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June 1996

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Topos-theoretic classification of Lie groups
by None 03 Jun '96

03 Jun '96

Date: Sat, 01 Jun 1996 12:14:20 -0400 From: Vladimir Trifonov <trifonov(a)ix.netcom.com> Those interested in real-world applications of categories are advised considering the following. There is a natural (and strictly defined) sense in which a Heyting algebra is assigned to a Lie group with a family of (proper and/or pseudo)-Riemannian metrics over it (V. Trifonov, Europhysics Letters 32, 7, pp.621-626, 1995): <http://www.ed-phys.fr/articles/euro/abs/1995/32/8/ep32801/ep32801.html>. Initially the result was motivated by the Four-Dimensionality Problem in physics (1.Why does spacetime look four-dimensional?). It turnes out, however, that it gives concise (the paper is letter-sized) and elegant solutions to at least four other difficult problems: 2.Why is the effective metric Lorentzian? 3.Why does it look like there was a big bang? 4.Why are only three matter generations observable? 5.Why does the arrow of time exist? To-date this remains the only solution of the Four-Dimensionality Problem and the Metric Signature Problem (2). For details see recent posts in sci.math.research and sci.physics.research. We need your expertise. Vladimir ______________________________________________________________________ http://www.ed-phys.fr/articles/euro/abs/1995/32/8/ep32801/ep32801.html http://weblab.research.att.com/phoaks/sci/math/research/index.html MAA; Gamma Systems Research Group, Inc. Date: Tue, 4 Jun 1996 22:23:14 -0300 (ADT) Subject: Preprint (announcement) Date: Tue, 4 Jun 1996 11:57:15 +0200 From: Anders Kock <kock(a)mi.aau.dk> The paper "Remarks on the Bianchi Identity" by A. Kock is available (compressed dvi or ps versions) from the Hypatia archive in London http://hypatia.dcs.qmw.ac.uk It uses synthetic differential geometry to understand the Bianchi identity in terms of combinatorial groupoid theory. Date: Tue, 4 Jun 1996 22:22:07 -0300 (ADT) Subject: 61st PSSL - Program Date: Mon, 3 Jun 1996 19:23:23 --100 From: Enrico Vitale <vitale(a)lma.univ-littoral.fr> 61st PERIPATETIC SEMINAR ON SHEAVES AND LOGIC Universite' du Littoral - Dunkerque - France Saturday/Sunday, June 8/9, 1996 The 61st meeting of the PERIPATETIC SEMINAR ON SHEAVES AND LOGIC will be held in Dunkerque over the week-end of 8-9 june 1996. We wait for participants on Friday evening or Saturday morning. On Friday evening there will be a registration desk from 17:00 to 20:30 at the second floor of the University building, 1 quai Freycinet. Talks will be held at the second floor of the University building. PRELIMINARY PROGRAMME Saturday morning session, starting at 9:45 - Y. Diers (Valenciennes): Various exemple of categories of algebraic sets - J. Koslowski (Braunschweig): A convenient category for games and interaction - F. Lamarche (Lorraine): On constants in free symmetric monoidal categories - M.C. Pedicchio (Trieste): Pseudo groupoids and congruence modular varieties Saturday afternoon session, starting at 14:30 - G.M. Kelly (Sydney): The category of domains for limits and weighted limits - L. Nison (Valenciennes): Categorie des anneaux rationnels gradues - D. Cubric (Cambridge): Yoneda lemma and normalization - T. Streicher (Darmstadt): Repletion as an inductive definition - W. Tholen (Toronto): Separation versus connectedness Sunday morning session, starting at 9:15 - P.T. Johnstone (Cambridge): Remarks on quintessential localizations - F. Borceux (Louvain-la-Neuve): Morita equivalences of arbitrary sketches - S. Lack (Sydney): TBA - P. Ageron (Caen): Accessible categories with (finite) pullbacks - A. Kock (Aarhus): Revisiting synthetic differential geometry Dominique Bourn Enrico Vitale Laboratoire LANGAL, Faculte' de Sciences, Universite' du Littoral 1 Quai Freycinet - BP 5526 - 59379 Dunkerque, France Tel.: 28.23.71.61 - 28.23.71.73 Fax: 28.23.70.39 e-mail: bourn(a)lma.univ-littoral.fr vitale(a)lma.univ-littoral.fr Date: Sun, 9 Jun 1996 20:41:04 -0300 (ADT) Subject: May 28, '96 corrected revised `... Makkai sketch resolution' Date: Sat, 8 Jun 1996 14:45:43 -0400 From: James Otto <otto(a)triples.math.mcgill.ca> Dear people, In the May 27, '96 revised `... Makkai sketch resolution' I implied that showing that f.p. presheaves over the opposites of signature categories have projective covers was not besides the point. But, as an anonymous referee indicated, it seems that it is as it seems that all f.p. presheaves have projective covers. This was fixed in the May 28, '96 corrected revised `... Makkai sketch resolution' which was linked then to the URL ftp://triples.math.mcgill.ca/pub/otto/otto.html Regards, Jim Otto Date: Mon, 10 Jun 1996 08:37:29 -0300 (ADT) Subject: Gilberte Van den bossche Date: Mon, 10 Jun 1996 12:51:19 +0200 From: Francis Borceux <borceux(a)agel.ucl.ac.be> Dear colleagues, I have the painful mission to announce that our colleague and friend Gilberte Van den bossche died saturday night, after a long illness. All those who visited Louvain-la-Neuve, all those who met her at various international meetings have always been impressed by her kindness and her highly developed sense of social contacts. She has produced in her scientific life very nice pieces of mathematics, on derived functors, locales, localizations, Gelfand rings and more recently, quantale theory. She was also an exceptional teacher, highly appreciated by her students. Her disparition is a great pain for all the members of the Louvain-la-Neuve category theory seminar, and a true scientific loss. Francis Borceux ============================================================================ Francis Borceux phone 32 10 473170 (phone, office) Departement de mathematiques 32 10 614205 (phone+fax, home) Universite Catholique de Louvain 32 10 472530 (fax, office) 2 chemin du Cyclotron 1348 Louvain-la-Neuve E-mail borceux(a)agel.ucl.ac.be Belgium Date: Wed, 12 Jun 1996 08:56:05 -0300 (ADT) Subject: Re: Gilberte Van den bossche Date: Mon, 10 Jun 1996 21:08:45 -0500 From: John Duskin <mthduskn(a)ubvms.cc.buffalo.edu> I can't begin to say how sorry I am to hear of Gilberte's death. She was one of the most delightful people that I had the pleasure of knowing at Louvain la Neuve. I will never forget her answer to my question (on learning that she was native to the other side of the linguistic divide), "What was she doing in Louvain la Neuve?". With a twinkle in her eye, she replied," Je suis espionne!" We will all miss her, not only for her mathematics, but for who she was as a person. Please give my sympathy to her family and friends. Sincerely, Jack Duskin, SUNY Buffalo Date: Wed, 12 Jun 1996 15:06:26 -0300 (ADT) Subject: Computational Category Theory Date: Thu, 13 Jun 1996 00:01:24 +1000 From: Michael Johnson <mike(a)macadam.mpce.mq.edu.au> Subject: Computational Category Theory Greetings. Shortly after the Sussex meeting there will be a meeting on computational algebra which includes a special session on computational category theory. In case you're interested the details are: Dates: July 17-20 Place: RISC (Research Institute for Symbolic Computation), in a mediaeval castle outside Linz, Austria Cost: US$ 215 for registration. Approx US$ 55 per night for accommodation Forms: Registration and accommodation forms, as well as lots more information, are available from http://info.risc.uni-linz.ac.at:70/0/conference/IMACS96/imacs.html The conference is sponsored by IMACS, and is called ACA'96 (Applications of Computer Algebra). It includes a wide range of other (interesting) subjects in computational algebra, as well as the category theory session. Speakers at the Computational Category Theory session will include: Ronnie Brown Richard Buckland Robbie Gates Bob Rosebrugh Makoto Takeyama Robert Walters and perhaps more! Please feel free to write to me if you'd like any further details. Mike Johnson. Date: Tue, 25 Jun 1996 18:01:59 -0300 (ADT) Subject: Isbell interview Date: Tue, 25 Jun 1996 15:34:37 -0400 From: Peter Freyd <pjf(a)saul.cis.upenn.edu> It looks much better on the web, but here's a plain text copy ----------------------------------------------------------------- Topology Atlas Document # topc-06.txt WWW url: http://www.unipissing.ca/topology/t/o/p/c/06.txt Topology Atlas, atlas(a)yorku.ca http://www.unipissing.ca/topology/ ------------------------------------------------------------------ AN INTERVIEW WITH JOHN ISBELL Interview from volume 1 # 2 of TopCom by K.D. Magill, Jr. I'm not certain when I first met John Isbell. I do remember seeing him at the summer meeting of the AMS in 1966 at Cornell University but I can't even recall if we spoke. I do recall that he and Dov Tamari were having a serious conversation at the time. Dov was chairman of the mathematics department of SUNY at Buffalo and was probably trying to recruit John but was unsuccessful. The "old" University of Buffalo, a private university with a very small mathematics department had, not too long before, been designated as one of the four university centers of the SUNY system and Dov had been hired, appointed chairman, and was given the mandate of building up the department. He did an extraordinary job in this regard during the three years of his chairmanship, bringing in some extremely able people. Unfortunately, however, interdepartmental problems surfaced, with the consequence that some of the most able of these people left after the third year of Dov's chairmanship. I was chosen to succeed Dov as chairman and was told that my major task was to continue to build on the foundation laid by Dov. In other words, a large part of my job was to continue to recruit outstanding talent. This brings me to what I remember as the second time I ran into John. That was in the fall of 1968 at the Indian Institute of Technology in Kanpur, India where we both were attending a topology conference. I remember promising John that I wouldn't ruin his conference by harassing him about accepting an appointment at Buffalo, but I also mentioned that I wouldn't be doing my job as chairman if I didn't at least bring the topic up. John said something to the effect that he wasn't terribly interested but he wasn't entirely uninterested either. I wasn't too optimistic but upon my return to Buffalo, I arranged for John to give a Colloquium talk. At that time, John was negotiating with several other universities, all of which were higher on his priority list than SUNY at Buffalo. But things worked out extremely well for us. To borrow a phrase from the Al Pacino character, Michael Corleone, in the motion picture The Godfather, I simply made John "an offer he couldn't refuse". John accepted a position and I am convinced that this was one of the turning points in the development of the mathematics department here at SUNY at Buffalo. We have since attracted other very talented people that very likely wouldn't have considered the place if John hadn't already been here. KDM: What information can you give me about your ancestors? When did they come to the USA and where did they first settle? ISBELL: Well. Your asking about my ancestors reminds me of the young man who liked pancakes... he had two and a half trunks full. My father collected ancestors, and he had several notebooks full. He pretended to follow some line back to the wife of Rollo the Norman, the founder of Normandy. Of course to Rollo too, but Rollo's wife ... in my father's genealogy, at least ... was a princess, an English princess named Gisela. So her ancestors are all there in the Anglo-Saxon Chronicle, right back to Sceof, son of Noah. The Bible never heard of Sceof, but it does have Noah, and it gives his ancestors right back to the beginning. So yes, I had them. KDM: That's good, but when did they come to the USA and where did they first settle? ISBELL: You don't mean all of them; I expect you mean the first American Isbell ancestor. That was my father's great disappointment. He tracked them back, through Alabama and Virginia, to 1659 in Caroline County, Virginia, and there the trail ends. The county courthouse burned in 1659, and when the smoke clears, there's old Throckmorton Isbell or whatever, paying taxes, suing his neighbors, and no clue to where he came from or when. KDM: What were their occupations? In particular, what were the occupations of your parents and grandparents? ISBELL: Father, army officer, mother, housewife. She taught school at first, up in the Virginia hills, but in 1917 she struck for the bright lights of Baltimore, and I think she had a pretty good time until they married in 1923. She was working as a typist. His father was in lumber. First traveling around buying it on the hoof, and then working at the central office in Nashville. Mom's father would be the only intellectual in sight, he was a printer, and he published a weekly paper when he could, in West Virginia and Virginia. KDM: Any of them still living? ISBELL: Yes, my mother is still living at home, with my older sister looking after her. Mom's 97. Frances is only 72, and she's in better shape than I am. KDM: We weren't supposed to get to siblings until later in the interview. Let me ask, when were you born ... and where? ISBELL: 1930, in Portland, Oregon. Dad was a captain stationed at Vancouver Barracks, Washington, at the time. KDM: Back to the siblings. How many are there and what are they doing now? ISBELL: Two, one of each. My brother is 71; retired from the Postal Service nine years ago after a blood clot in his aorta nearly killed him. He hasn't walked since, but I expect he's pretty happy. He, his wife, and both his daughters are now in the same small