From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4930 Path: news.gmane.org!not-for-mail From: Paul Taylor Newsgroups: gmane.science.mathematics.categories Subject: Famous unsolved problems in ordinary category theory Date: Fri, 5 Jun 2009 09:41:32 +0100 Message-ID: Reply-To: Paul Taylor NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1244224343 13403 80.91.229.12 (5 Jun 2009 17:52:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 5 Jun 2009 17:52:23 +0000 (UTC) To: Categories list Original-X-From: categories@mta.ca Fri Jun 05 19:52:21 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MCdaX-00022t-8d for gsmc-categories@m.gmane.org; Fri, 05 Jun 2009 19:52:21 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MCcuF-0000CY-HN for categories-list@mta.ca; Fri, 05 Jun 2009 14:08:39 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:4930 Archived-At: Rafael Borowiecki, under the alias Hasse Riemann, asked, > Are there any famous unsolved problems in category theory? Ronnie Brown's posting in response to this is a classic, and deserves to be printed out and pinned up in every graduate student's office! I particularly like the military analogy with the choice between a frontal assault and making the obstacle obsolete. The following point is especially important: > I was early seduced (see my first two papers) by the idea of > looking for questions satisfying 3 criteria: > 1) no-one had previously asked it; > 2) the question was technically easy to answer; > 3) the answer was important. > **** Usually it has been 2) which failed! **** I sent (a version of) the following reply to "categories" when Rafael first asked the question, but then asked Bob to withdraw it as I thought I could write it better. I put off doing so because other topics were under discussion, but by his posting Ronnie has obliged me to send it, since otherwise I would just be a chicken. So here goes: Do I hear taunts of "do you have a Fields Medal?"? These are a bit like those of "do you have a girlfriend?". Well, no, I admit it. I don't. I have a boyfriend (Richard), and some of you have met him. If you bear with me, you will see that this is not a completely frivolous answer, even though it is a personal one. My point is that there are analogies between being a gay man and being a conceptual--constructive mathematician: - They both involve long periods of self-doubt and pretence in the face of real and perceived discrimation. This is very much still real in the mathematical case, as evidenced by that fact that categorists and consructivists are largely to be found in computer science departments, excluded from mathematics in case they might corrupt the youth. - The result of this is a significantly delayed adolescence -- I have met gay men going through adolescence in the 50s or 70s. - Finally, there is pride in being who you are, and the recognition of "Honi soit qui mal y pense" -- that it is the people who think ill of it that have the problem. In the words of a song from "La Cage aux Folles" that is known as the "sweet potato song", "I yam what I yam!". Before I came out as a categorist, I pretended to be interested in difficult puzzles, I was in the British team in the International Mathematical Olympiad in 1979, but didn't do very well. I started a magazine called QARCH, whose total output in 30 years amounts to less than one of my papers now. I was taught as an undergraduate by the Hungarian analyst and graph theorist Bela Bollobas. He set problems for first year students problems that took three weeks to solve, if at all. (Bela is a mathematician of considerable stature -- so great that it took me five years to notice that he is 10cm shorter than me -- and I remember him with great affection, in case he gets to read this.) However, I hope that Bela (along with Andrej Bauer, Imre Leader and Dorette Pronk, who help organise IMO things in Slovenia, Britain and Canada nowadays), will forgive me if I say that there is something fundamentally unsatisfying about IMO problems. Once you have the solution, that is it. They are like crosswords or jigsaws or sudoku. After that I had my delayed adolescence (with an unsuitable boyfriend). I studied continuous posets instead of algebraic ones and categories instead of posets, just to show that I could. Somebody should have told me to get a proper job as a programmer, but they didn't have the guts to say it to me. (If graduate students ask me for advice nowadays, I do tell them to get proper jobs, and not surprisingly they (mis)interpret this personally.) Long after this, the first paper on Abstract Stone Duality was published on my 40th birthday, more or less. According to G H Hardy's depressing "Mathematician's Apology", and to the rules for getting a Fields Medal, I was officially finished as a mathematician. But it is pretty clear that I have been doing my best mathematics during my fifth decade. On the other hand, all of those gratuitously difficult problems had gone into the mix. Before I return to the question. please refer to number 6 in en.wikipedia.org/wiki/Hilbert's_problems which asks for the axiomatisation of physics. Even in this most famous collection of gratuitously difficult problems, we find a conceptual question. The first of Hilbert's problems is called the "continuum hypothesis", but is about smashing the continuum into dust. Elsewhere, he said "no-one shall expell us from Cantor's Paradise", but I regard it as a dystopia. I dream of some eventual escape, returning to the Euclidean paradise. There we would actually talk about lines, circles, compact subsets or whatever, instead of families of subsets or arcane algebra (or, indeed, category theory). I am looking for a language for mathematics that would look like "set theory" (as mathematicians, not set theorists, perceive it) but would yields computable continua instead of dust. More categorically, I believe that there is some notion of category that is very similar to an elementary topos, but in which all morphisms are continuous (in particular Scott continuous with respect to an intrinsic order). I also believe that these ideas are applicable to other subjects. When I have made the appropriate tools, I hope to be able to understand algebraic geometry, which was a complete mystery to me as a student. I am in princple capable of doing this, BECAUSE I am a categorist, by following the analogy between frames and rings. One version of this problem that I still cannot solve is a question that Eugenio Moggi asked me in April 1993, although I forget the exact words. We wanted a class of monos (I said they should be the equalisers targetted at power of Sigma) that was closed under composition and application of the Sigma^2 functor (ie taking the exponential Sigma^(-) twice). Another is how to embed the category of locales in a CCC WITHOUT using illegitimate presheaves (Vickers and Townsend) or the axiom of collection (Heckmann). When I wrote the original version of this posting a couple of weeks back, I thought I could solve this one. I am still hopeful, but it turns out to be a powerful question, cf Ronnie's (2) above. Notice that I give the principal formulation of the question in vague language, not as a Diophantine equation. The more specific the question, the more likely it is to have been the WRONG one. Asking an impertinent question is the best way of getting a pertinent answer. This still involves very difficult problems and hundreds of journal pages of formal proofs. But for me the problems serve the concepts rather than the other way round. This is the essence of what it is to be a conceptual mathematician. Ronnie Brown has told you a different story of his own, but with the same message. Many other experienced categorists (including the ones in higher dimensions, which Rafael excluded from his original question, for some reason) would do likewise. What about Fields Medals? People will get them, using my work, two or three generations down the line. Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]