Topos theory (contd.)

0:00 In fact, the three of us have come up with a good example of that. The Sotheby's spacecraft, for instance, seems to have fit. Now, most of the spacecraft, we have to be able to prove, with much effort, that each cell from the rock bed can be represented by an arrow and a signal to the sun. Right, but not the other way around, is that true? Not every such error comes from a certain kind of work. We think of, yes. So someone wants to really characterize the things you're making or what? Yeah, yeah, I don't agree. It's a lot of work. But what would you do if I found, I don't know if it's possible, but if I handed you a topos with an object in it such that Homs from the terminal object into it would be precisely equivalent to a Hilbert space? You know what I mean? Wouldn't that be a good candidate for a state object? Well, I can do that. I can take a set of Hilbert spaces. Thank you. Come back. No, no, no. A state, to me, is a either-way function. Oh, I see. Well, I see. It's not a form from... I'm confused. I guess we should now move on. Sorry. No, no. It's a discussion meeting, man. It's not a... So, we thank you, please, again, and... ...for the opportunity to be in a place where there is concentration and intuition at sea. I've also been born in the physics department. I've been studying physics for about 8 years in Excel, so I'm completely ready to model the classical and quantum physics.

2:30 It's impossible, so it's a very good company. And what I want to talk about is something I've been saying really good, especially in the last few months. In the last talk, pretty much all we used at the top was that the true value of physics is not the truth, it's not just false and true. The plan is as follows. I'm given a C-Style for A, it's called a model of quantum physics, which is what I'm looking at, and you pass in what you want. Now we're going to define the problem. I want to start with the example of a movement object, because I guess what we call movement objects. You know, you have a lot of concepts. You have to really know about them by the time you get there. So normally, you can set something like, you know, what happened in Persia. So when I come down to some concept, I have a multiplication. Why can't I add a new one? I can say, you know, whenever I add a concept or I add a category, I can say an object. When I say this multiplication is supposed to be associative, that's just a certain diagram that needs to be used. When I say this unit is really a unit, when I want to multiply by one, I get the same thing back. There's also just the diagram that is. So there I have, what, three equations, the diagram commuting, and then I want to make a sentence saying,

5:00 this equation holds, and this, and this, and that's the property I want on my object. So if I want to be able, in general, to make, to pinpoint an object in a category, and I say I want it to have... I want to have more involved notions on the category and need to have more structure on the category to interpret it with. So now we have a very simple language guide. I just had type m, a function symbol, m for multiplication, small m, and no relation symbol actually. If I want to have more than finite conjunctions, I'm going to get a bigger kind of logic, and I need a more structured category to interpret this. So the next example is, suppose you have finite conjunctions and unique existentials, so you would need something like that. The same with a group object, it's because every object has a unique inverse. For this, I don't just need a tensor product, I need a proper product, and I also need equal objects. And this goes on and on. If I want to specify objects or satisfy things that I can define in regular logic, that means I have not just a unique existential, but a normal existential quantification. I need even more properties of my category. As your logic grows, you also need the structure of your category you want to interpret it in to become more important. And the sum of this is a topos. This has, as Chris has been saying, it has only structural things here, but they're very powerful. The combination of these few things with finite limits and sub-object classifier really gives you all the first order and error of the logic you want. You have all the usual objectives and the lower implications of the HR principle that I want. Remember the plan was, I'm given a C-style algebra, I'm going to make a totals and insert them aside there and I'm going to pick an internal C-style.

