From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6169 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Evil in bicategories Date: Tue, 14 Sep 2010 18:12:18 -0700 Message-ID: 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: dough.gmane.org 1284594538 15569 80.91.229.12 (15 Sep 2010 23:48:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 15 Sep 2010 23:48:58 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Thu Sep 16 01:48:57 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Ow1ih-00045h-VD for gsmc-categories@m.gmane.org; Thu, 16 Sep 2010 01:48:56 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33149) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Ow1hJ-0007aV-R2; Wed, 15 Sep 2010 20:47:29 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ow1hB-0002Q2-KJ for categories-list@mlist.mta.ca; Wed, 15 Sep 2010 20:47:21 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6169 Archived-At: On 9/13/2010 1:04 PM, Steve Vickers wrote (in response to Jean Benabou's question as to whether CT is a religion): > I'm not so sure of the answer for general category theory, but Topos > Theory is undoubtedly a religion. At a monastery in Meteora, the monks > have erected a sign saying "O topos einai ieros", i.e. "The topos is > holy". And as Psalm 126 says, "He shall doubtless come again with > rejoicing, bringing his sheaves with him." For my part I'm not so sure of the answer for topos theory, but for general category theory I would reply in the same vein as I did when Paul Taylor opined to this list back in October of 1992: The strings, braids, etc., are a completely different matter. It is not belittling the work done on these topics, in Australia in particular, to say that they are of minority interest, compared to the use of the simpler forms of diagrams. It seemed to me that Paul was making a religious issue out of category theory that might have been sound English orthodoxy in those days but that came across as heresy to Australian ears. I therefore replied at the time as follows. ================================= Paul's representation of strings and braids as a flash in the category theory pan is not borne out by the literature. Strings have been an integral part of the arrow business for a long time, witness Psalms 21:12, "Therefore shalt thou make them turn their back, when thou shalt make ready thine arrows upon thy strings against the face of them." Besides their supporting role with arrows, strings are also known to every linguist and computer scientist to be an integral part of language, as Mark 7:35 attests: "And straightway his ears were opened, and the string of his tongue was loosed, and he spake plain." Commutative algebra has been synonymous with mainstream algebra for a long time. But noncommutative algebra has been more than just a cottage industry for many years, and moreover has found a warmer welcome in linguistics and CS than the commutative variety, providing us with an algebraic basis for Turing machine computations and formal grammar derivations, both of which can be conveniently laid out in the (oriented) plane, where they lend themselves to a 2-categorical formulation. But as Zadeh reminds us, we live in a fuzzy world, neither clearly commutative nor clearly orientedly planar, but somewhere in between. The laws of this in-between world are braidal, with planarity corresponding to the initial or discrete braids and commutativity to the final braids, which can pass through themselves without losing their identity altogether (commutativity without idempotence). Under these circumstances we can only keep our 2-categorical cool with braids, suggesting the following slogan: Braids are the rule, of which commutativity and noncommutativity are its two extremes. This appears to be a natural idea in both senses of "natural." It is a natural mathematical idea to suggest and pursue; and it appears to be one that can be found in nature, witness the Yang-Baxter equations arising early on in physics, whose braidal character is now clear and about which several mathematical physicists have started writing, e.g. John Baez's recent MIT lecture notes on "Braids and Quantization." Nor would I be accusing Ross of riding a bandwagon if I suggest that in five years' time he'll probably be interested in something else and drawing a completely different kind of diagram. Nor would I be calling Paul shortsighted if I suggest that in five years time many of us in both mathematics and physics, and conceivably even philosophy, will be drawing braids. (This is not to suggest that Jon Barwise's reaction to my explanation of linear logic last February would have been any different had I omitted the section on braids, which included five braid diagrams I had to do in ASCII that I am looking forward to being able to render in Taylorese.) As for Ross, I rather expect that in five years time he will be drawing whatever it is that those of us in the trenches will be drawing in ten years time, and one might hope that these too would appear in some diagram package, preferably in 1997 rather than 2002. Meanwhile others on the fringe of the expanding categorical cosmos will only just be learning to use commutative \square's. This brings me back to my first theme. Categories have been a pons asinorum for "the rest of us" for a very long time, ever since the exam in category theory given to the young lad who appeared briefly in the story of David and Jonathan [I Samuel 20:21-22]: And, behold, I will send a lad, saying, "Go, find out the arrows." If I expressly say unto the lad, "Behold, the arrows are on this side of thee, take them;" then come thou: for there is peace to thee, and no hurt; as the Lord liveth. But if I say thus unto the young man, "Behold, the arrows are beyond thee;" go thy way: for the LORD hath sent thee away. As it turned out the arrows were indeed beyond the lad, who "knew not anything, only David and Jonathan knew the matter," and the lad was sent off to the city [20:37-40], a drop-out who for all we know may have later become the Bill Gates of his day. Another biblical character who struggled mightily with the subject was Job. "For the arrows of the Almighty are within me, the poison whereof drinketh up my spirit: the terrors of God do set themselves in array against me." [Job 6:4] One imagines him tackling either metacategories or coherence on that occasion. Job was thus afflicted for a long time, during which he complained bitterly of his plight to his three friends and the Lord in three major jam-sessions. In the last of these, the Lord showed up in a Whirlwind evidently hoping to be able undo Satan's mischief and set things straight at last. After spending the better part of three chapters extolling the virtues of His nobler creatures and getting Job into the proper frame of mind, the Lord came to the whale, of which He said "The arrow cannot make him flee." [Job 41:28] That apparently did the trick: Job immediately apologized for his ignorance of the subject: "I have heard of thee by the hearing of the ear, but now mine eye seeth thee. Wherefore I abhor myself, and repent in dust and ashes." [Job 42:5-6]. The Lord then in unexpectedly firm tones told Job's friends that Job now understood the subject better even than they did and to treat him properly henceforth. And He gave Job twice what he had before. This came to 14,000 sheep, 6,000 camels, 1,000 oxen, 1,000 she-asses, 7 sons, and 3 daughters, so you can figure out what he had before, at least for the livestock. If Job's arrow anxiety lasted 25-35 years, that's around 2-3% p.a., probably a good rate for those days. But I digress. Anyway you can read it for yourself, you'll see it's exactly as I said. Coming from the electrical engineering side of the business myself, the advice to "Cast forth lightning, and scatter them: shoot out thine arrows, and destroy them." [Psalms 144:6] speaks more directly to me. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]