town where his wife grew up, Clarksville, Tennessee. One girl has two kids and a husband, a Professor of Spanish. My sister never married; she followed my father's line of work, Air Force Intelligence in her case. Besides shepherding our mother she's, not running the farm, but managing the property; they hire a farmer to do the farming. And she's a leading figure in the local branch of the Texas Historical Society. KDM: Which of these people had the most influence on you? ISBELL: That would have to be my father. My mother was a support, of course; her absence might have done God only knows what to me, but she wasn't absent, and it was my father who expected me to show something. He pulled me out of public school after the ninth grade, when I started to goof off, and sent me away to military school. That was ... KDM: OK but let's talk about elementary school first. Where did you go to elementary school? ISBELL: Where? That's Reno, grades 1 to 4; Minneapolis, 4 and 5; Gainesville, Georgia, grade 6. They keep moving officers around, you know. "Where are you from?" "Man, I'm not from, I'm an Army Brat. I'm here." KDM: Oh. Well, then, high school was mostly in one place, was it? ISBELL: What we called high school, I had all in one place, Staunton Military Academy in Virginia. Junior high school in Arlington. KDM: What were your favorite subjects? ISBELL: That has to be math. The only subject I took Saturday classes in. There was an enthusiastic young math teacher who organized a class of two of us, eleventh graders, to study analytic geometry. Then he overreached and tried to take us on to group theory. But he was a true Huntingtonian, like E.W. Huntington, the axiom man. He didn't see any need to tell us what groups were for, and he lost us after I think one lesson in group theory. KDM: You keep answering the following question which, in this case, is what teachers influenced you the most? ISBELL: Oh, well, I'm not sure Mr. Foster influenced me the most. His subject did. One reason I wasn't ready for group theory was that I wanted more analytic geometry than he and his book offered. Foster, and I think the authors, pitched it as showing the equivalence of geometry and algebra. Now I had pretty definite ideas about algebra, and geometry seemed a LOT bigger... any curve you can draw, not to mention disconnected graphs which of course you already get with xy = 1. How could they be equivalent? I did NOT have definite ideas about what a construction was. One of the Bernoullis said... of course I had no idea of this at the time... that the limit of a sequence should be thought of as the last or $\infty^{th}$ term. A sequence of three terms has a third term, he said; a sequence of ten terms has a tenth term, and a sequence of $\infty$ terms has an $\infty^{th}$ term. The task of the mathematician is to find it. Okay, so take an arbitrary curve; draw one, say your signature. A half-civilized algebraist, like Mr. Foster (Achilles Foster; he later got a Ph.D. and a professorship in Newark) would say you can't write an equation for it, and in that respect geometry is not reducible to algebra. But I figured you could write an equation for it, and I proved it. I cheated, of course. In fact, what I actually told all the kids who would listen was, I can write an equation to forge your signature. I would approximate the signature with a union of line segments. Ask them to imagine magnifying and getting a really close approximation, and they agreed that a union of line segments would make a good forgery of their signature. Then show that d(A,P)+d(P,B) = d(A,B) is an algebraic equation; with radicals, since the distance is the square root of the sum of the squares, but radicals are perfectly legitimate, you use them to write the quadratic formula. Finally point out that f(x,y)g(x,y) = 0 has for graph the set of all points (x,y) where f OR g is 0... the union... and you have it. Descartes freed from every blemish; algebra = geometry. No, the teacher who influenced me the most would probably be old Marshall M. Brice, the head of English. We groused about how Brice had no soul when he asked us on a test how many kisses the knight-at-arms took to close the eyes of La Belle Dame sans Merci... counting kisses seemed the opposite of poetry. But then I thought some more about it. I didn't get as far as Keats took his brother, I think it was; anyway there is a letter in which he mentions that the number... four... seemed about right, and at least treats the right and left eye equally. I did, however, begin to get the idea that if it's worth doing it's worth doing right. And Brice evidently took pleasure in making me look stupid when I came up with a remark intended to make me look smart. But God knows that's fair. And he did it very well. I was reading some Marlowe that year, my senior year, and Brice tried to get me to read Kyd. I'm not sure if any of Kyd's plays were even available, anyway I didn't even look, but I did take it to heart and started on Kyd when I had a college library. Kyd is to Marlowe something like Fourier to Laplace. Except that Shakespeare learned from Marlowe, while neither Newton nor Einstein learned from Laplace. But the point is, Kyd plays better then he reads, which is putting first things first. Just as Fourier's ideas are better than his definitions. KDM: When did you first become interested in mathematics? ISBELL: Interested? I think in the winter of 1936--37. In school, the numbers went 1,2, oh you know how they go; but when the temperature dropped to 0 it didn't stop dropping. The numbers kept right on going down. And that is much more vivid than looking at a lot of ants, or pebbles, and thinking a thousand, a million, and on and on. Just step out the door and there in your face is a number you had no idea of last week. It's not that I took up mathematical games, but the ontology of mathematics; that's not something you're going to forget. It must have been the same half-year or so that I got into fractions. Not in school, but in life. KDM: I would imagine that you were an exceptional student. Did any of your fellow students impress you in any manner? ISBELL: I think not in an interesting manner, for whoever might read this thing. KDM: When did you graduate from high school? ISBELL: 1947. KDM: Where did you attend college or university? What were your major and minor subjects? ISBELL: That's kind of a tangle. Five colleges, up to the B.S. I was longer at Chicago than any of the others, because after doing my fourth year there I started summer school, registering for one course with Irving Kaplansky. I minored in physics and majored in math. KDM: Did any of your professors have much of an influence on you? ISBELL: The one who influenced me most, I didn't take a course from. That was Saunders Mac Lane. I was reading their papers, and I got more