7:30 So first I will need to say in this kind of language. What it means for an object in a category in the topos to be a C star. So if you're having this language, it had types, it had function symbols, and from those basic formulas you can make larger formulas. If I fix, I want to interpret the type M in my language as the object M in my topos. And I fix, I want to have the multiplication function symbol as the M in my topos that I was talking about. And all of the rest of the bigger formulas are fixed because of the structure of this category. For example, if I have a formula with a conjunction in it, then it just gets interpreted by having the finite coordinates. So in particular, if I have some interpretation here, if I have any formula pi built up from the basic formulas, this gives me a sub-object of the three variables in it. So in particular, formulas closed, that means there's no three variables at all, and I get the empty... I get a sub-object with that. Let's call that object omega, right? So this means the interpretation of any formula gets a value, so to speak, in omega. So think of omega as being the true values of these formulas being true to all of them. As I said, there can be more than that. In particular, we can also wonder when is the interpretation of such a formula the full sub-object, when it's really actually true. If that's true, I'd say my topos D validates this formula. And a surprising thing happens if you go and check all the usual things in logic you have, like excluded middle, for example, P or not, P is true. And you check all the usual P and O axioms, it turns out that the logic of this internal language is precisely intuitionistic, which constructs it. Case in point, you don't have excluded middle and you don't have axiom of choice.

10:00 The quantum sea-style is crossed in the first course. It's given a set of some structure. First of all, it's a type of seas and an old equation. And in terms of physics, you should think of elements of this set as observables, you can imagine. The equation can be either commutative or non-commutative, and it's specified in the classical system, modeling by the sea-style. When it's not commutative, you get a quantum system, in a momentum, not beam, that an op is made. So when a student says, I don't believe you that quantum mechanics is true, and not this way to life, following an analogy, a very simple experiment at home, either you first take off your clothes and then take a shower, or you can just take a shower first and then undress, and the result will be very different. So there's multiplication, not necessarily, and you have some other structure which is not really important. Except for this norm here, notice that it takes value in R, and you want this structure to cooperate. So, as Chris has been talking about, the first course on how you actually construct art from the rationals, for example, one way to do it is, for example, when David came to us. So you say, I have the left half of my rational line, I have the right half, and they come very, very close to each other, zero distance between them. You want to formulate the zero distance between, you're using classical logic. I'd like to get rid of this art and see what a C-Solid is. So this is something Chris has been working for 30 years on. I don't want to function anymore, I just want to see. When the norm of an element of a C-style graph is smaller than some ration, I don't want to pinpoint exactly this real number, I just want to be able to approximate it from here.

12:30 So all that change from going from here to here is this R, and of course that C is the same thing. And then these, you have to of course reformulate these things and work together. Now we have the techniques in place. Now suppose I'm given a C-style graph in the real world. And now Niels Bohr says it's actually a pretty logical thing. If I want to investigate physics, and I want to be able to make a real theory, I should be able to tell my neighbor what I discovered, of course. But that means I have to write it down in classical terms, otherwise he can't read it. Mathematically, that translates to an access to the commutative part. Suppose A is this algebra we were given. We're going to consider all the subalgebras which are commutative. This is all the things, it's called the classical snapshots, the world views, the things you can actually get. They form a set, in the first instance. Of course, these things are ordered, partially ordered by inclusion. Some algebra can be bigger than another, or they can be totally one. So in this case, in this sense, they form a pre-order. We make that into the category by saying the objects are just victims of alchemists. They have an error from C to B, precisely when C is included in B. And then finally, I'm going to consider the category of functors from this context category to set what was associated to. I find a very canonical object in here. When I say what an object in here should be, it's a functor from C to A to set. So I should first of all say what it does on the commutative part of A. Well, I have to give a set back, so let's just give a C back. And then I have to say what it... it also has to act on morphisms for it being a functor, so that means I have to say what it does.

15:00 And finally, we have an inclusion from C to D. You have to make a function from C actually to D. Well, that's C to D, so you have to cannot call it inclusion. So this is a fine, proper object in this category. Now, it turns out, if you formulate these requirements I gave you for being a C-style algebra in a constructive way, inside this topos we've been thinking, it turns out this object actually satisfies them. And not only that, it's commuted. So that means... We were given this seed sample from the outside, we made a topos, and by moving inside the topos, it all of a sudden became commutative. Moving, restricting yourself to this topos we built, kind of represents moving from the possibly quantum things to the classical things. I made a picture of this. So this is the topos we're all living in. Let's just for a moment say it's set. Bring an algebra in here, then we made a smaller one, picked an object, and it turned out to be an algebra again. And we can reason about this inner algebra in two ways, either we're in this area, so we're still classical mathematicians, or into here. That's what Warren would say, that's what physicists do here, and here they can only see the commutative parts. So suppose now we take the role of this internal observer for a while. Now the thing about commutative species diagrams is they're totally characterized by a Galvan duality. They all look like continuous functions on some. So here I have stated the classical version of this. It's an actual orderly duality in the categorical sense. On one hand we have the compact house door space. That's the other thing. What just are is a problem. Topological spaces can be a problem because they don't have points. You need the action of choice. So we have to do something about this. The trick, and it's a very old idea actually, is not to look at the topological space as its points, you only look at the open sets.

