July 2003 - Categories - categories.org.au

Godel and Bernays on CT.
by Noson Yanofsky 31 Jul '03

31 Jul '03

Recently Volume 4 of Kurt Godel=92s Collected Works came out. Volume 4 and 5 are correspondences. There is some discussion of category theory that I thought might be of interest. The relevant portions have been extracted. First is Godel=92s letter to Bernays. Then Bernays=92 response. Finally there is Solomon Feferman discussion. Letter 47. Godel to Bernays (9 January 1963) on page 221: =93... I found it interesting that you speak on P. 199 of the =93newer abstract disciplines of mathematics=94 as something lying outside of set theory. I conjecture that you are thereby alluding to the concept of category and to the self-applicability of categories. But it seems to me that all of this is contained within a set theory with a finitely iterated notion of class, where reflexivity results automatically through a =93typical ambiguity=94 of statements. Isn=92t that also your opinion? I=92ve heard, by the way, that someone has formulated the axioms of set theory with the aid of the concept of category and that this has perhaps even been published. If you know something about it, I would be very grateful to you for relevant information.=94 Letter 48. Bernays to Godel (23 February 1963) on pages 229-231: =93That in the enumeration of the domains of mathematics in which the classical methods come to be used, I named the =93newer abstract disciplines=94 in addition to analysis and set theory was conceived in the sense of a distinction between categorical and hypothetical mathematics. Abstract axiomatic topology and algebra can be pursued in such a way that one indeed uses concepts like that of natural number and real number as well as set-theoretical concepts and theorems, without undertaking a strict incorporation into set theory. As to the difficulties associate with =93categories=94, of which Mr. Mac Lane spoke in his Warsaw address, I have only a rough idea of the requirements in question. All the same, it seems to me that the difficulties rest at least in part on the fact that it is exclusively the extensional characterization of categories that is being considered. In an analogous way, for example, if we characterize the number 5 by means of the class of five-element sets, there is an impediment to forming a class with the number 5 as [an] element. To be sure, categories are also effectively given by means of axiom systems, and the multiplicity of the axiom systems that have to be considered can presumably be represented as a set, and certainly as a class. Your idea that a set theory with a finitely iterated notion of class is suitable as a framework for the theory of categories seems very plausible to me. If axiomatic set theory is being extended in any event, then the other novel infinitistic investigations should also find a place in the extended framework. Just recently I received a manuscript of the dissertation submitted ny William Hanf, in which the author expresses the opinion in an introductory chapter that for his investigations a set theory with a finitely iterated notion of class does not suffice as an axiomatic framework. =93That someone recently formulated the axioms of set theory with the help of the notion of category I learned for the first time from your letter.=94 This is Solomon Feferman=92s discussion of the correspondence on pages 59-60: =93Foundations of category theory. The foundational aspects of the subject of category theory came up for discussion in Godel=92s letter of 9 January 1963 (number 47); his point of departure was a remark in Bernays 1961a to the effect that =93the =91newer=92 abstract disciplines of mathematics=92 [are] something lying outside of set theory.=94 Godel assumed (mistakenly as it turned out) that Bernays was =93thereby alluding to the concept of category and to the self-applicability of categories=94. With reference to self-applicability, what Godel presumably had in mind are examples such as the category of all categories, for which there is no straightforward set-theoretical interpretation. He went on, interestingly, to suggest that such cases of self-reference could be handled through =93typical ambiguity=94 applied to an extension of set theory by classes of finite type. Actually, something like that had been pursued in the Grothendieck school of homological algebra (e.g., in Gabriel 1962), which assumes 9instead of higher types) the existence of arbitrarily many =93universes=94, i.e., collections of sets closed under various standard operations, such as the stages Va in the cumulative hierarchy for a strongly inaccessible. But for typical ambiguity, one would need that the properties (in the language of set theory, or higher type theory as suggested by Godel) of any universe used are the same as in any other universe. Kreisel suggested in 1965, pp.117-118, the required applications could just as well be taken care of by use of the reflection principle in set theory without the assumption of inaccessible cardinals; the idea was spelled out and verified to a considerable extent in Feferman 1969, though not all the problems were dealt with thereby. =93Also in letter 47, Godel said that he had heard =93that someone has formulated the axioms of set theory with the aid of the concept of category and that this has perhaps even been published,=94 and he asked Bernays if he knew anything about it. The first publication of an axiomatic theory of the category of sets was by F. William Lawvere in his 1964; since that appeared in the Proceedings of the National Academy of Sciences, it is possible that Godel had heard of its submission through one of the other members of the academy. In any case, Bernays=92 response in letter 48 was not really helpful in dealing further with either question concerning category theory, and the subject was not pursued further in the correspondence.=94

