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

0:00 A book by McNamara and Reyes, The Logical Foundations of Cognition, Oxford University Press, 1994, but I must get this book by Canani of his writings, Scritti Linguistici, Matematici e Juridici, ISBN 88-222-52535. And it's, again, it is scritti, linguistici, matematici e juridici, must study particularly his common categories, his common cultural patrimony paper, and, for instance, his theory of quantum sets, and they send that to Steve French and get his reactions, présentation géométrique, for semantic relations. I really do want to study the categories as a common cultural part for many. Geometrique de evaluaciones, Semantique représentation de la espèce, Metrique de la espèce, Canary book, as well as the notes of Lovier's lecture this morning. Textos de matemática 35, Alcumas, Teores de Galois, Boscorpos, Edos, Anés, by Francisco Cerds, the Spanish version of one of his books.

2:30 Handbook of Categorical Algebra, Volumes 1, 2 and 3, published by Cambridge University Press in the Encyclopedia of Mathematics and its Applications. So it's volumes 50, 51, and 52, again it's Handbook of Categorical Algebra 1, Basic Category Theory, ISBN 0-521-44178-1, hardback. The second volume is Categories and Structures, that's volume 2 of the Handbook of Categorical Algebra. And the ISBN is 0-521-4417-X, hardback. The Categories of Scheibs, Volume 3, again of the Handbook of Categorical Algebra, published by Cambridge University Press in their Encyclopedia of Mathematics and its Applications series, Volumes 50, 51, and 52. ISBN 0-521-44180-3. And that is dedicated to Bill Lafayette for his kindness and his genius. This is Galois Theories by Francis Bourser, B-O-R-C-E-U-X, and Georgiana Lidze. It's Cambridge Studies in Advanced Mathematics, Volume 72. The ISBN is 0-521-389-8.

5:00 The next book is Mathematics is Francis Bourceux and Dominique Bourne. Malchef, protomodular, homological and semi-abelian categories. Malchef is M-A-L apostrophe C-E-V comma protomodular, homological and semi-abelian categories. Published by Kluwer. Academic Publishers ISBN 1-4020-1961-0 Book is Teoria de Faschiz Uno Spazio Topologico Bressanoni September 1991 by Francis Bossert. It's a leaflet, I'll try and copy that, but most important urgent is to copy the Canani book, both the geometry of spaces and Ask Alberto Peruzza to get that for me, the geometry and semantics and the categories of common cultural patrimony. Francis Poussin, Fasci, Logica et Topoi, Quaderni dell'Unione, Matematica Italiana, Pitagora, Editrici Bologna, 1989. There should be an ISBN. Yes, there is. The ISBN is 88-371-0486-3. Published by Pitagora, Editrici, Bologna, 1989. I'll try and obtain a copy of that. Carrière de centre de logique, 12 aspects de la dualité en mathématique, sous la direction de Pivan Prague, Université catholique de Laval, Département de philosophie. Interesting, I must see if I can obtain that. Published by the University of Louvain-Laneuve at EISBN is 2-87209-687-6, and there are articles by Prahn on the theory of sex and duality, Thierry des Ensembles, Dualité, and the left-right duality in the theory of categories by Borseur himself, and by Chembler.

7:30 History, oh, that's Karen Chimler, the history and prehistory, history or prehistory of duality, reflections on spherical triangles with and after Hilbert. This is Madame Chimler, the expert on Chinese history. Yes, they're all records of the contributions, the four contributions to a day organized by the University of Mons at Heine on the theme of duality in mathematics. Yes. ISBN 2-87209-687-6. Ask for service anywhere. I can obtain a copy of that. The book is Introduction to the Theory of Categories and Functors by I. Bucher, B. U. C. U. R. and A. Delano, D. E. L. E. A. N. U. A Wiley Interscience publication. Introduction to the Theory of Categories and Functors. In the Wiley Interscience series Pure and Applied Mathematics, it's copyright 1968, there was no ISBN in those days, the SBN was 47011651X, published in 1968. The notion of a category, dualities, subcategories, examples. It seems a very interesting book, giving a very general introduction, for instance, into abelian categories, growth in deep categories, before the discovery of topo, of course, before the cognition of the construction of the topos, or certainly before the law of the attorney, the axiomatization of topos. But by way of the kind of background machinery that was being deployed in the formalization of topos theory, it's a very interesting and probably important book.

