. He does give this example of the proposition, or whatever he calls it, is a vector space together with a given subset. Well, that's very quantum logical, isn't it? And that's how we're supposed to think of propositions in quantum logic, as subspaces, vectors, subspaces. The conjunction of two set propositions is that you take the tensor product and the vector space is a given product. I hope you had the introduction of the tensor product explained to me. I mean, what the motivation was in... Basically the same thing. Yeah, but this is a much deeper way of thinking of where it comes from. The key point is to get the closed structure on the common commissary. That thing in turn has closed deadlines relating it to the original two categories. And the linear one, yes. Thanks, Bill. Thank you very much. Could I get one of your cigarettes? Oh, sure. Thanks. Oh, no, I think I... Well, no, I mean... Yeah, I do. You know, I do. Occasionally. I do show that, but... No, it's just a lot of concentration. I was going to say, well, we must... Oh, I should think they've probably broken for coffee, haven't they? I think they have, in fact. 10.45, isn't it? Well, just let me get the... Now, have you got the... Yeah, no, I haven't got one of those myself. So can I just take one of them? Right. And then you take the rest. We have to mail one to George Rousseau. Yeah, I promised to. I'll just take, well, it doesn't matter which one I take. Let's take, we'll take the top one. Yeah, that's fine. Let's make sure it's all there. There's a staple down there. Yeah, actually, I've got a stapler, but I haven't sorted them out yet.

1:00:00 It really is going to give me time like this. I mean, well, well, you know, I'm just thinking you've got all these very distinguished colleagues of yours around who you really ought to be sharing your ideas rather than, you know, dumb clucks. But thank you very much. You were very touch. You also learned a lot. I really do want to understand the CCR, the derivation of CCRs better. Let's go through that another time. Just let me... Do you need to give it to him? Oh, I've got Gonzales stuff here too. I said I'd... I said I'd give that to Wesley Furr to take back to Cambridge. Just see what the writing order is. Yeah, there it is. I've got some questions on the back of it that I wanted to... Well, no, actually you were very... No, no, I'm not just saying this. You were actually very clear. It was just that I wasn't up to concentrating hard enough because of the fact that it was midnight. No, I probably had one too many glasses of wine and I'd been thinking about what Anders had been saying about SDG earlier in the evening. Where is the... And also trying to understand about this linear logic and quantiles, which I've just been hearing about for the very first time. But with a lot of mystification thrown in. It's popularly founded in actual physics, you see. It's like Schutzenberger with these non-communicable lattices, he just, well, let's make it formal generalization, which is not a bad idea, I mean, Engels talks about the virtues of this heavy science, but one, you know, one should eventually try to, after 50 years, you see, there's not, there's not. 652, 53, 53, yeah.

1:02:30 1936. Yes. And it's 65 years now since Heisenberg. I think. Yes. And still. It should be called the Leibniz rule. 5 in 1932, that maybe there was something else there, but Stone proved the theorem in his book in 1932. Every irredentation of the abstract is actually isomorphic to multiplying by functions and differentiating. What was the proof? I don't want to go through everything you said last night about the CCRs. I do, but another time. But what was the, you said that at the time that you wrote the, sorry, the time that you wrote the Continuum Mechanics introduction was spring of 1174. No, no, I don't. There's one thing I'm going to ask you to send me a copy of. Well, what I'll do is go out and buy the book when I get the chance. Yeah, that's a very good paper. I don't have any more. But I'll certainly go out and buy it. But I don't have the introduction, no. But this derivation of the CCRs was something that you'd seen later. And it came from... was it actually the Stern paper that you were talking about? No, it wasn't. There was another paper. No, it was a paper by Mark Riefel. That's right. More categories. He was a student together with me at Columbia, so he's quite well-versed in my category theory, so I'll be careful when I do this. But what was the, I mean, what, um, what does he, what's that paper, what was it in that paper that you saw that there was just one instance of this overall construction with intensive and extensive quantities? Yeah. Ah, yes, that's right, it's Cavera's instance. Yeah. Well, because, you see, from the C-star algebra point of view, they used the Hermann Weyl...

1:05:00 In other words, instead of talking about, they talk about the, P is an unbounded operator, so they put it entirely in terms of bounded operators in order to fit it into a C-star algebra, so they already have it in a global form, not a differentiated form. Yes, I mean, P is as in P cubed, in the differentiated form, just the standard conjugate quantities. In particular math, it's not merely linear, it's linear with respect to all the variables, the intensive quantities on the end, the numbers sort of inherited that formula. Well, anyway, we'll talk more about pushing forward. In fact, pushing forward is exactly what I'm going to do now. Shoot! Yeah, sorry, I just want to make sure I don't set fire to the University College of North Wales, which I was just about to do. That wasn't a very clever thing to do. Keep you going, Fred. Most likely, yeah. Somewhere where I could stay, I'd rather be alone. Sure, why don't you have him come and stay with me? Okay. That's okay. And I would have asked you last time, but Colin was already staying and I only got one spare room. I would like him to come to the regular room. Yeah. Well, why don't we give him a ring? I could give him a ring with socks. Perhaps I could invite you both to dinner. They're very nice. Well, I was going to ask you if you'd let me invite you and Fatima together. She might be leaving tomorrow morning already. Oh, but she's going straight back to Milan. To Zurich. Oh, to Zurich. Oh, I see. No, I didn't realise. When you said she was leaving, I thought she was just leaving to go down to London maybe to spend two or three days. Oh, that's the seminar room. Yeah, I'm going to have a look there later. Copy, no.

1:07:30 Yeah, I might do that from Conway this afternoon. I love his number. I could also invite his wife. Yeah, I'm sure she'd like to. She's a very nice person too. But if for any reason you can't make it, I'll... Anyway, you want to go? Right, OK, I'll see you in the coffee room.