from Mac Lane's papers than anyone else's at Chicago. I didn't see much of him during the academic year, but he engaged me in a conversation when he first saw me wandering around Eckhart Hall and then, we at least nodded when we passed, and then when Chicago rejected my application for an assistantship and I was sweating out my Texas and Oklahoma applications, Mac Lane wanted to know what I was going to do. Well, I wanted to know what I was going to do, too. I had spent three years in crummy colleges down there (one of those was Washington in St. Louis, but it didn't show me anything except an interesting German professor... he thought Coriolanus, Shakespeare's Coriolanus , was the perfect play; I guess Coriolanus is a sort of civilized Siegfried), and I didn't want to go back there, but I needed an assistantship. There was more. This was Joe McCarthy's second act... on a Shakespearean plan of five acts. Act III would be when Eisenhower held his nose and rode McCarthy's train to the White House. Well, Northern and California universities were firing leftists; Oklahoma was firing Quakers. To hold a faculty position in a state school there you had to sign an oath that you would shoot commies. Ainsley Diamond had been a noncombatant officer in the Air Force in World War II, but he wouldn't sign to shoot them. It didn't say 'shoot', it said 'bear arms'. I didn't mind that for myself, but when Oklahoma A & M fired Diamond, Nachman Aronszajn quit too, and they moved to the University of Kansas. Mac Lane urged me to stay in Chicago. I could certainly get a job to support myself, and I could study even if I couldn't pay tuition. I took an assistantship at Oklahoma A & M instead; but when I blew up after six months in Oklahoma I came back to Chicago to try to work out what to do next. KDM: I'm surprised Chicago turned you down. ISBELL: Well, I had a B average. Counting the D Chern gave me. But they don't like D's in math. And getting a D with Chern was not a smart move. I was fascinated with set theory and game theory and I goofed off in differential geometry. KDM: What did you do after Oklahoma? ISBELL: I went from Chicago to Lawrence, Kansas, and got myself considered as a candidate for an assistantship for '52-53. But it turned out that the Oklahoma A & M chairman, Wayne Johnson, had been urging A.W. Tucker at Princeton to do something about me. And somehow or other Tucker decided I should have a job at George Washington, where he was a consultant on an ONR project, and he took me over. A respectable job in Washington, for me in 1952, was like a non-teaching job in Baltimore for my mother in 1917; I was off like a shot. And Tucker decided, in due course, that I would do. KDM: What areas outside of mathematics interested you? ISBELL: You mean intellectual areas? No heavy interest outside of math. KDM: Were you involved in any extracurricular activities? ISBELL: As an undergraduate, none. Activities, yeah, but no organized extracurricular activities. In Nashville, my junior year, I played a lot of pinball. KDM: When and where did you begin graduate school? ISBELL: Well, my program of math courses for my undergraduate major didn't include the courses I took in Spring Quarter, so I guess I began graduate study at Chicago in March 1951. But I became officially a part-time graduate student only in that summer, and a regular graduate student at Oklahoma A & M in the fall. KDM: Who were the faculty members there that impressed you the most? ISBELL: That would be O.H. Hamilton, a 1938 R.L. Moore student. Under Hamilton's direction, I started to write what was supposed to be a master's thesis. I couldn't stick that year out, but before I left Stillwater I sat in a hotel for I think three days and finished the thesis, and Hamilton said I couldn't get the assistantship back but if I wanted to do something about getting enough credits he could okay it as a master's thesis. That was encouraging, I was doing something right; but it was even more encouraging seven months later when J.H. Roberts accepted it for the Duke Journal. KDM: Who, among your fellow students, impressed you the most? ISBELL: Well the best graduate students at A $amp; M had just left to go to Kansas with Aronszajn. I got to know them that year, with Al Jennings having married a Stillwater girl, and then my trying to follow the crowd to Kansas. I think you could say Al Jennings. He seemed to me clearly a pro. He was not world-class like Mac Lane and Andr\'e Weil; but like Mac Lane, he seemed to be basically satisfied with the world. It was an attitude I hadn't seen much of. Wayne Johnson had it, but Wayne didn't strike me as a pro. Hamilton was a pro, but he didn't seem satisfied, more resigned. KDM: I seem to remember that your doctoral dissertation was in game theory. Who was your thesis director? ISBELL: Tucker. I don't think he expected it when I went to Princeton, and I certainly didn't, but I was sick and tired of being a student and I saw how to finish a game theory dissertation in two months. KDM: How old were you when you received the Ph.D. degree? ISBELL: 23. KDM: You told me who supervised your first paper. What was the title, where was it published, and when? ISBELL: "Homogeneous spaces", Duke J., 1953. KDM: So that was topology? ISBELL: Oh, yes. My first four papers, I think, were in topology. The second was a joint paper with Mel Henriksen, partially repairing an error in Ed Hewitt's very influential paper on "Rings of real-valued continuous functions, I". That second paper was published in the Proceedings in 1953. We wrote it at Kansas in June '52. The third was my sequel to Henriksen-Isbell; I saw how to get the complete result that Hewitt had claimed: all residue class fields of rings C(X) are real-closed. I did that in December '53, after finishing my dissertation. Then I think the fourth was my T\^ohoku paper on "Zero-dimensional spaces", written in the summer of '54. That was a nice paper, but it appeared at the same time as a closely related and clearly better paper by Hugh Dowker, something like "Local dimension of normal spaces". KDM: So did your collaboration with Mel Henriksen go right on from 1952? ISBELL: Well, nothing like continuously. Obviously when we worked on matters of interest to both of us we exchanged preprints if not earlier previews. I remember sitting on a hillside at Fort Bliss, after I was drafted in October '54, reading Gillman-Henriksen "Concerning rings of continuous functions" while waiting for orders to attack the Blues or the Purple People-Eaters or whatever they were called. But we were together in Madison in the summer of '55 and didn't collaborate. Our second and third joint papers were written only when we were both at the Institute for Advanced Study in 1956-57 and they put us in the same office. KDM: You were drafted in October '54 and you