Session: FW Lawvere & Anders Kock (contd.)

0:00 For the wave equation, you can talk distributions of compact support, and that's a fairly typical case, although the usual bases for the wave equation, like e to the i theta, are not common, but still. However, when you come to the heat equation, it's inevitable that things immediately fill out all of space, and so you have to have unbounded, I mean, non-compact support of extensive quantities. Incorporating that theory into the basic theory, which first, the first thing that pops out of it is compact support with divisions. So I would like very much to understand better the technique you used on that. Because I also have a suggestion about it, which I talked about in Brussels in honor of Franz C. By changing the notion of function, there's prejudice that rings should have unit elements, and so human functions in fact should have unit elements. The idea is that support, interestingly enough, the idea of support for an extensive quantum is very general, logical, and character. Whereas for an intensive quantum, they always bring this idea of what is commonly called zero. It's zero here and it's not zero there, and so forth. But really it shouldn't have anything to do with this special measurement called zero. It's a matter of whether you have genuine dependence or not. That idea can be incorporated, sort of clear excessively, but you can also incorporate it intensively.

2:30 And then you find that the corresponding functions, the pieces of smooth functions, There are a number of different types of functions that can be applied to a particular type of function. For example, a function called a line has a value at the two different values at the end points, and it also has a part which is the compact support. Therefore, if you subtract off, you get something like this. Sometimes it's often zero. The constant function 1 is certainly included. I didn't say it very clearly, but the constant function 1... So you consider an equivalence relation on the space of functions rather than considering the ideal of functions with... The idea is that you need the notion of large subspace. The large subspace is one for which there exists some compact space that can be useful for everything. At least that's the way of defining an example. Another very general construction is that if you have any subspace or any space, The components of the first phase, and algebraic and positive, that always puts you on the collapse algebraic point, but this is another discontinuous structure because if the value has one point, and one component rather, then my result would be the same instead of the algebraic and positive, but in general it would be somewhat different, what I call the exterior value. The exterior value in the sense that it retains... All the information in X outside L, the L itself, has been reduced not necessarily to a single point, but because the L has several components, it can be reduced to at least three points.

5:00 Those are just pinched to points, and you retain the rest. Okay, so you can consider, of course, the function space R, for example, R to the power X minus L. And then you take the limit over there, and that's a directed union, so you still have an algebraic sum. And then when you're functional, you do the inverse of it. They can measure the r-measure on a non-compact group as a property that gives rise to a whole, and it really consists of a whole bunch of measures that subset into one. You split the element down to the inverse, and so you rescue the order of rings with one. And at the same time, a different notion of test functions. And so these extensive quantities are going to have information about the end, in addition to the aversion. Well, I think it's critically important vector spaces where you have the notion of a box and you can simply say that a function has to exist, a box, such that the function vanishes outside. The internal language, when you interpret it by a sheep's semantics, to be...

7:30 It means that you consider functions which uniformly bind with support, something like that. It is suboptic for us and we are. It's not a subplane, it's a submodule. We have these tapes of Grothendieck in 1973. Oh, yes. Well, no, they're, they're, they're bills. I'm just involved in the project of getting them, getting them online. Yeah, yeah. We're, we're working together to get them online. No, no, no, no, no, no. Physically, they are in bills. Physically, they're in Buffalo. No, no. Physically, they're in Buffalo. Yes, yes. No, they're bills. They're in Buffalo. Well, yeah. It's just that Jack, having had a stroke, is no longer able to contemplate carrying out his longstanding project of somehow transcribing and making available these things. So he turned to the... That's why I say they're in Bill's... So that's the background. So now, to get on with the... You still have to do it. I still have to do it. The topologies are the natural domain of general topology, which I agree with. And then he says, well, there are generalizations achieved on a space, which I don't agree with. I mean, it's not the only original example. He's starting off at that level. It's a course. Suddenly he's listing off various examples, and he says radon measures. And so that brings up this question.

10:00 In the course of the course of the course of the course of the course of But the thing is that functions of compact support on U are uniquely functions of compact support on the whole space as well. So the functions of compact support on U are not varying with U in the same interesting way that restricting functions does. It's just a trivial sort of extension. They're all pieces of this one big system of functions on X. So that's how it is. When you, when, equipped with these inclusions, when you take them into your dual, you no longer get inclusions over storage, surject necessarily, or anything, so it's an interesting thing to achieve, but you're using the, this sort of strange functorality of only along the post set of subspaces or something, but really, really they're all global functions. They're just global functions with a certain property that the support lies in you. So, in fact, this is the way that sheaf theory started, was that you have global functions and a notion of support. So that's somehow what's involved there. Have you discussed such issues with Marta Bohr and Jonathan Funk? Because they know a lot of, I mean, not so much for the functional analytic case, but say for the locale case, where they have locale distributions. Which turn out, essentially, to be closed subspaces and all that. Well, not exactly, because the closed subspaces are still extensions. It's a co-variant function.

