From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5347 Path: news.gmane.org!not-for-mail From: Paul Taylor Newsgroups: gmane.science.mathematics.categories Subject: A well kept secret Date: Thu, 10 Dec 2009 14:49:02 +0000 Message-ID: References: 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 1260494317 7073 80.91.229.12 (11 Dec 2009 01:18:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 11 Dec 2009 01:18:37 +0000 (UTC) To: categories list , Original-X-From: categories@mta.ca Fri Dec 11 02:18:30 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 1NIu9M-00083H-0f for gsmc-categories@m.gmane.org; Fri, 11 Dec 2009 02:18:28 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NItZv-0002gC-WC for categories-list@mta.ca; Thu, 10 Dec 2009 20:41:52 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5347 Archived-At: I'm not too sure what the context was, but Andre' Joyal said on 7 December, > Category theory is a powerful mathematical language. It is extremely good > for organising, unifying and suggesting new directions of research. I completely agree. > It is probably the most important mathematical developpement of > the 20th century. It is too early to tell. [Comment attributed to Zhou Enlai (Chinese Communist leader 1949-76) when asked his opinion of the French Revolution.] > But we cant say that publically. I think we should be wary of slapping ourselves on the back too much. The fact is that category theory alienated the rest of the mathematical world. Since the damage had been done in the 1970s, well before my time, I have never managed to work out how this happenned, or who was responsible. Probably it was the result of haughty claims about being the "most important mathematical development", and about being the foundations of mathematics before any serious technical work was done to justify this. Of course the ignorance and arrogance of mathematicians outside our subject had a lot to do with it too. Indeed, I believe that there is nothing wrong with pre-1980 category theory that cannot be attributed to the fact that it was done by pure mathematicians, and nor is there anything wrong with the post-1980 subject that is not the result of its having been done by computer scientists. However, discussion on that is not going to get us very far. What is more relevant and able to be fixed is the point in Andre's title, that category theory is a WELL KEPT SECRET. Secrecy, like charity, begins at home. For example, the notion of ARITHMETIC UNIVERSE was one of the most insightful developments of 1970s categorical logic. It captures exactly what is taught as "discrete mathematics" to computer science students (and is relevant to combinatorial mathematics), namely products, equalisers, stable disjoint sums, stable effective quotients of equivalence relations and FINITE powersets. It is the least structure that is capable of constructing the free internal gadget of the same kind, so the original idea was to prove Godel's incompleteness theorem categorically. Recently I was looking though the archives of the "Foundations of Mathematics" (FOM) mailing list at cs.nyu.edu/pipermail/fom/ and, amongst all of the personal abuse directed at Colin McLarty and Steve Awodey, came across an interesting argument against category theory, namely that the notion of elementary topos was merely an aping of the axioms of set theory. Arithmetic universes answer that objection extremely well. The work on arithmetic universes was done THIRTY SIX YEARS AGO, and many people since then have been nagging the author to write it up, indeed I myself have been doing so for half of that time now. I don't want anybody to read this as a personal attack -- it is simply an example of a general phenomenon, albeit an important example because of the importance of the material. Anybody in my generation or younger can cite lots of examples of "well known" "folklore" results that were supposedly discovered in the 1970s but have never been written up. The worst thing is that any younger person who is so impertinent as to write out a proof of one of these results has their paper rejected. To give another example, the theory of continuous lattices is crucial as background for my work on Abstract Stone Duality. I asked exactly the people who should have written it whether there was an introduction to continuous lattices suitable for analysts. There isn't, so I had to write my own. In this, I stated without proof that the evaluation map Sigma^X x X --> Sigma is continuous (when the topology Sigma^X is itself given the Scott topology) iff X is locally compact, and in this case Sigma^X is itself locally compact and obeys the adjunction Yx(-) -| Sigma^(-). The referee quite reasonably asked for a reference to a proof, but, so far as I can gather, no such proof exists in the literature. Two more examples: when is some Australian going to write "2-categories for the working categorist"? Where is the textbook on universal algebra based on monads? So, to answer Andre's question about why category theory is such a well kept secret -- it is because category theorists KEEP it as a secret. Each of us can help to leak this secret by doing two things: PUBLISH (= make freely available on the Web) all of the papers that you PRIVATISED by handing them over to commercial journals. WRITE textbook or encyclopedia accounts of your work for resources like the "n-cat lab", ncatlab.org/nlab/show/HomePage Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]