17:30 The newest function is not. The function from the set of points to the other set of points is continuous when the inverse image of an open set is open. And we can just translate this into if you only have the open sets, so you get a lattice of open sets on that. The last is both the set on the giving end, and then the last is morphism going back. The key things are called frames, or locales, depending on which direction you want to put morphisms. And this form of the scale-factorality is something Drisnolty has been working on for a very few years, and they have a proof of it which works for any growth-in-leap totals, and then does what they're saying. They generalize it to any elementary totals. So that's great, we can apply this to our internal commutative system onto the network. So when you apply it to R, where it's the inner C-style algebra, you apply the scale of punctuality, you get topological space and locale. So this is kind of the phase space, the configuration space, which tells what the world is like. Where this algebra is the whole . And the point about this thing is it's very funny. If you go to the second theorem, it says these things don't have global sections. But now this thing is a topological space, or at least the open sense of it. It actually has no points at all, if it's a proper quantum thing or not. You have a kind of phase space, but you can't pinpoint the way exactly things are. You can only approximate it and set it around. So now we're here, we're giving an algebra, we made a topos, and inside there we have very funny objects. The other holy grail, you want to do a state thing, you have a proposition, you want to pair them, you can tell it. Probability, for example, to see whether it's true. So this is the last part we're going to talk about. So let's start with dates.

20:00 You have this external algebra, suppose you have the state on it. We have to convert it somehow to the internal world first, before we can talk about the concepts. But that is where the base base lives. Now, a state is just a function from the algebra to the complex numbers. It happens to be positive and linear. Likewise, you can talk about integrals of any. An integral on that is also a function which is linear and positive. So it's not surprising that we can prove that a state on the external thing is precisely a one-to-one correspondence with an integral on the internal thing. A state on the outside we can consider in the inside world. The thing as a measure of a measure says how big the set is. And you can also make an evaluation. It doesn't say how big any set is, but it says how big an open set is. You don't get a number out, I think it is, but you get only approximated from below, which is this funny symbol here. The whole space is a size one, so to speak, we have to set the size zero, which only gives the lower part of the rational line, just precisely a real number of the sides. The thing to take home of this slide is I'm given a state on the external algebra that gives me a map internally from the space space into kind of the interval zero one. And the other thing is these observables. This is where the fascination came from. The observables are just self-adjoined elements of the external algebra. First we have to convert them again to the internal world. But this is a bit hard to do. So we've hidden a little a in here, and we have to say what it does there. That means we have to say what it does on each commutative subalgebra. But the problem is, this element of a might not be in C. So the best we can do is approximate it. That self-adjoined part has this partial order you can use. So the best we can do is look at all the smaller things, which are in C, and all the bigger things, which are in C.

22:30 To find out the type of what this object is, we need to find the left-right arrows there. Remember when I talked about how we usually construct delicate wheels as cuts of the rational line? Well, this arrow precisely does that. If we have any partial order of the line here, here, I can consider the left halves and the right halves, and I want them to be open. I don't want them to end precisely at a point. But I've now dropped the requirement that they have to come infinitely close. These things are, I hope, not significant properties. These things is, this is what I mean by this right here. Of course, what you cannot do is you take the rational line, you take the consummate without the significant property. That gives you topology on information, topology, so to speak, pointed in this interval here with more and more information that gives you the open approximations. Yeah, so now we may... A little A here into an internal version of approximation, but we're going to go further. At the point of this approximation, I can let loose my Galvan duality again. So that gives me a great map, a continuous map, from the base-base to this rigid wheel, where every external interval gives us such a map, every state gives us such a map, and when I have another thing about this rational object in there, is when I have an external interval, I can just make an internal rational. There are these three kinds of maps, and physicists would really like to know what the value of these compositions is. Well, I can just compose them, and I get the map in my topos from the terminal object approximative of your 1-interval probability, right? But we can even go farther, and you can see, you can check whether this probability invariant is precisely 1.