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Conference announcement - Ramifications of Category Theory
by alberto peruzzi 30 Jul '03

30 Jul '03

Florence, Italy November 18-22, 2003 University of Florence Ramifications of Category Theory A Workshop including a featured lecture series by F. W. Lawvere A Workshop on the Ramifications of Category Theory will take place in Florence, Italy, from November 18 to November 22, 2003. The workshop will pay attention not only to issues internal to category theory, and in particular to topos theory, but also to manifold relationships of category theory with topology, logic, philosophy, physics and theoretical computer science, as witnessed by the list of invited speakers. The aim is that of providing a recognition of present-day frontiers of category-theoretic research and a perspective on future research in the field, with emphasis on the foundations of mathematics and the applications of category theory. The workshop is also intended to honor Professor F.W. Lawvere on the 40th anniversary of his Doctoral thesis on Functorial Semantics for Algebraic Theories and to help bring into clearer focus the philosophical consequences of his work, in particular the deep unification of geometry, algebra and logic and of the concepts and tools of these areas of mathematics which have stemmed from it. Professor Lawvere will give a series of lectures, one central topic of which will be quality, in its relation to space and quantity. Intensive and extensive aspects of quality, together with an adjoint characterization of it will be examined. The general goal of his lectures will be to concentrate some essentials from research in algebra and geometry to arrive at precise philosophical formulations as a guide to the pursuit of learning and future research in mathematical sciences. For further information (in English and in Italian) Visit the web page under construction: http://ramcat.scform.unifi.it/ Alberto Peruzzi Dipartimento di Filosofia Via Bolognese 52 50139 Firenze Italia alper(a)unifi.it

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CMS Summer 2004
by rjwood＠mathstat.dal.ca 29 Jul '03

29 Jul '03

The Summer 2004 Meeting of the Canadian Mathematical Society will be held at Dalhousie University, Halifax, Canada from Sunday, June 13 to Tuesday, June 15. Readers of this list may be interested in some of the research sessions of the meeting: Topos Theory, Organizer: Myles Tierney (Rutgers) Hopf Algebras and Related Topics, Organizer: Yuri Bahturin (Memorial) Algebraic Topology, Organizers: Keith Johnson (Dalhousie) and Renzo Piccinini (Milan) RJ Wood Summer 2004 Meeting Director

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Second International Symposium on Formal Methods for Components and Objects
by cat-dist＠mta.ca 28 Jul '03