were in Madison in the summer of '55? ISBELL: On furlough. I was invited for the four-week Summer Institute, but I only had two weeks' furlough time earned. I think maybe I had less: two weeks a year, and they were giving me an advance, letting me have two weeks after nine months. KDM: Did the army keep using you to attack Purple People-Eaters? ISBELL: Oh, no, that was just basic training. Then they sent me to Aberdeen Proving Ground and I wrote papers. When the Army permitted. We had a little celebration the day the Labs had advertised a lecture by Al Jennings on his dissertation and people came from as far as Penn, in Philadelphia, and from the University of Maryland, and they went to the room where the talk was to be and some asshole had to get up and say "I'm afraid Dr. Jennings has KP today. The lecture is canceled". KDM: I seem to remember Don Johnson telling me that you and he and Mel had published a joint paper and you and he hadn't met at the time. ISBELL: Yeah. I think that's much less unusual now than it used to be. The way that happened, Mel and Don had a forty or fifty-page joint paper which came out in Acta, and they had started another one that was just a stub when Don left Purdue and went to his first job at Penn State. I came in to Purdue for the year, and got copies of both, and I saw what to do with the stub. It didn't make forty pages, but it made a paper. [MH: The Acta paper was Don Johnson's doctoral thesis, he was the sole author, and it was about 50 pages in length. Don and I had written two papers on archimedean lattice-ordered algebras that appeared eventually in Fundamenta, and the three of us wrote the third one for Fundamenta to which John refers.] KDM: When did you become interested in Category Theory? Evidently, by the time you completed your book entitled "Uniform Spaces". ISBELL: Summer 1951. That is, after the Spring Quarter was over or practically over. I had gotten as far in Mac Lane's papers as, NOT the big category paper of 1945 with Eilenberg, but the new 1950 paper "Duality for groups". I didn't know how to do research in category theory for years after that. It was very hard, before Kan introduced adjoint functors. Homological algebra, and to some extent abelian categories, were well enough licked into shape that some questions were clearly worth working on; but papers in general category theory before Kan were heavy on the axioms and light on the theorems. KDM: You seem to have a very fruitful collaboration with both Bill Lawvere and Steve Schanuel. Would you to comment on this collaboration and on these two people? ISBELL: The first comment I have is that what I have with Bill isn't a collaboration. We have never even started a joint paper. We have talked about ideas, certainly about ideas in several of my papers, and I acknowledged Bill's help in those papers, and I think it went the other way a few times, but we don't work together. I think the best recruiting move I ever made was going on leave to Italy in 1973 to try to sell Fatima on Buffalo after Bill and Fatima turned down our 1971 offer. Talking with Bill, I often feel like a fly buzzing around a cow. (It seems to me I can liken Bill to a cow, if I'm just a fly myself.) On any easy question, I'll probably see the answer first. But his thoughts seem to move on a level where I don't function, I can barely see down there. Steve, now, I can work with Steve, and we have done five or six joint papers. The reason there haven't been fifteen is that I don't know enough. I am particularly proud of our joint paper on number theory. Steve knows at least ten times as much number theory as I, and one would think we would never start a collaboration in that area. One would be right; we didn't. I wrote a sort of semi-paper and was pleased enough with it to send a copy to Paul Bateman and to show it to Steve and Tom Cusick. Paul wrote back greatly improving my results. But the exchange of letters took more than a week. Meanwhile, Tom had shown me that I had extended a result of Dirichlet... a ridiculous little extension, but an extension. That gave a way of looking at the problem, so suddenly it was fifty times as interesting. I could do a tiny bit more with my methods, and, wham! Steve found the right method. It was in a paper of Walfisz, and it finished the job; Dirichlet had less than half solved the problem... probably not losing any sleep over it... and Isbell and Schanuel finished it. That's in Proc. AMS 60 (1976). 65--67, if any of your readers want the theorem. It's a very nice theorem, but there's a little story about it that must be told. At the next annual meeting of the AMS, the first evening I go into a restaurant with a couple of other guys, and there is Paul Erd\" os coming in with a couple of other guys. (Maybe gals, I don't recall). So I say "Paul! let $j = o(n)$, etc. etc." I get through the hypothesis, and I pause for breath. And he tells me the conclusion. So I say, "Oh, my God. Is that your theorem?" "No," he says. "It's a nice theorem. I never heard about it; but if there is a theorem, and that is the hypothesis, then this must be the conclusion." Oh, and Paul Bateman. His letter greatly improving my results was crossed by a preprint of our paper with much better results than his. KDM: You have published an enormous number of papers on many different topics. How many papers have you published by now? ISBELL: Math Reviews on-line counts 146 under my name. Six or so non-papers; corrections, a paper in Alexander Soifer's Geombinatorics, and the like. But I have about that many non-reviewed publications, at least three of them in the Monthly. And then I wrote three of John Rainwater's dozen or so papers, all of M.G. Stanley... that's four... and all of H.C. Enos, that's two. If you want an enormous number of papers, look up Erd\"os or Shelah. KDM: You have an extraordinarily broad knowledge of mathematics in general. Often, when I begin to do research on a particular topic and I don't know if something has been done, I ask you and if you know of no reference, then I know I have to go home and prove it. This exceptional knowledge is certainly a strong point. What do you regard as your other strong points? ISBELL: Ugh. Answering a question like that is a no-win shot. One can't help remembering the story of the enthusiastic disciple of Peano, telling Henri Poincar\'e that Peano's symbolic calculus or whatever it was "gives the mathematician wings". "Alas, poor Peano! Ten years with wings; and not to have flown!" Well, besides a fairly decent memory and a fair share of low cunning, I may have gained more than I lost by having an inability to close. I return a day after mailing off a submitted paper, or four months after, or twenty years after publication, to look at a solved problem again and see if there is any more in it. It costs time, but