August 2007 - Categories - categories.org.au

Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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Re: Teaching Category Theory
by Jeff Egger 09 Sep '07

09 Sep '07

--- Bas Spitters <spitters(a)cs.ru.nl> wrote: > > > You see what I really have in mind is not so much topos theory > > > (which you might have suspected at first), but FOLDS.=20 >=20 > Could you give me a link to more information about FOLDS? Sorry for not explaining! FOLDS is an acronym for First Order Logic=20 with Dependent Sorts, with which I knew that Tom is familiar (having=20 discussed its pros and cons with him back when we were both students). > Google was not very helpful. Googling the whole phrase does produce satisfying results,=20 but to save you the effort, I can point you (all) towards http://www.math.mcgill.ca/makkai/folds/ Cheers, Jeff. Ask a question on any topic and get answers from real people. Go to= Yahoo! Answers and share what you know at http://ca.answers.yahoo.com

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Journees Jean-Yves Girard
by Pierre-Louis Curien 01 Sep '07

01 Sep '07

This is the last announcement for the **** Journees Jean-Yves Girard **** http://www-lipn.univ-paris13.fr/jyg60/index-fr.php 10-12 September, IHP, Paris The final programme is now available on the site. On-line (free but obligaotry) registration is open (on the site) until Tuesday, September 5. Best regards, Pierre-Louis Curien

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Teaching Category Theory
by Tom Leinster 31 Aug '07

31 Aug '07

Dear all, Glasgow is just now introducing a Masters-level mathematics programme, and I'm teaching the Category Theory course. I'm looking for suggestions on a particular aspect of teaching it. It's a question of "size". Most of the times I've taught category theory previously were at Cambridge, where the students are exposed to ZFC-style set theory as undergraduates. Every year there'd be a few people who'd really worry about the set-theoretic validity of category theory: "doesn't Russell's paradox forbid a category of sets?", etc. I'd tell them, essentially, not to think about it; one can make a distinction between "small" and "large" collections, and experience shows that this suffices. Not a profound answer, but there you are. At Glasgow I'm going to have the opposite problem. Undergraduates here do no set theory of any kind. So, for instance, there's no reason why they should have heard of ZFC, or that there are collections "too big to be sets". Be careful what you wish for: after years of telling Cambridge students to forget their set theory, I now have students with no set theory to forget. And the question I'm having trouble answering is this: what do I need to tell them about sets? I can't tell them nothing, as far as I can see. For instance, I want them to know that the category of groups has "all" limits; but of course, Grp doesn't really have all limits, only small limits, so they'll need to know what "small" means. Later, I'll want to teach the Adjoint Functor Theorems. A rough and ready solution would be to tell them that there is a distinction between "small" and "large" collections, otherwise known as "sets" and "proper classes". This would necessitate giving them an example of a large collection, and I guess the obvious choice is the class in Russell's Paradox. But then I'd have to tell them that this is exactly the kind of thing that they shouldn't be thinking about! It's hardly satisfactory. There's probably a better solution involving an axiomatization of the category of sets (along the lines of the Lawvere-Rosebrugh book), or at least a listing of some its properties. I have two difficulties here. One - which readers of the list may be able to help me with - is that I haven't figured out how this would work in practice: for instance, how it would feed into the statement above on the completeness of Grp. Does anyone have experience of this? The other is that I haven't got room to be too radical, as the syllabus is already set (categories, functors, transformations; adjunctions, representables, limits; monads and/or monoidal categories). In a way this is an ideal situation: a classful of minds innocent of ZFC, able to come at set theory in a completely fresh way. I'd very much appreciate suggestions on how best to use this freedom. Tom

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Undelivered Mail Returned to Sender
by MAILER-DAEMON＠neptune.ucc.ie 31 Aug '07

31 Aug '07

This is the Postfix program at host neptune.ucc.ie. I'm sorry to have to inform you that your message could not be delivered to one or more recipients. It's attached below. For further assistance, please send mail to <postmaster> If you do so, please include this problem report. You can delete your own text from the attached returned message. The Postfix program <vk1(a)neptune.ucc.ie>: Command died with status 126: " exec /usr/local/bin/procmail". Command output: sh: line 1: /usr/local/bin/procmail: No such file or directory sh: line 1: exec: /usr/local/bin/procmail: cannot execute: No such file or directory