28 Jul '03

(We apologize for the reception of multiple copies) *********************** CALL FOR PARTICIPATION ********************** Second International Symposium on Formal Methods for Components and Objects (FMCO 2003) DATES 4 - 7 November 2003 PLACE Lorentz Center, Leiden University, Leiden, The Netherlands URL http://fmco.liacs.nl/fmco03.html OBJECTIVES The objective of this symposium is to bring together researchers and practioners in the areas of software engineering and formal methods to discuss the concepts of reusability and modifiability in component-based and object-oriented software systems. FORMAT The symposium is a four days event in the style of the former REX workshops, organised to provide an atmosphere that fosters collaborative work, discussions and interaction. The program consists of keynote and technical presentations, and contains an exquisite social event. Speakers' contributions will be published after the symposium in Lecture Notes in Computer Science by Springer-Verlag. KEYNOTE SPEAKERS Desmond D'Souza (Kinetium, Austin, USA) E. Allen Emerson (University of Texas at Austin, USA) Andrew D. Gordon (Microsoft Research, UK) Yuri Gurevich (Microsoft Research, USA) Tony Hoare (Microsoft Research, UK) David Parnas (University of Limerick, IE) Joseph Sifakis (Verimag, FR) TECHNICAL PRESENTATIONS Albert Benveniste (IRISA/INRIA - Rennes, FR) Frank de Boer (CWI, NL) Egon Boerger (Pisa University, IT) Werner Damm (University of Oldenburg, DE) Razvan Diaconescu (IMAR, RO) Gregor Engels (University of Paderborn, DE) Jose Luiz Fiadeiro (University of Leicester, UK) Jan Friso Groote (Eindhoven University of Technology, NL) Jean-Marc Jezequel (IRISA, Rennes, FR) Bengt Jonsson (Uppsala University, SE) Yassine Lakhnech (University of Grenoble, FR) Rob van Ommering (Philips Research Laboratories, NL) Amir Pnueli (The Weizmann Institute of Science, ISR) Willem-Paul de Roever (University of Kiel, DE) Jan Rutten (CWI, Amsterdam, NL) Philippe Schnoebelen (CNRS, Cachan, FR) Natalia Sidorova (Eindhoven University of Technology, NL) Heike Wehrheim (University of Oldenburg, DE) Jeannette Wing (Carnegie Mellon University, USA) REGISTRATION Participation is limited to about 80 people, using a first-in first-served policy. To register, please fill in the registration form at http://fmco.liacs.nl/fmco03.html. The EARLY registration fee (BEFORE September 15, 2003) is 375 euro for regular participants and 250 euro for students It includes the participation to the symposium, a copy of the proceedings, all lunches and refreshments, and a social event (with dinner). ORGANIZING COMMITTEE F.S. de Boer (CWI and Utrecht University) M.M. Bonsangue (LIACS-Leiden University) S. Graf (Verimag) W.P. de Roever (CAU) For more information about participation and registration see the FMCO site above or consult either F.S. de Boer (frb(a)cwi.nl) or M.M. Bonsangue (marcello(a)liacs.nl)

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Louis Nel in the news
by Peter Freyd 28 Jul '03