one-thirtieth of the time it pays off, and I may have gained more than I lost by it. KDM: How many doctoral students have you had? Who are they and what were their dissertation topics? What are they doing now? ISBELL: If we had been doing this three years ago, only 39 years after my doctorate, I would have asked you to omit the question; but I had a student to be proud of in 1994. The first was Richard Yoh, Ph.D., 1973. I would have to look up his dissertation title. It was something around adjoint functors and algebraic theories, and as I recall it boiled down to about a six-page paper in a decent journal. Yoh got a job at Florida, and started something I thought more interesting, on semigroups $S$ and $S$-sets. As I recall, he published one paper on that and submitted another, which he couldn't get published. Anyway he came up for tenure at Florida and was fired. He was co-manager, as I understood it, of a car dealership in Gainesville, and at least a year after he left the university he was still doing that, maybe full-time. Then there was Miroslav Klun, Ph.D., 1974. Steve Schanuel chaired Klun's committee, because I was on leave in Poland at the time; but every major topic in the dissertation was well started before Steve took over. Klun was a very touchy man. He kept getting into quarrels with other graduate students, and with faculty members. As far as I know, only with three faculty members, but these were three independent cases, at least two more than one could reasonably expect. I was the third. The last I heard, Klun was still not speaking to me; and he was teaching at Northeastern. I don't think he had tenure, and I don't think he ever published the bulk of his dissertation. It was on models of the affine part of group theory. Some of the dissertation was on inverse semigroups, and he published that in Semigroup Forum. The next student, oficially my second, was Madhav Tamhankar in 1976. Tamhankar's dissertation was a very nice paper in Algebra Universalis. Almost all of it was proving a theorem I had published in J. Algebra in 1967; but my proof didn't work. The beginning of the line was A.A. Albert's 1940 theorem that in an ordered division ring, every element algebraic over the center is in the center. Albert had a restriction, I'm pretty sure the whole thing had to be finite-dimensional over the center, but that was just because... try to remember how antediluvian 1940 was, they hardly knew Bourbaki!... Albert didn't call an extension an algebra over a field unless it was finite-dimensional. (Infinity was transcendental.) Anyway, Bernhard Neumann pointed out in Math. Reviews that Albert had proved what I said. Tamhankar's theorem is that in an ordered ring, a division subring algebraic over the center is in the center. You have to work much, much harder to prove that. Tamhankar had one-year jobs, or maybe less, for about three years in Canada and the U.S., and finally went back to India where he had a secure job teaching engineers. I think that after leaving North America he published no more research. Now, 1994: Till Plewe. Till is going to be heard from, in fact he has already started. His dissertation at least doubled knowledge on spatiality of locale products of spaces. Nice products of spaces are spatial, e.g. products of two factors one of which is locally compact (theorem of Dowker and Strauss) and countable products of complete metric spaces (my theorem). Beyond the dissertation, I have a writeup in Topology Atlas on the state of knowledge in descriptive locale theory, which says there are two and a half (good) theorems in this very new subject; one is mine and the other $1{\frac{1}{2}}$ are Till's. There are two further important papers, one joint with Anders Kock. Till is on a postdoc at Imperial College, London (with Steve Vickers). KDM: When and under what circumstances did you meet your wife? ISBELL: Yeah. I think Joan was similar material to my mother and me; but maybe she was behind the door when they gave out stubbornness. She was a sorority girl; that was compulsory. She dropped out and married when she had a proposal from a man already earning a living, and she had kids. And kids. She put her foot down at five. This was possible, because it was 1953. Joan fought her way back to college, and when we met in 1960 she had a new divorce and was one course short of a B.A. So her real life centered on campus. We were actually introduced by Don Silberger, who got a Ph.D. under Anne Morel in 1971 and is now a professor at New Paltz. KDM: How many children do you have? What are their names and ages? ISBELL: Margaret is 34, John Claiborne 33, Brecht 31. I have to tell you about Brecht's name. There we were with five kids, six kids, seven kids. There was no fear of Maggie getting lost, she being my first. Or Clay: my first son. But the new one was going to be just another kid. So like Walter Shandy's son, he needed a name of power. Unlike poor Tristram Shandy, Brecht got the name of power. KDM: What are the professions of your children and their spouses? ISBELL: Maggie has been working for the city of Bristol. Bristol, Avon, England; her husband Tony Thornborough is a journalist there. Neither of them has been very well paid, but they have two kids and Maggie is starting university in the Fall '96 at Bristol University. In education; she looks forward to working with kids instead of people who are already beaten. Now, Clay; he's the one who has a profession. Assistant professor of French at Indiana, Bloomington. His author is Germaine de Stael. Brecht graduated from MIT with a major in God knows what (political science with a side of English). He's been doing miscellaneous office work, as what used to be called a Kelly Girl. He wants to get into music publishing. KDM: Do you have any grandchildren and if so, do you get the opportunity to visit with them on any regular basis? ISBELL: As I said, Maggie has two kids; Z\'elie will be four this summer, and Alexander is two. I saw them in August '94 in Switzerland, August '95 all over the West (Chicago, Grand Canyon, Yellowstone), and this August we're taking a holiday in Wales. KDM: What are your mathematical plans for the immediate future? In particular, what areas and problems have your attention at the moment? ISBELL: Not a whole lot. I have two little papers to write up; one of them, I keep looking for a handle to turn it into a considerable paper. These are in descriptive locale theory and what could be called descriptive category theory. There is also an unsolved problem Ivan Rival has accepted for Order, and I would like to make him bring it out with a footnote saying 'The proposer has just solved this problem', but it doesn't look very likely. Let me advertise Krull dimension of metric spaces. People keep saying Krull dimension doesn't work except for weird Zariski