Example of toposes 1st talk / MW notes — books (& others)

0:00 In any case, the graphs of the one elements are the identity elements, these appear on the one, and the multiplication rule is that D-I-D-J equates with two, which say that these two are includes, in the sense that each one completely absorbs the other, or in other words,

2:30 Constant element is one C, such that Cx equals C for all that. You have one in various ways in terms of curves. So, this monoid is very, very special in that it has a property. Every element is either an identity or else constant. Your identity is Ux. So, actually, it turns out that there's a whole class of examples of this property that... When it comes to generalized sense, everything is either an identity, or a constant, or a relatively constant, it means that you have a three element identity, it says x, y, x, and x, y is all x and y, more noise in more general categories that satisfy an identity of that sort, have many properties in common with this one example, on the other hand, they are multidimensional.

5:00 All of these kinds of information are in the intuitive sense one measure long. Reflective graphs are just many topos U at the sets. So you take the contravariant representations and it's more or less U. You write this right, and then you have the interpretation as an example, or if you like, two standard examples. Operation extends from that, and we could consider this total as equal.

7:30 So that gives us two canonical objects, one of which is the terminal object. In any finite monoid like this, if you take the product of all the elements and split that, you get the terminal object. These two objects are, one of them is the one that came from the splitting, and the one that's already here, you might call it, for example, I for interval, but it has two points. I think that by composing the unique map here, I get, these are the splittings, the category that I hang my splittings in are the impotent and the natural-motion morphemes, and then the out-of-the-map, the Charles T. Henry. You can think of it as inside the pre-sheet category, you just take all retracts. Retracts are almost like representativeness. Again, if you look at concrete examples, this almost turns into vast areas of mathematics, because if you go to open sets in Euclidean space, that's an example of that operation. We're going from free objects to projected objects. One of the things about reflexive graphs is I can think of this object, I, as an arrow.

10:00 What it really is, is the identity map on I. And these two points are these two dots. The two morphisms are simply placing the dots, the arrows. The thing about this very simple picture is that it's the generic picture, at least in the case where U doesn't have too much structure, and in general the plus or minus is a full picture. So if you take any object X, then of course there are the equivariant morphisms, the natural maps, and these one can think of as figures. Figures of this shape in this object are just being mapped, except that usually figures of special domains such as 1 or 2 are. Composing such a figure, these two points, you get two points of x. So actually, on the other hand, since every representable is just made up, you might happen to have its two endpoints the same, which in case you divide it by the root.

12:30 So as well as dots and arrows that connect to dots, any such object has pictures of that sort, can be represented by a picture of that sort. But the picture is not exactly unique because you can look at it around in all sorts of ways without changing the size. Essentially, this is what's called a reflexive graph for networking. Why are they called reflexive? It means that at every dot, there is a canonical loop. There may be lots of dots, lots of loops. By category, they could say that, well, this is the basic thing. The basic thing out there. Everything else is gotten by moving together. In a sense, it has three elements. The two of them are together. Interestate is the word that's used in common ethereal topology. It begins with retraction, which are retractions, in other words.

15:00 Then, when you look at the Chinese, you want to say, well, this serves, for example, the United, I think, they're compatible with the legacy, not with the previous left, that's the only thing, you know, inverse emitter, the substitution, that's the star, that's what the star is composed of.

17:30 And then there is the, you know, that lower star, including this pair of atoms, that lower star is, indeed, left exact and required for the very strong reason that it preserves all. But there is no, there are two, this is a common for the two. These are actually two different subcategories, which are isomorphic as definitely not true for a point set with the only arrow point and I guess with the loop.

1:07:30 This is the graph, that is the graph.

1:10:00 If you have a map from the truth to the force, you have inconsistency of the logic. Yeah, but these are not maps. This is all the object that you are defining. This is not a diagram. Yeah, but this is not true and false. This is what he is saying. We have to assign something to something that is innate, so if they are innate, we go to that point. If the R is outside A, we get to this part. What I'm against is an E. Ah, yeah, okay. Because the notion of... No, he called them true because, I mean, when you classify a sub-object, you always measure every point in a sub-object. Yes, but you have also the force in the omega, and so it doesn't make any logical understanding of what you are doing. That's the problem with the purely logical. I mean, that's why you don't want to get too hung up on the labels. In any logical sense, if you have the... But I mean, you don't know what you have. In order to understand what you have here, you should work out the internal, the ideological structure. Exactly. But in any... I don't understand what you said. Just that in... I understand that, but what I'm saying here is that in the picture the specified element, which is true, also are points, right? But you can do the x, x and y and it's better. Traditionally they are called true and false and they are identified by an f of 1, 2 and 2 of n, right? Just think of the algebraic structure of the Boolean case. No, but that's not what that is doing. So what that is doing is realizing this thing, which is the point of the loop, into this thing, which is this object.