10:00 Compatibility, Stability, and Sheathes by J. L. Bueso, B-U-E-S-O, P. Jara, J-A-R-A, and A. Verschoren, V-E-R-S-C-H-O-R-E-N, published by Decker. Marcel Decker, Inc., New York. The ISBN is 0-8247-9589-X. A unique and self-contained reference book on compatibility, stability, and sheaves in the series of monographs and textbooks, Pure and Applied Mathematics. Okay, the next book is Lecture Notes in Pure and Applied Mathematics, Volume 46. Again, published by this publisher, Decker. Marcel Decker, Inc., New York. And it is entitled Banach Modules and Functors on Categories of Banach Spaces by Johann Sigler, C-I-G-L-E-R, Victor Lossert, L-O-S-E-R-T, and Peter Michor, M-I-C-H-O-R. The ISBN is 0-8247-6867-1. Croll, C-R-O-L-E, Roy L. Croll, C-R-O-L-E. Categories for types, Cambridge Mathematical Textbooks, the ISBN is 0-521-45701-7. The next book is Dietz, sorry, D-I-E-R-S, Diers, D-I-E-R-S, Categories of Commutative Algebras, by Wes Diers, D-I-E-R-S, it's in the Oxford Logic Monograph Series, Oxford Science Publication Series, I should say, the 1992, the ISBN is, the ISBN, the ISBN appears to be,

12:30 This here is the ISBN 0-19-853586-4, a study of the properties and structures of categories of qualitative algebras. Next book is Alexandru Dimcha, D-I-M-C-A, Sheaves in Topology, a university text, ISBN 3-540-20665-5. The next book is Categories et Structures by Erisman, yes, Erisman Categories et Structures in the series Travail et Recherche Mathematique under the direction of Andrei Lichnerovich, published in Paris in 1965, interesting quotation from Goethe, Faust, Charles Erisman, published in 1965. By Dunod, Paris, D-U-N-O-D. It doesn't have an ISN, so the only way of obtaining it would be to obtain the entire book and to photocopy it into the theory of fiber categories and the work of... In what sense was it distinct from the Grothendieck school? Need to understand that. So categories are structures. Next book is also by Erezman. And that is entitled, it's the oeuvre complete et commente, so the complete works of Giles Erezman with commentary, topology, algebra, geometry, differential, published in AMIAR 1984, edited by Madame Erezman.

15:00 It consists of supplements 1 and 2 of the volume in Cahiers de topologie et géométrie différentielles, so I think it has the paper by Lorvier, the 1983 paper by Lorvier. It doesn't appear to have an ISBN. Yes, it's Ehrensman's Collected Works, publi avec les de la midiste M.I.G.I.S.T. Dans le cadre du programme mobilisateur n°6, promotion de français, langue scientifique et diffusion de la culture scientifique et technique, AMI en 1984. Yes, this has got a whole section here on internal categories and vibrations. I'll have to ask if there's any way I can buy a photocopy of the whole thing. The next book is Categories, Allegories. Peter Freyd and Andrei Shedrov published by North Holland in the North Holland Mathematical Library series. The ISBN is 0-444-70368-3. Next book is Calculus of Fractions and Homotopy Theory by Peter Gabriel and Michel Zisman. Published by Springer-Verlag. ISBN is... don't have an ISBN. Published in 1967, so it doesn't have an ISBN. Published by Springer-Verlag in their Ergebnisse der Mathematiker und ihre Grenzgebäute, band 35, 5-7-9, Library of Congress, catalogue card number 67-10470. But no ISBN.

17:30 Théorie des faciaux by Roger Gaudemont, G-O-D-E-M-E-N-T, Théorie des faciaux, publication de l'Institut de Mathématiques de l'Université de Strasbourg, and it is by Herman, H-E-R-M-A-N, ISBN 2-7056-12521. Published back in 1972, so it's interesting that ISBNs must have come in about that time. This book is Herlich and Strecker, Category Theory and Bacon. Yes, Category Theory and Introduction, Horst Herlich, H-E-R-L-I-C-H, N. George, E. Strecker, S-T-R-E-C-K-E-R, Allen and Bacon, Boston, Library of Catalog. Library of Congress, catalogue card number 78-182352, and it's a very good general introduction to category theory, which we should try and obtain. It was published in 1973, so interestingly they hadn't introduced ISBNs in 1973 in the US, and foundations, sets, classes, and conglomerates, categories, sections, retractions, and isomorphisms. Yes. Foundations, categories, special morphisms and special objects, factors and natural constellations, limits and categories, adjoint situations, set-valued factors. Yes. The purpose in writing the book was to present the theory of categories at the earliest moment at which the reader can appreciate it, that is, as soon as he becomes reasonably acquainted with set theory, modern algebra and general topology.

20:00 I really do want to retain and study that. Very nice looking box of intermediate level introduction. Hurlick and Shtack category theory, published by Allen and Bacon. Try and obtain it. Categories and groupoids, Philip J. Higgins, Van Nostrand, Reinholdt, ISBN. Doesn't look like it had an ISBN. The number on the back is 442034067. It's a relatively early book. Le langage des catégories by Peter J. Hilton. Translated from English by J. C. Mathis, M-A-T-T-H-Y-S. Collection Formation des Maîtres, ISBN 2-7124-0110-7. Topos theory by Peter Johnston. Stone spaces by Peter Johnston. Which work you have, I think. But just for the record, the ISBN of Johnston's book is 521-23893-5. Then Johnston, Sketches of an Elephant, Volumes 1 and 2, which you also have. Then Basic Concepts of Enriched Category Theory, G. M. Kelly, London Mathematical Society Lecture Note Series 64, Andre Joel and Iker Murdyke, Cambridge University Press, ISBN 0-521-55830-1, which breaks down to which Steve Aude is also going to be talking about.