spaces, because all Hausdorff spaces have Krull dimension zero. But that is because everybody has been defining Krull dimension wrong, until 1979. In 1979 Ram\'on Gali\'an gave a modified Krullian definition of dimension and showed that it agrees with all definitions of dimension for separable metric spaces. I gave an equivalent definition in 1985 and showed that for all metric spaces, it's >= ind and <= dim. For all known examples, it's equal to ind. (Metric examples. For compact Hausdorff spaces this is in a knotty area where Pasynkov has done something. See the Sancho de Salases' paper (Juan and Maria Teresa) in Proc. AMS 105 (1989), 491--499; they don't cite Pasynkov, but they cite my 1985 paper, which does cite Pasynkov.) But I called my definition 'by Lebesgue or Krull, out of Menger-Urysohn'. The main problem was my being brainwashed; they tell you for thirty years that this silly definition copied from algebra is topological Krull dimension, and if you aren't careful you believe it. In fact, Gali\'an had the definition right: The Krull dimension of a topological space is defined as the minimum dimension of any lattice of open sets which is a basis for the space. The dimension of a lattice is what Krull said, the maximum length of a chain of prime ideals. (A chain with two points has length 1, one link.) In 1986, Juan Navarro Gonz\'alez sent me a copy of Gali\'an's paper (incomplete, but all I have ever seen). So I asked him, "How do you prove our definitions are equivalent?" And he said, "Why, you proved it." And you know? He was right. What none of us has said in print is that of course this is how you transpose a definition of dimension from algebra to topology. Is the inductive dimension one more than the worst dimension of a boundary of a neighborhood? No, you take the best neighborhoods. Is the covering dimension the order of a bad covering? No, you take the best coverings you can get, get enough of. So for Krull dimension, you take the best lattice of open sets you can get enough of. (What the traditional definition actually uses is not prime ideals of the lattice of all open sets but prime principal ideals; that's different, but no better.)... The problem is to determine whether Krull dimension = inductive dimension for all metrizable spaces. People have been working on it without knowing it, because they've been trying to get into the gap between ind and dim and see what's there. Prabir Roy proved more than thirty years ago that there's a GAP there, and hardly any more is known now. KDM: Do you have any plans for writing any more books? ISBELL: Not bloody likely. Date: Thu, 27 Jun 1996 12:38:46 -0300 (ADT) Subject: Re: Computational Category Theory Date: Fri, 28 Jun 1996 01:00:14 +1000 From: Michael Johnson <mike(a)macadam.mpce.mq.edu.au> An update on the Computational Category Theory section of the computer algebra conference following Sussex: As well as the six speakers already announced, Bill Lawvere will be speaking on "Graphic toposes, n-categories and resulting problems of computer algebra". The details are still: Dates: July 17-20 Place: RISC (Research Institute for Symbolic Computation), Linz, Austria Cost: US$ 215 for registration. Approx US$ 55 per night for accommodation Forms: Registration and accommodation forms, as well as lots more information, are available from http://info.risc.uni-linz.ac.at:70/0/conference/IMACS96/imacs.html I'll attach the seven abstracts here: ************************************************************************* <> Ronald Brown The Axiom computer algebra system applied to computational category theory (joint work with W. Dreckmann(Bangor and Stockholm)) Abstract: The Axiom language is based on what are called: (Axiom) categories, domains, packages, objects. An `Axiom category' consists essentially of a signature. The representation of objects, implementation of operations, and expression as output form, is carried out in the domain constructors. The advantages of the Axiom language and system are discussed, and illustrated in terms of the code for directed graphs, free categories, and the category of finite sets. It is argued that this type of system allows for the development of code for the interaction of examples and abstract algebraic systems, and code which is relatively easily modified, and sufficiently general to cope with new examples. That is, the code approximates more than is usual to the standard ways of writing mathematics. <> Richard Buckland CASE for concurrent systems based on the computer algebra of n-categories Abstract: Richard Buckland will describe a concrete application of computational category theory in which a system designed to perform computations in n-categories is used as the engine for a computer assisted software engineering (CASE) tool for concurrent systems. The skeletal structure of higher dimensional categories has been found to provide a useful model of concurrent computation. However calculating with such higher dimensional algebraic objects is, as is well known to those who have done such calculations, surprisingly difficult. This difficulty arises both from the combinatorial complexity of all but the most trivial or low dimensional examples, and from the problem of verifying the reasonableness of procedures or results since we have little native intuition when dealing with high dimensions. To assist in this difficult task of high dimensional calculation a computational system has been developed to perform calculations with Schemes (schemes are the skeletal structures underlying paths in omega categories). This talk will briefly describe the algorithms for calculating with Schemes and will then show how, in a nice piece of circularity, this computer-aided mathematics has as an application the engine of a CASE tool to specify computational systems. <> Robbie Gates Generic solutions to polynomial equations in distributive categories Abstract: Following a remark of Lawvere, Blass exhibited a particularly elementary bijection between the set of binary trees, and seven tuples of binary trees. A "set of binary trees" may be abstracted to "an object $T$ of a distributive category with a given isomorphism $T^2 + 1 \cong T$". Particularly elementary in fact means "constructible from the given isomorphism using distributive category operations". In this context, it is seen that there is no such bijection for pairs, triples, ..., six-tuples of binary trees. The author has described, for a given polynomial P, the free distributive category containing an object $X$ and given isomorphism $P(X) \cong X$, and the isomorphism classes of this category. Since distributive operations correspond to the operations used to build straightline circuits/programs from simpler circuits/programs, the results of the author may be interpreted as proving the existence/non-existence