25:00 You can only approximate, and this is an abbreviation for a role, and it's bigger than 1.1 over n. But you can just consider it. And that gives you a proper truth value as a morphism from the terminal object to the truth value. So it turns out this phase-based sigma is a very great thing to have. So this topos-chrysalidation has been talking about. You can make it based on a system, on some more classical system, it doesn't really matter. And via very molecular constructions you get internally stuff which looks intuitive and it actually gives you these truth objects. So it's not really a weird thing to consider. They did a lot of hard work to find this underlying signal, phase space, but it really rolls out if you look at it properly. Thank you for having me. No, no, no, you can do this for any thing you can frame in geometric logic. For example, when you take any group, you take all its abelian subgroups, and generally this gives you an abelian subgroup. The problem is going back up. You can't really do that, I think, so I don't see how it's all going to work like this.

27:30 Yes, so you do have a department to read, a department to learn more, to use the set. Learning how to use the set will help you correct it. This one? So I was wondering if you could give us a sense of what you're trying to find here. That way you can get an even more structured version of the quorum. I'm one of the persons myself who already collects these things and builds them. You'll have to justify it some way or the other. I'm not sure which way is best either way. Okay, so now we've closed this first session and we've finally got respect. We'll go up to the eighth floor and I just recommend you to follow the locals for those who are very sore. Can you hear me on the mic? Well, the purpose is to move me, I think. I see it for the guy making the video, of course, I'm sorry. Can you hear me now? Good. That's all I'm asking. And they're so composteric. Okay, so I'm going to talk about, I was asked to give a survey talk on the model approach to the category of quantum mechanics.

30:00 Possibly not exactly the name I would have chosen, but... What is true is that Chris was actually mentioning, well there were two Chris's this morning, Chris Hyshaw was mentioning in his talk that there are already several different approaches within the compost theoretic approaches to physics, but more broadly we can already see that there are several approaches being called the categorical approaches to physics. There's the work of Lewis Crane, who was speaking at this meeting, and Myers, Golem, and others using high-dimension categories. And there's a category of applications, operations that don't want to be called Hilbert space. There's work that I'll be describing, which is a programme of work that I started with Robert Booker, and the other people that I've contributed to by now are hearing other talks about it today as well. And we're using the certain kinds of mathematical categories with additional structure, as we'll see, and of course the topos-theoretic approach that we heard about this morning. The idea is to give an introduction to this kind of work that we've been doing. In the group of Oxford, we've got various colleagues elsewhere also contributing, which could be called the rival, it could be called linear, and perhaps to be doing linear quantum mechanics sounds quite good, quite appropriate. In fact, it would turn out that the fact that we're in the rival categories, although that's a very basic part of the setting, is already quite significant. In fact, if we're talking about logic as well, it's the other. Aspects of the general theme of this workshop and this series is that if we're talking about logical inference, even before we introduce any connectives, it's quite important to know what the significance of the humble comma is, and that's something like the significance of whether we're in a Deloitte or a Cartesian setting. So, anyway, let's continue with this. So let me just make a couple of brief comparisons with the other approaches, just for something to keep in mind as we go through. So a comparison with the topos approach.