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Maxwell Institute Colloquium on Khovanov Homology
by paul＠ma.hw.ac.uk 29 Aug '07

29 Aug '07

Maxwell Institute Colloquium on Khovanov Homology Friday, 16th November, 10AM - 5.30PM ICMS, 14 India St, Edinburgh, UK As part of the series of Maxwell Institute colloquia we will be holding a one day meeting on Khovanov homology in November. The speakers are M. Khovanov (Columbia), M. Mackaay (Algarve), I. Smith (Cambridge), C. Stroppel (Glasgow/Princeton) and B. Webster (Berkeley/Princeton). The meeting is supported by the Maxwell Institute and the Glasgow Mathematical Journal Trust. We have some very limited funds available to help cover expenses of UK participants. Please email khovanovhomology(a)googlemail.com if you wish to attend, as space is limited. More details can be found at http://www.ma.hw.ac.uk/~paul/MIC. I. Gordon, C. Stroppel and P. Turner

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PSSL86: Getting nearer
by Francois Lamarche 28 Aug '07

28 Aug '07

Less than two weeks are left before the 86th Peripatetic Seminar on Sheaves and Logic, http://www.loria.fr/~lamarche/psslHomeEN.html to be held at the Institut Elie Cartan in Nancy on September 8-9. A list of participants is now available at http://www.loria.fr/~lamarche/listPart.html It is incomplete: some people who have told me they were coming have not registered yet. As you can see we still can accomodate a small number of talks. In the PSSL tradition, people arriving not too late Friday are welcome at Le Petit Cuny, 97 Grande Rue, where some of us will hang around starting about 7pm, to have drinks and sample the hearty local (well, Alsatian) cuisine, in particular Flammekuche, our very own answer to pizza. There will also be a banquet on Saturday evening for which those who want to attend will be asked a moderate sum. Looking forward to seeing you in Nancy François Lamarche

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Categories in geometry
by zoran skoda 28 Aug '07

28 Aug '07

Dear colleagues, the webpage of the conference Categories in geometry and in mathematical physics, Split, Croatia, Sep 24(23)-28, 2007 is at http://www.irb.hr/korisnici/zskoda/catconf.html Questions should be addressed via email zskoda (AT) irb.hr Zoran Skoda

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Preprint: Cubical cospans and higher cobordisms
by Marco Grandis 27 Aug '07

27 Aug '07

The following preprint is available from my web page: M. Grandis Cubical cospans and higher cobordisms (Cospans in Algebraic Topology, III) Dip. Mat. Univ. Genova, Preprint 559 (2007). http://www.dima.unige.it/~grandis/wCub3.pdf http://www.dima.unige.it/~grandis/wCub3.ps Abstract. After two papers on weak cubical categories and collarable cospans, respectively, we put things together and construct a weak cubical category of cubical collared cospans of topological spaces. We also build a second structure, called a quasi cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has lax identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree. The previous two papers of the same series are available at: http://www.tac.mta.ca/tac/volumes/18/12/18-12.dvi (ps, pdf) http://www.dima.unige.it/~grandis/wCub2.pdf Marco Grandis Dipartimento di Matematica Università di Genova Via Dodecaneso, 35 16146 Genova Italy e-mail: grandis(a)dima.unige.it tel: +39 010 353 6805 http://www.dima.unige.it/~grandis/

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Ph.D. scholarships at the IT University of Copenhagen
by Lars Birkedal 26 Aug '07

26 Aug '07

A number of Ph.D. scholarships are available at the IT University of Copenhagen, including some in the areas of the Programming, Logic and Semantics (PLS) Group with research in programming languages, automated reasoning, logical frameworks, type theory, semantics, category theory, domain theory, distributed and mobile computing, business processes, concurrency theory, electronic voting. Please let potential students know. Deadline for application is October 22. See http://www1.itu.dk/sw66047.asp for the official announcement. Best wishes, Lars Birkedal Head of the PLS group.

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