28 Jul '03

On a recent warm summer's Monday evening, I met Louis Nel, club founder and current president for a game and plenty of instruction. * * * "It's a game with a magical touch," says Nel, a retired Professor of Mathematics at Carleton University with a PhD from Cambridge. "When I first saw the game, I was enchanted. My reaction was very similar to when I fell in love. There's a real kinship between croquet players who feel this. And it is definitely a different experience than meeting another toothbrush user." When he started playing croquet, Nel "knew nothing about the game at all. For me it was just an image out of Alice in Wonderland." But when the soft-spoken South African -- who is now a Team Canada player and ranked 18th in North America -- bought a toy croquet set for his four children from Canadian Tire, he was hooked. "The children moved on and thought of it as nothing more than a good game, but I saw that there was a lot more to croquet than a toy set. So I went to nearly every sports store in Ottawa and no one knew anything about the game." Nel pursued his passion for a real croquet set and once he'd found one he drove all the way to Toronto just to look at it. He didn't even buy it. But when he did finally buy a set, it had a copy of a croquet magazine in it and that was the lead he needed that set him on the path to founding Croquet Ottawa. For a year, Nel drove to Montreal once or twice a week to play at a Westmount club, but he realized that it was no long-term solution. He needed a club nearby and he realized that they only way to get it was to create it himself. * * * Nel tells the story of a bunch of lawyers who came to play at the club for an office outing. As one young man lined up with a glint in his eye, he hit his opponent's ball to the far side of the court. "So sorry, Boss," he said with a huge grin. This captures the essence of the game. Croquet can be highly competitive and really quite nasty, but at the same time it's lots of fun. Copyright 2003 CanWest Interactive Ottawa Citizen July 26, 2003 Saturday Final Edition SECTION: The Citizen's Weekly: Style; Pg. E3 LENGTH: 1391 words HEADLINE: Serenity on the green: Croquet is all grown up and making its presence felt in Ottawa SOURCE: The Ottawa Citizen BYLINE: Hattie Klotz BODY: At the noisy corner of Bronson Avenue and Gladstone Street, where trucks thunder by and a film of city grime and street dirt covers almost everything, there's an oasis of calm. Sparkling like an emerald in the dust, the smooth, green lawns of the Central Lawn Bowling club form a square, bounded by chain-link fences, at this busy intersection. It's here that Croquet Ottawa makes it home. Croquet, you say? Yes, that childhood game that entails hitting a ball through a hoop, once played on lawns and at cottages across the country, has some serious players right here in Ottawa. While the rules of the game vary according to geography (there's a North American version, an international version, a version that is most popular in Egypt, a cottage version), three times a week, the members of Croquet Ottawa gather at the club on Gladstone or at Elmdale Lawn Bowling club where they also have playing privileges, to enjoy a game or two. Croquet is the perfect way to pass a couple of hours. On a recent warm summer's Monday evening, I met Louis Nel, club founder and current president for a game and plenty of instruction. As soon as club members measured out the correct distances on the court -- croquet is nothing if not a game of precision -- and hammered in the hoops, two games got underway side-by-side. We played Golf Croquet, a version of the game that was new to me, but proved simple and easy to grasp. There were plenty of exciting moments during our fast and fluid game. And I found that as I concentrated on each of my plays and their consequences several turns down the line, everything but the game faded into the background. I didn't notice the noise from the street. I couldn't imagine that I was in the middle of the city. It was enough to be simply playing croquet. The sound of mallet on ball and the satisfying thud when struck cleanly and correctly, absorbed my attention. I realized croquet is more than a game for children and elderly aunts. It is a game of strategy. It is a mental game, somewhat like a cross between snooker and croquet. It's a game for the soul too, as there's a serene, timeless quality about playing as the sun sets and the lights come on around this small square of green in the middle of so much concrete. There's a certain contemplative side to croquet, since it's not athletically demanding. And it's always played in beautiful surroundings on velvet-smooth lawns. "It's a game with a magical touch," says Nel, a retired Professor of Mathematics at Carleton University with a PhD from Cambridge. "When I first saw the game, I was enchanted. My reaction was very similar to when I fell in love. There's a real kinship between croquet players who feel this. And it is definitely a different experience than meeting another toothbrush user." When he started playing croquet, Nel "knew nothing about the game at all. For me it was just an image out of Alice in Wonderland." But when the soft-spoken South African -- who is now a Team Canada player and ranked 18th in North America -- bought a toy croquet set for his four children from Canadian Tire, he was hooked. "The children moved on and thought of it as nothing more than a good game, but I saw