From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5364 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: re: A well kept secret Date: Sat, 12 Dec 2009 23:01:20 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1260746806 8634 80.91.229.12 (13 Dec 2009 23:26:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 13 Dec 2009 23:26:46 +0000 (UTC) To: categories list Original-X-From: categories@mta.ca Mon Dec 14 00:26:38 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 1NJxpj-00065T-Oy for gsmc-categories@m.gmane.org; Mon, 14 Dec 2009 00:26:36 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NJxLE-0004oz-1K for categories-list@mta.ca; Sun, 13 Dec 2009 18:55:04 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5364 Archived-At: Michael Barr wrote: > First, I was around all through the 70s (and most > of the 60s) and I have no idea what categorists did to earn the opprobrium > described below. My experience with CT may give some insight here. When I joined the MIT faculty in 1972 it was already 8 years since I'd taken Max Kelly's category theory class, and the only bit of it that I had retained by that time was that categories weren't required to have an underlying set functor (and I couldn't in 1972 even phrase that much of my recollection in the language of CT I used just now). Now skip the next two paragraphs (which are only there to preserve chronological order) unless you want all the boring details. In between I spent a year taking mathematics honours (fourth year), in which Max taught us topology (he had planned to teach us algebraic topology but realized we weren't prepared for it) and we got many other courses from John Mack (number theory), T.G. Room (axiomatic geometry), Bruce Barnes (group theory), etc. I then spent a year doing physics honours (a second fourth year, I did maths honours first because I wanted to be a theoretical physicist and had sensed that without maths honours the physics honours year would be insufficient grounding for a theoretical physicist). But then the next year I noticed that computer science was an as-yet-untapped gold mine of important yet easily solved problems, whence my career move from physics into CS. (A pity in some respects since I've always been good at solving hard problems once they engage my interest and the problems in physics had by then become quite hard and therefore should have been right up my alley.) So I became a CS grad student at Sydney then Berkeley then Stanford, and then Don Knuth's postdoc in 1971-72, and then spent a few months at IBM Yorktown Heights in a visiting faculty position in 1972. In 1972 the only person in the whole of 545 Technology Square (a 9-story building on the "other side of the tracks" from the main body of MIT) who talked about categories was Mitch Wand. Mike Fischer was his advisor. I lived out Mike's way and commuted to work with him much of the time, a half hour ride each way, so we got to discuss many things, but I don't recall category theory ever coming up. We mostly talked about algorithms and program verification and programming language design and group theory and other technical things, along with the vegetable gardens we were growing in our back yards as a joint project that also involved Albert Meyer. We both were totally oblivious to politics, which never came up. And it never occurred to either of us that we should discuss CT. While still a grad student Mitch gave a graduate course on CT. I didn't attend any of it, being rather busy as a junior faculty and having no occasion to, but I would occasionally hear feedback from those who did. The general feeling seemed to be that this was "mathematics made difficult," a way of obfuscating the obvious. I had no reason to defend CT at the time and simply accepted these reports as putting CT in the same ballpark that Rene Thom's chaos theory was later put by some of its detractors. In 1979, finding logic problems becoming more challenging, I (re)discovered algebra by way of universal algebra. I learned UA from Rasiowa and Sikorski, which I found to my surprise I could speed-read (must have been the excellent Sydney algebra courses), and successfully applied it to the logic problem I'd previously been stuck on. In 1983 I realized that category theory was the algebra of functions. I tried very very hard to understand Chapter 1 of CWM, which seemed far more obscure than universal algebra. Speed-reading that chapter was out of the question for me. Eventually I gave up and moved on to Chapter 2 and beyond, and after that it was just as easy as universal algebra. ---------- So I would say that the opprobrium could well have originated from the impression that CT was obfuscation, which Chapter 1 of CWM did nothing to dispel. Two four-syllable words beginning with the same letter, one leading to the other. ---------- So what do I think today? Well, I would rank three related concepts as being of fundamental but not equal importance, in the following order, most important first. 1. 2-categories 2. Dense functors 3. Natural transformations The algebra of 2-categories is barely algebra, it is really the associativity intrinsic to geometry. If you cut a string, even one with colored ink marks on it, in two places you can't tell after the fact in which order the cuts were made. If you cut a painting by Picasso vertically then horizontally into four pieces, the same holds even though the painting has depreciated. These are respectively associativity and middle-interchange. I hardly recognize these as algebra, they're geometry as far as I'm concerned, but they're the algebra of 2-categories. They suck, e pur si muove. Dense functors are important because they expose what is "natural" about natural transformations as an instance of 2-cells. To see how this works see http://boole.stanford.edu/pub/yon.pdf , "The Yoneda lemma as a foundational tool for algebra." I imagine Steve Lack et al have some equivalent way of describing this viewpoint which I'm still waiting to hear about (my 1962-1965 classmate Ross Street promised he'd get back to me on this but that was a while back). Meanwhile I've received enthusiastic feedback about it from Ronnie Brown and also a response from William Boshuk ("very enjoyable pamphlet"), though that's it so far. I've felt for at least 15 years that the notion of natural transformation as traditionally defined is a complicated concept. This I believe whether one thinks of them as a category theorist or (in their manifestation as homomorphisms) as an algebraist. Either way the idea is subtle. This subtlety of the concept is why I don't rank it higher. My ranking makes it ironic that transformations that are called "natural" should end up third. But then that's just my ranking, YMMV as they say. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]