of isomorphims constructible by straight line circuits/programs. The talk will briefly describe the connection between distributive categories and circuits, the bijection presented by Blass, the techniques used and results acheieved for the general case, and some speculation on future directions. <> F. William Lawvere Graphic toposes, n-categories and resulting problems of computer algebra Abstract: In 1989 I proposed pre-sheaves of a special kind as algebraic, displayable descriptions of hierarchical systems. The multi- dimensional graphs underlying n-categories (i.e.ball-and- hemisphere complexes) form a central example, but finite monoids satisfying the identity xyx = xy play the pivotal role. New algebraic results involving these toposes include an intimate connection with Coxeter groups and a characterization of the commonly-occuring case in which the lattice of sub-toposes is a total order. Computer graphics should be capable of displaying the geometric realizations of these objects as computer algebra should be capable of solving the relevant word problems for a known graphic monoid. Whether more than a few such monoids are needed to govern the modeling of the access-algebra of HYPERTEXT or similar hierarchies remains to be seen. <> Bob Rosebrugh Database tools for category theory Abstract: We have developed an interactive system in C to store and manipulate finitely-presented categories and functors among them. The tools allow, among other things, determination of equality of composed arrows, sums, products and so on. Also implemented are calculations of right and left kan extensions of finite-set valued functors. These include computation of finitely presented limits and colimits of finite sets. The talk will indicate some of the algorithms used and demonstrate program output. (This is joint work with M. Fleming and R. Gunther.) <> Makoto Takeyama Universal structure and categorical reasoning in type theory Abstract: Computer implementations of type theory have been used for studies of various formalized objects in both computer science and mathematics. Category theory can provide a unifying and clarifying framework for such studies. Universal structure is a classification scheme for categories with extra structure that is aimed at machine-checked, interactive development of categorical proofs in a type theory implementation. The traditional classification, (essentially) algebraic structure on categories, tends to force unnatural thinking and awkward computation for this purpose. To avoid the problems, we define universal structure in terms of universal properties, universality being the most important and most central concept of category theory. We express universal properties by representability of appropriate functors / modules, so we develop some basic theory of representability. Many instances of universal structure are of great interest to computer scientists, such as cartesian closed structure, fibrations with extra structure, limits, colimits, and natural numbers objects. We use our definition of universal structure to develop a ``category theory layer'' on top of the LEGO implementation of type theory, and discuss the difficulties encountered. <> Robert Walters The Todd-Coxeter procedure and the computation of left Kan extensions Abstract: Not currently available Date: Fri, 28 Jun 1996 09:22:20 -0300 (ADT) Subject: E-address needed for Vera Trnkova, Charles Univ., Prague Date: Thu, 27 Jun 96 18:53 EST From: Fred E J Linton <0004142427(a)mcimail.com> "Subject: " line says it all -- if you have an e-mail address for Vera Trnkova please send it to me at fejlinton(a)mcimail.com -- and perhaps to Mike Barr at barr(a)triples.math.mcgill.ca for incorporation into his mail-reforwarder. [Or even: post here (?).] Many thanks. -- Fred [E.J. Linton] Date: Fri, 28 Jun 1996 15:44:46 -0300 (ADT) Subject: Re: E-address needed for Vera Trnkova, Charles Univ., Prague Date: Fri, 28 Jun 1996 09:48:42 -0700 From: Vaughan Pratt <pratt(a)cs.Stanford.EDU> Since Vera was on Structdir I sent her a test message and learned that Bitnet no longer knew her machine. Turns out 20 of the 40 or so bitnet addresses on Structdir, listed below, are in that condition. I would appreciate more current addresses for any of the following, preferably not bitnet if possible. More generally, if you encounter any nonworking address on structdir please let me know. -- Vaughan Pratt almada: matjoao%ptearn.bitnet(a)frmop11.cnusc.fr (Teresa Almada) bracho: bracho(a)unamvm1.bitnet (Felipe Bracho) brunner: hbrunner(a)mun.bitnet (Herman Brunner) carter: nicola(a)mcgill1.bitnet (Nicola Carter) comer: comers(a)citadel.bitnet (Stephen Comer) diers: diers(a)frcitl81.bitnet (Yves Diers) faro: v068p76v(a)ubvmsa.bitnet (Emilio Faro) feldman: d_feldman(a)unhh.bitnet (David Feldman) ferreirim: mimafer(a)ptearn.bitnet (Isabel Ferreirim) frei: bitnet.arfr@ubcmtsg (Armin Frei) gerstenhaber: gersten(a)penndrls.bitnet (Murray Gerstenhaber) golasinski: mg001(a)pltumk11.bitnet (Marek Golasinski) jenkins: maj(a)qucis.bitnet (Mike Jenkins) lubarsky: r_lubarsky(a)faudm.bitnet (Robert Lubarsky) lubliner: coby(a)ucbcevax.bitnet (Coby Lubliner) jmckay: mckay(a)conu1.bitnet (John McKay) pollara: pollara(a)csearn.bitnet (Victor Pollara) sankappanevar: sankah(a)snynewba.bitnet (Hanamantagouda.P. Sankappanevar) saramago: matjoao%ptearn.bitnet(a)frmop11.cnusc.fr (Maria Saramago) trnkova: trnkova(a)cspguk11.bitnet (Vera Trnkova) Date: Sun, 30 Jun 1996 12:03:04 -0300 (ADT) Subject: new address Date: Sat, 29 Jun 1996 12:42:54 -0400 (EDT) From: F William Lawvere <wlawvere(a)ACSU.Buffalo.EDU> Dear Vaughan, Here is my new unix address which I will be using from now on. Thanks, Bill Lawvere Date: Sun, 30 Jun 1996 23:20:29 -0300 (ADT) Subject: Monoidal adjunctions Date: Sun, 30 Jun 1996 17:28:47 +0100 (BST) From: A.Jung(a)cs.bham.ac.uk Consider the following definition from Eilenberg and Kelly 1966: (C; x, l, r, a, I) and (C'; x', l', r', a', I') are monoidal categories. A functor F from C' to C is called monoidal if there is a natural transformation n: FA x FB --> F(Ax'B) and a morphism p: FI' --> I commuting in a suitable sense with l, l', r, r', a and a'. What we have proved is the following: LEMMA Let (F,G) be an adjunction between (symmetric) monoidal categories C and C'. If F (the left adjoint) is a (symmetric) monoidal functor such that n consists of isomorphisms and p is an isomorphism then G is also monoidal and unit and counit are monoidal natural transformations. The proof is pretty straightforward. Yet, we could not find such a lemma anywhere in the literature. Can anyone help with pointers to existing work? Achim Jung Mathias Kegelmann Eike Ritter School of Computer Science The University of Birmingham Edgbaston BIRMINGHAM, B15 2TT England

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