32:30 So, as we just said, if we're setting the model rather than Cartesian categories, Cartesian then extending up to the root structure of topos, which means on the logical side that we're in the realm of some kind of linear or resource-sensitive logic, so to speak, as opposed to intuitionistic logic. Perhaps a more subtle point is that, in some sense, as we heard in Chris's talk, the topos approach is very much concerned with propositions that we make about. There's a lot of emphasis on propositions and the structure of propositions, whereas in some sense what we're looking at is the structure of the physical processes themselves, and that's what we want to directly model in the kind of categories we're dealing with. And this actually goes back again to quite the deep. Distinction at a logical level, a kind of distinction between something like a more model-theoretic view of logic, where we're using logic to describe and talk about structures, make propositions about structures, as opposed to thinking of logical type theories and the like, thinking of the type-out correspondence. We think of logic as a structure in itself, as directly exhibiting the kind of structure that we want to start with, so we could spend the whole hour talking about that, but that would make us a slightly different idea. There are also different kinds of connections with geometry. Because we're focusing on the processes themselves, we'll see that geometric considerations abound. We may think of our objects in the category which we may think of as proofs. We want to think logically of seeing some kind of direct geometry of proofs of the sort of diagrammatic categories that we've discussed. And we can contrast that with the key role of geometric logic and connoisseurs, so two different kinds of interface between logic and geometry. I think that's a fascinating point to compare. And perhaps ultimately to try and synthesize the common good elements or ingredients of both these approaches.

35:00 And to make some sort of brief comparison in the end, that has mostly, I think, been focused on the aim of quantum gravity, and in some sense thinking of quantum gravity setting as Chris has often emphasized on cosmology, but thinking of the whole universe and then the outside of the universe to observe it. We're quite concerned, as we'll see, with operational aspects, when we're not taking any particular philosophical stance, certainly the interplay between quantum and classical kinds of information is very important for something we won't be able to describe. And for example, since we're interested also in applications in quantum information and computation, And for similar kinds of reasons, we've written in methods which are compositional, which goes along naturally with the fact that we've just been studying open rather than closed systems. And the point of an open system is that it's not the whole universe, there's some other larger system that may ultimately be seen as a part of it. We've just been studying interactions between this part and others. So these are partly important. These kinds of applications include mathematics, but they also have a foundational significance as well. Okay, so now we're picking up this idea that we're not taking any particular philosophical stance. We aren't necessarily looking at things either as an operational point of view or a process point of view. One can sort of make some nice discussion about thinking of operations or thinking of processes In this kind of instrumentalist scenario, we may think of operations that we're performing in a quantum computer or maybe in a lab and so on, or we may just think in a more sort of objective way of the unfolding of quantum processes. But it is those quantum processes themselves that are interested in describing some very passionate. So let's firstly have a brief review of what quantum operations look like and how they can transfer those from a very classical point of view. And by the way, I should say, I know there are world experts on probably all the topics I'll be covering sitting in the room here, but since there's a diversity of backgrounds, I'm not particularly assuming I would open anything and it would be great introductory and stuff like that.

37:30 So, classically, here I'm focusing just on the computation point of view, quantum information, classical versus classical information point of view, and it's certainly fair to say, I think, that the work on quantum information has re-energized the discussion of the foundations of quantum mechanics, which has been a bit dormant for some time, and gives a new angle to thinking about some of the questions. So, of course, from the point of view of classical information or classical computation, we have the ordinary classic bits, very much like the sort of classic, well, like the classical truth values, booleans, that we learn hearing about from a logical point of view, we can do things like making copies of or reading them as many times as we like, and manipulating them in order to catch things that take the grounds of a classical form of computation. So when we go to the quantum picture, we now have this sphere of values. We have a two-dimensional complex vector space and the equal space of cubits. And the kind of operations that we have available are performing measurements and... These are performing data transformations which are now restricted to be unitary transformations. These are, in a sense, highly restrictive. There are a lot of them because we have continuous data in the sphere. But they're essentially just rotations of the sphere, which must be reversed. And as far as quantum measurements are concerned, they're very different from the kinds of tests that we can do in classical information. So, if you think of applying a measurement to some useful story, applying to some state vector, that we measure in some basis that we choose, this is exactly a particular case of the choice of a commutative sum algebra, of a... We see two aspects to this. The outcome is not certain, it's probabilistic.