12:30 The closed subspaces are construed as a co-variant function. Right, right, yes, but what they really, they don't consider them. The question of the support of the intensives. Of course, the closed spaces, closed hot spaces are extensive in quantities, but they construe them via the Riesz paradigm as a dual to the test functions, which are the open spaces. I don't know whether they consider... The question of complex support test functions. Now you see that what you're saying is correct, but it's an analogy with the sort of standard case of real valued functions and... Compact support, the compact support question doesn't come up as such in this topos valiant situation because you know you can add up infinite things and still get things where you can't do that so that particular sort of problem doesn't hit you in the face but no but this extra ingredient you see that you can have You make something that sounds intensive, namely functions, into a covariate factor, first of all, of your parameters base, and then take the dual and produce a contivariate thing, in other words. And that's why distributions can be construed as a sheaf. Yeah, well, for distributions, it's the same as for Rado measures. It's just a smooth case and a continuous case. Yeah, yeah. You have a sheaf of distributions. You have a sheaf of distributions. Exactly, exactly. So it's the same mystery, you see. If all you know about is pre-sheafs in my story of intensive and extensive in a similar way, this is a big mystery. How could extensive become partly contravariant? It isn't globally contravariant. It can't. We can pull a distribution back along a general map, but along an open inclusion, there's a notion that that is the distributions that are difficult to come back. There's something similar to that in the Decian homomorphism, where there are these long-wave homomorphisms.

15:00 What are they called again? Decian. Decian. G-Y-S-I-N. Dimensions, different coefficients, different values, and so on. There are many which follow, you know, terminally from the general idea of co-varying the fonts of various thunkers and one being acting on another and, or being dualization. But there are some that produce maps which sort of seem backwards from that general point of view. I think they usually depend on compact support or something like that as well, certainly distributions as a sheaf. It does seem that the whole issue of the supports and the roots of intensive and extensive quantity, respectively, is an extremely, a topic which has ramifications across virtually every branch of mathematics, including logic. In other words, it really is a kind of, you know, the key to... Exactly. This is a special... ...piece of logic there, too. These things are really global. Among the global things, you single out those who support lies in a certain part. Is it a covariate factor?

17:30 Exactly. So that's logic, so to speak. It's independent of the character of the quantities you're talking about, the particular character you could sell. This is exactly the kind of thing I wanted to try and get some clarification on why I suggested that a general discussion of extensive and extensive quantity would be a useful part of the workshop. This has already been very useful just going into it through this. Specific point of entry through radon measure. The same way we categorize the pre-sheaves, the idea of pre-sheaves, as to the A-off, that's most correctly viewed as a covariant functor. Any functor from A to B can induce geometric morphism, in fact an essential geometric morphism, pre-sheaves on A and pre-sheaves on B. And A and B, the unit embedding is Dirac embedding. A more primitive, you know, the more primitive or general idea would be, well, first let's consider S to the A, which is a contra-varying functor of A in the obvious way because of Cartesian closure, and then we'll look at Ham of S to the A into S, where Ham means functors that preserve addition and are continuous.

20:00 But the hom of s to the a, common s, is equal to s to the a-op if a is small and s is complete, blah, blah, blah. So somehow there's already the idea of a Fourier transform there. I mean, we're representing these extensive quantities in the naive sense, i.e. functionals on covariant content, as actually functions on something else, namely a-op. The circle group as opposed to the integer group, you know, it's very, it's serving that role of representing the opposite kind of quantities. So in a way this is, in some cases, a more natural way to appreciate topologies. It's also a way that exists no matter if A is big. See, I'm also insisting on this idea that we're down with illegitimacy. Do you think you can write this down? Yeah. You could rub off a bit if you wanted, Bill. I think everything except the far right-hand corner, which I couldn't quite read, is all taken down. You might as well just say that these are the punctures that have the right adjoints, but that would sort of become phonological, so I'm trying to emphasize in a somewhat more general case, so go along. By the way, preserving code limits, I think that was continuous linear, because to preserve finite sums, it's like linearity. To preserve multiplication by a fixed set, that's homogeneity.