1:12:30 It's saying that this is going true, is sending this thing here, and false is sending this thing here. So then you have the internal, the structure of a Heine-Haswell on omega, and you know that you cannot have the true is less than false. That's true. But the fact that there's an arrow here does not mean that this is less than that. There's an arrow here, it's something else, and in order to understand what the order... What the eigenadge of the structure on omega is, you have to work it out, but it's not obvious by the, it's not in use by these others. That might be a conclusion, maybe you should have specified that. But that, I think, I know there is the thing that makes you confused, but it nevertheless makes sense to call this two of these false, because this is the image of the full map along this, and of this object along the true map, and this is the image of this. Yes, but there might be a computer which can think of those. But your point is absolutely right. It's making, it's thinking in terms of the images of the mass. Yeah, yeah, yeah. As was the case with the brilliant numbers, it's just that they're obviously one plus one. But it's also the standard way. Yeah, exactly. But obviously, you know, he does have a very... So he had these arrows and he said, ah, you have to put this, because there might be an arrow in my A, which goes, which is outside, and that I understood. What I didn't understand is why doesn't you see the possibility that there is an object outside with a loop which is inside? There is an arc, there have to be included also the nodes of the arc, the vertex of the arc, and that's why this is important.

1:15:00 And now I understand how this works. Of course, you don't have to have all of them. Yeah, no, you might have to have all of them. And the N there. Yeah, yeah, yeah. And they go to the intestine. Yeah. It would be still identified with a loop because the points go down. Yeah, yeah, yeah, yeah. So do you mind if I ask you, did you get a decent set of notes? Because it's impossible in the back. No, no, no. Well, I'm sure they're better than anything I could get at the back, because the back you can't see when he writes on the lower half of the board. So would it be possible for me, not right now, but at some point in the future tomorrow, well, I won't give any justification or anything, but they'll be better than anything I was able to get at the back, because you can't see when he writes on the bottom half of the board at all. So, um, it's just a little bit, but the other two I was able to get, but this one is really difficult to get because of the way, oh, this is much better. Yeah, I think there is. I think it's in the next house where the, um, can I probably give them back to you at, um, 2.30? It's embarrassing, eh? I won't. You don't know my name. I'm sure they're much better than anything, they certainly look much better than anything. We'll step to the next panel, OK? Yeah, there is one now, we'll do it in a minute.

1:17:30 Applications of Sheaves, 900, Boss, Soren, van den Bosch, 38, Boss, Soren, van den Bosch, 1888, Pedicchio, Rossellini, category theory, the classifying spaces, classifying topology, CT2000, International Summer Conference in Category Theory, Abstracts, July 16th, 22nd, was the Scuolo, Stiva in De Logica, Cassena, 18th and 22nd of September, 2000, Catmatt2000,

1:20:00 Proceedings of the conferences, categorical methods in algebra and topology, Herlich and Post, H-E-R-R-L-I-C-H, and Post-P-O-R-S-T, Spapiera, Mathematica Arbeit, Spapiera No. 48, SANS, E-Post-P-O-R-S-T, categorical methods in algebra and topology, Bulletin de Société Mathématique de Belgique, 1989, Applications of Categories in Computer Science, London Mathematical Society Lecture Notes, Series 177, edited by M.P. Foreman, Peter Johnston, Johnston and Pitts, so Foreman, Johnston and Pitts, Lecture Notes in Computer Science, S.P.L.S. Bangalore Fairlock Lecture Notes in Computer Science, 389.