40:00 And also, whatever the state was before we performed the measurement, all we know after the measurement is that it's now one of these projection postulums where if we perform the same measurement again, we immediately will get the same answers and collapse the state down to one of the possible answers. So measurements change the state and in some sense destroy information. This is the predictive aspect of quantum measurement. So this is from a, if we're thinking from the point of view of what we can do with quantum systems for information processing purposes, this sounds extremely bad news, we have some unreliable and destructive measurements and tests, but of course we get richness in other ways that isn't possible in the classical information processing or physical perspective. So we get in particular quantum entanglement. We have both states and the R state. We have pairs of qubits, which may be in entangled states, and what that means is that if you measure one of the qubits, you may get the answer either zero or one, and yet the probability is a uniform one, but I haven't had any sort of internalizing process. But if we do, whatever answer we get, we get the answer zero here, that by the collapse of the entangled state, if someone were to look at the other qubit, they would get the same answer, that the planet Toyberg, the PPR state, has an anticorrelation, that we get zero in one, and we would get one in the other twice the first. And this possibility arises inevitably because the compound systems in the quantum setting are represented by the tensor product, because states are closed under this. In general states, which are states enclosed under superposition, a general state in its compound system is some linear combination. For example, by setting some of these coefficients to zero, we can arrange tight correlations, as identified here, between what happens in one of these qubits and one there. So superposition can be used to encode correlation.

42:30 And this is, of course, Einstein speaking at a distance. And even if the particles are spatially separated, there's still this, in principle, instantaneous impact on the other. And Bell's theorem is saying that this kind of non-locality of effects of measurements is an essential feature of quantum mechanics because you can't rig up a local, realistic theory. Concepts that the Leinstein 4-bit axiomatic of physical theories have to be to achieve the degree of correlation that we can in quantum mechanics. The Bellin equality sets the bounds for the correlations that can be achieved in local realistic theories, and those bounds are exceeded or violated by quantum mechanics. And all the physics tests so far of the... Well, inequality goes on with very subtle ruling out of possible subtle assumptions and so on, but nevertheless all tests confirm that quantum mechanics is right and that quantum mechanics and the world is essentially enveloped in this sense, and that's one of the ways of expressing why quantum mechanics is a positive thing. Okay, so that's a little taste of what we're trying to capture here. So what I want to do now is just proceed by layers and start by asking a sort of general question. What would a general formalism for describing physical processes look like? What would be an appropriate place for describing this view of operations or unfolding of physical processes? So what do we need to have in such a formalism? So following this operational process philosophy, which actually fits rather well with the categorical philosophy that Chris was reminding us of in his talk this morning, the structure would be carried by the operations or the actions themselves, not in the elements. We should look to, say, preordaining what kind of structures we're dealing with in terms of the elements. Now, a point that may not seem so evident from a physics point of view, but is extremely evident from the experience gained in logic, computer science, category theory, and so on, is that the great deal is gained if operations are tight, if you distinguish different kinds of operations in terms of what kind of input they can be applied to and what kind of output they produce.

45:00 In fact, we're going to start writing the systems and sets of products in Hilbert space as we've really dropped some of the inventory types going through right there. There has to be some basic reflection of time, so we're going to talk to be as general as possible here. From an operational point of view, we must at least have the idea that we can firstly perform one operation and then perform some other operation after, getting up dressed, having a shower, or vice versa. And a basic reflection of space that we should be able to describe compound systems. In just the way that you can see there, there's a few metabolic items and systems spread out in space, and be able to talk about operations which are localised to part of a compound system, and being able to perform operations independently on different parts of a compound system. Alice is here, and Bob is there. Alice will be able to do something locally in her part of the world, and Bob in his part. So what we in computer science refer to as parallel problems. So that's the question. Yes. It doesn't convey the total space that you spoke of, sequential, time, space. What's that wrong? What are you doing? Is there a big difference between the two? Well, I mean, I would say this is very neutral. I mean, we should be able to express the things I describe, but it doesn't imply anything about sine-optimality or anything like that. Actually, the notion of parallelism I have here is independent of this, or even of the, in terms of classical computation. It's a sort of standard point that we don't order actions in government. Yeah, but you've got a common ground system. A common ground organization. A common ground organization in space. You don't need common ground. Well... You take those parts from time and space.