22:30 Scalar of ingenuity. And then to preserve filtered co-limits. That sounds like actual continuity, you see. I mean, in a way, the whole idea probably is these limits. Limits and co-limits is a little bit of a distortion of classical and mathematical prejudices, because certainly the limit of a constant ought to be the same thing. Whether it's Cauchy or D'Alembert or whoever, you know, it's a limit. It should be that if you apply it to the constant case you get the same thing. So in other words, limits and co-limits of the model, if that's applied to the case of a connected A, those are really limits, whereas these more general ones, maybe they shouldn't be called something else. In fact, I tried to call them roots, and I called them roots in my thesis. Anyway, so we separate the case and thesis, and this is the Houghton case. Further experience, this could, all this could be tuned for rigs and concepts and stuff like this, but the actual category is the further important ingredient, which is reflexive co-equalizers, so that sort of goes together with filtering co-limits, in the sense that it's these two kinds of co-limits that commute with finite products, and hence algebraic structure of all sorts is compatible. And, in fact, conversely, things that are compatible with both of these are essential on the grid. So, this is reflexive. So, reflexive and unfiltered. Put those together, all that continues. So, you have continuous, you have linear. Put those together, it's the same thing. It's a very light function. So, anyway, the idea is, of course, obviously, without any size considerations or anything, There is a delta here that when we evaluate given in the object A, there is a particular function of evaluation in the A, and the evaluation in the A preserves all this stuff.

25:00 Because all this stuff in a function category is precisely computed point-wise, so if you evaluate at a point, it's going to be preserved. So therefore, you have this. Because you have our major, so to speak, that is the homophony, because you have this, and then the, this you have A, A, A, A, that one, reduces the other one, because this restricts, you restrict along this, you restrict along that, you wind up in S to the A up. In the sense that every full continuous is a continuous linear extension, a con-extension, of its restriction to these characters, these sort of like characters in the theory of harmonic analysis, and so on. Restricted to another character on e to the, apply a functional to the function of e to the ix. You get the sort of spectrum generated by these characters, and you go back and integrate that by Fourier series and Fourier integral, and you get the original function back. So it's like a spectral analysis, but you can recover the thing from a spectral analysis, the thing being intensive.

27:30 But the collage analysis goes further if you have a tensor product on A, and you have multiplication and convolution. Convolution, Fourier transform of convolution is the product and so forth of all those articulations by applying the pension product which is A to this point was this. The thing is that this construct just depends on having a Cartesian closed category. It doesn't become a two-dimensional category notion of adjoint. You represent it to that extent and you need those additional communities to re-represent these. Manifestly extensive quantities as intensive quantities on this other object, A-up. The thing is that, you know, in category theory, A and A-up are more or less the same thing, you just reverse the errors. Whereas in analysis, to go from A to A-up is a much bigger step. You go from the integers to the circle, from the circle to the integers. And that's the most basic case, i.e. the integers indexing of a function on the circle. There's a big basic problem of analysis in that, from the restriction, but it is just a really extensive block. It probably applies to any kind of function, apply it in particular to the sinusoidal or exponential functions.

30:00 That gives you a sequence of coefficients, but then can you re-construct it and do it legitimately at the next level? When that stuff comes in, it actually could take infinite sounds. Yeah, yeah. This is punctual. This associates that. It's also a double globalization monad. Oh, correct, yes. Yeah, so it has also two monadal structures. In other words, there's a... Yeah, of course. You know, what that does... What time do we have to do it? It's not clear. It's just from A plus B into S, A up. The two multiplications are going to coincide, but if A is large, they might be different. Oh, restricting along delta. See, I see. This was really restricting along delta, so... I understand the intuition right. So you're saying that each pre-sheet is like a generalized... Hold on. Extended function. No, an extended pre-sheet is an extensive one, not a function of any kind.

32:30 Yeah. You don't see the elements as... The elements in here as extended functions. There are extended functions on S, maybe, but I'm thinking S fixed and A variable. But see, the delta doesn't depend on having unative embedding. It's the restrict along the unative embedding. Delta itself, viewed in terms of this delta, is semiologically more primitive. And indeed, it's sort of like the R-measure. I'll vote for that. In fact, if you try to...

35:00 There are many constructions that make sense in that context. Yes, that's clearly independently a very, very fruitful viewpoint. One of the things I'd like to understand this afternoon is what Sammy Eilenberg had so, you know, why he was so down on double dualizations. He just said that many constructions in the algebra theory... And so forth. They make an unnecessary... Okay, okay. Okay, right. So in other words, people appeal to it unnecessarily in ways that, you know... Okay, I should like to understand that better. In a context where the double loom of the thing really is the thing itself,

37:30 by complicating matters by talking about that instead of talking about the thing itself... Sure, sure, sure. ...it's only in the context where you have... Sure, sure. Sure, sure. Shall we make a move? What what time should we come back, Richard? What time is it now? I'm sorry. It's ten to two already. Wow. Okay. Is it okay if we say to come back at three o'clock? Because we have to finish with this room because they're having the philosophy seminar here at, well, we should be able to do what, about 4.30? Yeah, of course, of course it's a party. There's on category theory, so people will... Yeah, we can stay if we want to, but for us to use the room for this, for our discussions, we should be out of here by 4.45, so if we come back at 3 o'clock, it gives us an hour and three quarters. Is that okay? You know, I've got to be here for this seminar, but people who didn't want to go to the seminar must carry on talking and use my office for that. Thank you for your attention.