1:22:30 Pitt and all category theory and computer sciences, category theory and computer science, lecture notes and computer science 393, Springer-Voelak lecture notes and computer science 393, categorical methods in computer science, research in operational quantum logic edited by David Moore, Alex Wilsey, W-I-L-C-E, published by Kluwer in Fundamental Theories of Physics series and Kluwer Academic Publishers, ISBN. O0-7923-625A-6, Bulletin of the Bulletin de la Société Mathématique de Belgique, 3A term, Excel 1.1 vs. Google 1.1990, for Algebras in Non-Commutative Geometry and Physics, edited by Stéphane Kynepiel and Freddie van Oosterijstijen. That's HOP for Algebras in Non-Commutative Geometry and Physics. That's ISBN 0-8247-5759-9, by Marcel Decker, New York, volume 239, a series of monographs and textbooks in pure and applied mathematics. But anyway, it's Stephan Kynapel, that's C-A-E-N-E-L, and Freddie Van Oystein, that's O-Y-S-T-A-Y-E-N, Proceedings of the Conference on Categorical Algebra, La Jolla, 1965, Categorical Foundation, Apology, Algebra, and Sheaf Theory, edited by Maria Cristina Pediccio, P-E-D-I-C-C-H-I-O, and Volta, D-H-O-L-E-N, Encyclopedia of Mathematics and its Applications, 97, Cambridge University Press, ISBN 0-521-83414-7.

1:25:00 Algebra, topology, and category theory, edited by Cameron Tierney, I think. Yes, this is the Eilenberg first shrift volume. Algebra, topology, and category theory, a collection of papers by Samuel Eilenberg, edited by Alex Heller and Miles Tierney. Academic Press, New York, 1976, 50-8, CIT Mathematica, Belgique, EMXXX111. Fascist, Gillis, Mohn, Series B, 1981, Proceedings of the Model Theory meeting held at Brussels during the 2nd to the 4th, 1980, and Mohn's during the 5th, 1990, Institute for Research in Mathematical Sciences, Fields Institute, Communications, Galois Theory, Hopf Algebras, and Semiabelian Categories. Edited by George Yiannilidze, J-A-N-E-L-I-D-Z-E, Rodriguez, P-A-R-E-I-G-I-S, and Walter Tolland, E-H-O-L-E-N. Editors, published by the American Mathematical Society, ISBN 0-6218-3290-5. Second part of the table, Abstract and Concrete Categories, edited by Yuri Adamek, A-D-A-M-E-K, Horst Ehrlich, H-E-R-R-L-I-C-H, and George Strecker, S-T-R-E-C-K-E-R.

1:27:30 ISBN 0-471-60922-6, published by Wiley Interscience, Bar and Wells, Thomas' Triples and Theories, Springer-Volag, New York, ISBN 0-387-96115-1, locally presentable and accessible categories. by Jiri Adamak, A-D-A-M-E-K, and Jiri Rosicki, R-O-S-I-C-K-Y, London Mathematical Society Lecture Notes Series 189, ISBN 0-521-42261-2. Now to the other side of the table now. An index and other useful information too. Lecture Notes and Mathematics, Structures, Categories, Faiskel. Sursets, structures, categories, and sheaves. In French, ensembles, structures, catégories, faissures, fa-i-s-c-e-a-u-x. Serge Vasilak, V-a-s-i-l-a-c-h. Le Presse de l'Université Laval, Quebec, Masson, ISBN 2-225-46849-4 and ISBN 0-47746-6469. University Press, Introduction to Categories, Homological Algebra, and Schieffker Homology by Strucker, Jan R. Strucker, Cambridge University Press, the ISBN is 521-21699-0.

1:30:00 Then there's Ernst Schubert, S-C-H-U-B-E-R-T, categories, translated from the German by Eva Gray, Springer Verlag, 1970, ISBN 0-387-0578-3-8, and also ISBN 3-540-03-8. Paperback edition, I think I have a hardback edition of that. Okay, continuing along the shelf on the other side. Scritti Linguisticae Mathematicae e Juridicae. Canani, Ugo Berni Canani, C-A-N-A. Leo S. Olsci, O-L-S-C-H-K-I, Editore. 2003, the publishing house of the University of Florence. Scritti Linguisticae Mathematicae e Juridicae by Ugo Berni. Categorises of common cultural patronage include De la Secola, Matematica, Diploma, Supplemento. Others are less obvious. Philosophy to court, the affinity of indiscernible contributes to performing a paper on them.