47:30 Um... Well, uh... There is a point there. Of course, in some sense, in the end, we knew what quantum gravity should look like. We might have a different... But there's certainly, what I'm saying here, there's nothing, for example, that is at all contradictory to relativity. And actually, the kind of general notion of process I'm describing here does fit very well with the kinds of process models... There have been developed in computer science discrete process models that are very deeply informed by relativity, and I'm particularly thinking of Petri Mat. So sequential is important? So sequential, because in some circumstances we can, following a timeline, we can definitely say that one thing was before another, because there is a causal relation. That's what it looks like. It takes a long time. There's no description of the time. This is really meant to be very neutral with respect to such assumptions, and it definitely accommodates, I mean one of the interesting things that seems to be the most common through the times is that we already have an important issue, even in sort of classical computation, everything is discreet, to not be able to give a total time. And that's really quite important. So we have these partially ordered models, but the partial order is something like the causal order, and some of it is very directly taught by the sort of cosmic thinking. So this is just meant to be a sufficient language to be able to talk about some things. So in a sense, the kind of series of power I'm talking about here are opposite extremes. One, where you definitely have a causal precedence because you're on one single set of time. Projectory and other cases where we make no assumptions. So you can think of it as a course of independence. That may be a better reading. I mean, maybe I'm a little bit careless. But remember, this is only a very abstract reflection of these things, any particular view of these things. And in fact, the kind of setting that I'm suggesting in the following of these desiderata is the very general notion of symmetrical logical categories. So, not a great deal of here, not a great deal of category theory, but I mean, just to make the point that because categories are not committed to thinking of things in terms of elements,

50:00 they just have arrows in their behaviour and their composition and so on, so that frees, that liberates the possibility to have categories in which the arrows are not functions, but may be processes of various kinds, which is, again, good to be in order for you to use the computer science in various contexts. So indeed, we can think of a category as just providing us a language where we have type processes or operations and the ability to have one operation before, after, and after. So that's just the idea of composition. And in logic, we can have an interpretation where we think of the objects as propositions and the arrows as proofs. So we're not talking about provability though, it's just about proof, so it's still a kind of process view of proof. In programming we can think of the objects as data types and the arrows as programs. And the idea is that in quantum mechanics we can think of types of systems, kinds of systems, that's what I prefer to say, and the kind of operations or processes that can be performed which transform one kind of system into another. It may, of course, lead us to the same kind of system that we spoke about, but that would be a special case. So that's just having a category, and a symmetric-rhenoidal category will then have, of course, binary functor, which up to, go here and back from isomorphism, is associative and commutative, so it's possible to sort of elide the isomorphisms for associativity for most purposes, but one has to be more careful about symmetry, and symmetry is actually... This is the most interesting one to put to because then one gets very interesting geometric possibilities in regard to having grading rather than symmetry. So here we're just looking at symmetric monoidal categories so we have the ability to interchange components of the usual parameter algorithm. And the sense in which I meant about some independence, it becomes manifest when we just look at what follows from bifunctoriality.

52:30 So we have a system that you've described in terms of components, A1 and A2. I have a process F1 that takes you from a system of kind A1 to a system of kind B1, and similarly a process F2. The product of these two things is independent. There is no defined order of doing one rather than the other. And reader science has provided us with a sort of classic idea of parallel composition of independent paintings. And, as we've already suggested over here, There's a kind of logical component to this as well. Even though we have so little structure, the fact that tensor is not a cardinal product already has quite profound implications. We can't reconstruct an element of a tensor in its components. We don't have pairing, we don't have projections, or we can say we don't have diagonals, but we have projections. And the absence of diagonals... It means, essentially, that we don't have the ability to copy a state, and that we don't have projections, that we don't have the ability to erase one. From a logical point of view, this means that our comma is a resource-sensitive comma, and we don't have, as corresponding to the structural rules of contraction and weakening. The main point of this work was that there were already rules you have to write down about manipulation of common, that you keep the same premises and arrange premises in a practical way. And of course they're valid in intuitionistic logic as in classical logic, but in this Lanoitian or linear sense they're no longer general. So such categories provide a setting for resource-sensitive logic such as linear logic, and in a sense they're already building in.