that there was a lot more to croquet than a toy set. So I went to nearly every sports store in Ottawa and no one knew anything about the game." Nel pursued his passion for a real croquet set and once he'd found one he drove all the way to Toronto just to look at it. He didn't even buy it. But when he did finally buy a set, it had a copy of a croquet magazine in it and that was the lead he needed that set him on the path to founding Croquet Ottawa. For a year, Nel drove to Montreal once or twice a week to play at a Westmount club, but he realized that it was no long-term solution. He needed a club nearby and he realized that they only way to get it was to create it himself. Croquet Ottawa was formed in 1994 with founding members Ken Shipley, a co-team Canada member who stills plays at the club -- his Team Canada sign remains above the door in the clubhouse -- and Robert Armstrong, who has now moved to Northern California, "but who gave me the nudge of encouragement and was a constant spiritual presence and enthusiasm," says Nel. "When you're starting something like this, you need some luck and for me, Ken has been a real shot in the arm. Another is Dean Chamberlain, a very gregarious man who has recruited many players." Chamberlain is vice-president of the club and Shipley began playing 10 years ago when he played at a garden party and, "had bad knees from tennis so was looking for something else to do," he says. Croquet Ottawa now counts approximately 10 regular players among its members. They make up a snapshot of Canadian demographics. There are a couple of Canadians, a South African, an Englishman, a Ukrainian and an Italian. The evening I played, our foursome consisted of me, a Brit, Stuart, a Canadian, Louis, South African and Marco, an Italian. Only in Canada could this happen. They don't play croquet in Italy. "I was looking for a summer sport," explained Marco, "as I snowboard in the winter and I just didn't know what to do. A friend suggested I try this and ..." Well, four years later, Marco plays at a competitive level, with a glint in his eye and a killer instinct. But in spite of the opportunity for really vindictive and venomous play that croquet presents -- it's all too tempting to hit another player's ball for six -- "everybody is very relaxed and at peace in the club," says Nel, "and we've never had a moment of strife." Nel tells the story of a bunch of lawyers who came to play at the club for an office outing. As one young man lined up with a glint in his eye, he hit his opponent's ball to the far side of the court. "So sorry, Boss," he said with a huge grin. This captures the essence of the game. Croquet can be highly competitive and really quite nasty, but at the same time it's lots of fun. Before the birth of Croquet Ottawa, there were other traces of the game in the city, "principally on the lawns of private homes in Rockcliffe," says Nel. But there's also a croquet court in Rockcliffe Park. Directly opposite Ashbury School, the lawn is a hidden gem. Enclosed by eight-foot tall cedar hedges, it has a newly refurbished clubhouse, flowerbeds overflowing with day lilies, picnic tables and benches under fruit trees and scattered pieces of stone sculpture. "It used to be a lawn bowling club," says Brian Murray, "that saw it's heyday in the 1950s and '60s. It was a very busy part of Rockcliffe social life. But it became primarily a croquet club in the 1980s," due to the enthusiasm of Patrick Murray, former Mayor of Rockcliffe Park. "He had the lawn specially seeded and mowed. We measured it out to international standards and brought in experts from Saratoga, New York and West Palm Beach in Florida. We played tournaments with these clubs and we used to have a diplomatic tournament. There's even a huge trophy. It's a giant croquet mallet, about 10 feet tall." But with the amalgamation of Rockcliffe with the City of Ottawa two years ago, the club died. "The minute Rockcliffe lost its autonomy, the city really didn't have the funds to put into the upkeep of the lawn," says Murray. Now, during the winter the land is turned into a skating rink for children, which means that the grass is destroyed come spring. Croquet is a game that is dear to many people's hearts. It's easy to grasp and can be played at many levels. It's a welcoming game and so are the members of Croquet Ottawa. While we're playing, a man and a small boy stop their bikes and peer through the fence. "What are you doing?" chirps the boy. "We're playing croquet," someone replies. "Ah, yes," sighs the man, "I used to play when I was a child." A couple of minutes later the pair have cycled around the fence to the club gate. They lean their bikes up against a bench and watch for a moment. Within minutes, Ken Shipley has given the pair a hoop, a mallet and a brightly coloured ball and they're practising happily at the end of the court. Croquet Ottawa is holding an Association tournament at the Elmdale Lawn Bowling Club on July 26 and 27. Information on the club can be found at http://www.magma.ca/ [tilde] acna/co.htm Hattie Klotz is an Ottawa writer. GRAPHIC: Photo: Chris Mikula, For Style Weekly; Canadian champion croquet player and president of Croquet Ottawa, Louis Nel lines up a difficult shot. 'It's a game with a magical touch,' he says.; Photo: Chris Mikula, For Style Weekly; Canadian champion croquet player and president of Croquet Ottawa, (Louis Nel) lines up a difficult shot. 'It's a game with a magical touch,' he says.