22:30 Monoids, Acts and Categories, Walter de Gruyter, Expositions in Mathematics, edited by Mattie Kilp, K-I-L-P, Ulrich Knauer, K-N-A-U-E-R, and Aleksandra Wien-Mikhailov, ISBN. So, it might be good to have some opportunity to ask questions about mathematics on a confidential and anonymous basis. I mean, people may, can be a bit intimidated to do that as a whole group. Keep in mind that we have several houses. Each house has a living room. Exactly. And that's, there is always the possibility of working with small groups in one of the living rooms. So the next one you guys are live, I think, and then... Okay, so I'm going to put myself on the line, and then when somebody else is going to ask a question about my talk, it's not fire. And if nobody's got fire, I'll leave the books on the line. You guys are all both, let's put it on the line. Yes, excellent. The same old question I was about to ask, if I'm helpful if I'm an actual doctor,

25:00 and I want to give the right amount of screen time, it's my rent, and I want to give everything to the project as well. Or you can use the examples to say what maps actually arise between those examples and fascinating things that you both lead to, to this same notion which seems to be absolutely tenable and it is a bit of a surprise. If you start from the examples, well the first examples of income are huge sums of local spaces, which is perhaps another way to talk about GameCom. A continuous function of x and y needs to be taken as a model. Given a function of x and y, what sort of functors do you get between the class of G?

27:30 The answer is you get two of them. First of all, a level of open sets, and it goes back again backwards. And this, thinking of these ordered sets as categories, this is a functor, it's not a preservative. But it's better than that. It preserves. There are three co-limits here. Those co-limits have become cheese when you have both two and that is actually going to be tomorrow's lecture. Get two funtos out of this. One of them is the one that's called the direct image. It's not the best name that could be chosen for this. Direct image does suggest the left-hand one whereas this is the right-hand one. This is simply composition with a code there in place. This is the operation that humans do. There's a sheet on x that's particularly attractive, functor on it, and composes that part minus one, and as a result, itself, is an extension of x that sits inside a sheet from x to one.

30:00 These are totally categories, and this is actually just some extension. So, what we get from that example alone, from every way you look at examples, is that the examples of topos are arising in nature from, say, these next category notions.

37:30 They come up as the inverse images of local homomorphic. Oh, sure. I mean, if you have them, then you're not lost out. But the combination of the corresponding topos is the same. It's also worth pointing out that the minus one on the blackboard has the radii joints, because the minus one, there's unions which are in those things, so you've got a miniature example of the geometric morphism right there.

40:00 Yeah, indeed, that's what I was going to say, because this is just the same as this, and one doesn't think of the direct image at this level to the same extent. As long as you take the probing part, you don't really know what's there, because it's guaranteed to be there whenever you want it, but it's already useful enough to change the base. You were saying that the Gaussian dystopian system has all co-limits, finite limits, and the work of these co-limits commutes with finite limits. And so that this... The definition of geometric morphism lies by nature, because these functions deserve what they have to do, what they have to preserve. Is there, included in the definition of geometric morphism, the fact that these decodings exclude these functions? Is there, included in the injunction, or in any kind of open-mindedness between topics,

42:30 maybe that you're as quickly as the geometric morphism? I don't know if anybody thinks this is true. It's always there, or you don't think about it. It also has a left angle. It's saying that if you have a left angle here, the smallest open contains each image in one, except in Y, that might be a small image. So you need a bit more of a receptive image here, and I think more can be stated in terms of being stronger, also between sub-objects of any object, whatever.

45:00 It's saying that you get the same sort of left angle here. So if you can take sub-object f and push it down to a sub-object of A, which is like taking a hand that's at the size of its reciprocity condition, you can take it as a definition of what we know, that for every object A, not just for the terminal object A, but for every object A, the co-dominant top has this left air joint, inverting in terms of the five sub-objects. It's complicated to verify, and it turns out there are all sorts of nice, just the existence of these really funny left air joints.

47:30 In short, the data at the start preserves the multiple operations that aren't used to preserve that same implication, universal quantification, and it's sort of obvious why this tells you that it preserves universal quantification, because, you know, it's just the same at left-back points. The Provenient Interest Proposal Condition says that the left-back point commutes with the intersection of the fixed object, and so that tends to be good at the same point. It's an implication by a fixed object, and then again the existence of the left-back point is an implication. And further, it's equivalent to the fact that Sondra, and I don't really understand it myself,

50:00 first of all, preservation up to a monomorphism of higher order, and so these are the things I'm talking about. Well, if you want to say anything more about my explanation of why, it implicitly comes from topology. Open, subjective maths in topology are useful things to have around, although I think it's somehow slightly reflective of the other one. And the same is true of the toposystem and the descent theorem, which enables you to reconstruct them precisely, essentially because you've got this back here, a rather mysterious connection, which remains a rather mysterious connection to me. The spatial effect is an interesting problem, although, so you cannot be completely additive.

1:05:00 It's called an image, acted by some gelatine solvents, and the predictive reaction on them is the one.

1:12:30 Just wanted to have a comment about that, because there are things that need to be talked about and about to make the difference and we've been hoping.