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preprint: Normed combinatorial homology and noncommutative tori
by Marco Grandis 23 Jul '03

23 Jul '03

The following preprint is available in ps, at: http://www.dima.unige.it/~grandis/Bsy2.ps ___________ Normed combinatorial homology and noncommutative tori Marco Grandis Abstract. Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of "noncommutative topology", agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity stricter by introducing *normed* cubical sets and their *normed* directed homology, formed of normed preordered abelian groups. The normed cubical sets associated with "irrational rotations" have thus the same classification up to isomorphism as the well-known irrational rotation C*-algebras. MSC: 55U10, 81R60, 55Nxx. Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups. Marco Grandis Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis(a)dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/

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CT04
by John MacDonald 22 Jul '03

22 Jul '03

Please circulate to all interested colleagues: SECOND ANNOUNCEMENT INTERNATIONAL CATEGORY THEORY CONFERENCE (CT04) July 18-24, 2004 University of British Columbia Vancouver, Canada This conference will be held on the University of British Columbia campus. It will begin with a reception at 6pm on Sunday July 18, 2004, and will end at 1pm on Saturday July 24, 2004. All those interested in category theory and its applications are welcome. The CT04 website is http://www.pims.math.ca/science/2004/CT04 Currently only the link to Visitor Information has been activated. This contains information which may be helpful when making travel plans. Links to Registration and Accommodation will be activated during September 2003. A further announcement will be made when these links are active. This conference is being organized with the help of the Pacific Institute of Mathematics(PIMS}. John MacDonald

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Re: Compatibility of functors with limits
by Tom LEINSTER 21 Jul '03

21 Jul '03

Mamuka Jibladze wrote: > It just occurred to me that there is something closely related > in lattice theory; unfortunately I cannot give a reference, but > I remember that one calls a subposet P' of a poset P > relatively (co)complete if whenever a subset of P' has an upper > bound in P, it has a least upper bound in P'. This is quite similar, but not the same. I'll take the dual concept (glbs rather than lubs), since it was respect of limits that I wrote about originally. Take an inclusion P' into P of posets, and take a diagram D in P', which might as well be just a subset of P'. Then to say that the inclusion of P' into P respects meets for D is to say that join {lower bounds of D in P'} = meet D where both join and meet are taken in P. (I'm assuming that P is complete; if not, respect of meets for D also asserts that the join and the meet exist.) In my first mail I described, vaguely, respect of limits as meaning that the limit of the image is "no bigger than it needs to be". Order theory is (unsurprisingly) the context in which this makes the most sense: the greatest lower bound of D in P obviously needs to be greater than all the lower bounds of D in P', but that understood, it's minimal. The dual of Mamuka's statement is that, with D and P' and P as above, if D has a lower bound in P then it has a greatest lower bound in P'. Here's an example where meets are respected but this condition (= relative completeness?) fails. Let 0 be the empty category. For any category C, the unique functor 0 ---> C respects limits if and only if C has an object that is both initial and terminal. So if 1 is the one-element lattice then 0 ---> 1 respects meets, and the subset 0 of 0 has a lower bound in 1 but no lower bound in 0. Tom

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chain complexes
by John Baez 21 Jul '03

21 Jul '03

Dear Categorists - Who first showed that an internal category in the category of abelian groups was a 2-term chain complex of abelian groups? What's a good reference? Who first showed that an internal strict omega-category in the category of abelian groups was a chain complex of abelian groups? What's a good reference? (Of course for "internal X in the category of Y's", I am willing to accept "internal Y in the category of X's" as a substitute.) Best, jb

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Higher-Dimensional Algebra VI: Lie 2-Algebras
by John Baez 20 Jul '03

20 Jul '03

Here's a new paper that studies categorified Lie algebras: Higher-Dimensional Algebra VI: Lie 2-algebras John C. Baez and Alissa S. Crans The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a linear natural transformation called the "Jacobiator", which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang-Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L-infinity algebras in the sense of Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This paper will soon appear on the mathematics arXiv, but the PDF version on my website looks a tiny bit better: http://math.ucr.edu/home/baez/hda6.pdf

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