Having criticised (ticked off) certain senior categorists for giving terse and uninformative answers to naive category theory questions, perhaps I ought to give more helpful responses to TeX questions. LaTeX 3 is a general and extensive re-write of LaTeX. I don't fully understand the terms of reference, though they started as making amstex work with LaTeX. Some very amateurish CDs were in amstex, so we may take it that the present objectives are to do (ie make compatible and recommend a package to do) that much, but more professionally. There were CDs in amstex because (presumably) whoever at AMS specified it considered that they were already an important mode of expression IN MATHEMATICS AS A WHOLE. Rectangular diagrams without 2-cells are very commonly found. We all know exactly what the idiomatic usage of such diagrams is and that we would be lost without them. Moreover the fact that Mike Spivak, Steven Smith, Kris Rose and I have put a lot of time into developing the matrix notation of TeXercise 18.46, that Francis Borceux came up with the same idiom without reading the TeXbook, and that large numbers of users have found these five packages convenient are evidence that this is the most appropriate way of rendering such diagrams in machine- (and, of course, user-) readable form. 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. 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. Meanwhile others on the fringe of the expanding categorical cosmos will only just be learning to use commutative \square's. With such a major piece of programming as CD drawing, "what you see" (type) and "what you get" (see on paper) are completely different things and are to be thought of (designed) separately. For example, the diagrams in my thesis (as submitted to Cambridge in 1986) look pretty horrible, but substituting a recent version of my diagrams package (and, I confess, a little bit of global editting) yields an aesthetically far superior result for (almost) the same input. Later (say post Sept 1989) sources are adapted more easily. I have continued to support the same input language whilst developing the output. With the exception of a few details, the original language design decisions turned out to be good ones. Those who have asked me in person "how do you do this in TeX" will have found that I have tricks up my sleeve which I use in my own papers but am reluctant to explain. Other things have had loud "PROTOTYPE" warning messages in them. This is because any piece of code I give out I have a duty to maintain (in the sense I've mentioned, of continuing to parse the input and producing similar or better output), so until I'm sure that I'm happy with the input language I won't release things. Consequently I sometimes get overtaken. So the short answer to Ross's comment (at the risk of compromising my own typographical principles and contradicting Mike Barr's recent [private] comment to me, "Hell, at least we agree on TeX!" :-) ), is, you're right in sticking to MacDraw for what are for the time being ad hoc diagrams. You can use epsf.sty to import them into LaTeX documents. That is after all the modern equivalent of getting the engraver to do it. When the idiom has become clear, and you've got me sufficiently interested in the subject to want to draw the diagrams myself, then I'll write your macros. Paul ==============================================================================
From: Paul Taylor <pt@doc.ic.ac.uk> Date: Mon, 5 Oct 1992 16:08:17 +0100 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. 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 ==============================================================================
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."
(Apparently Vaughn just acquired the King James Version on CD ROM.) The Hebrews and Greeks wrote in strings, as we do, but the Egyptians and Chinese wrote in pictures. Freyd has worked out the beginnings of a diagram-based language, although his statements still appear as strings of diagrams. (Egyptian cartouches, I would argue, are truly not strings.) Atish Bagchi and I will show how logic can be based directly on graphs and diagrams (not strings of them) in our forthcoming paper, "Graph- based logic and sketches." In this case, "forthcoming" probably means in a few months. I will talk about it in Montreal this weekend. (This is a commercial.) Several times lately Vaughn has posted interesting messages on one forum or another and I have answered by criticizing minuscule parts of what he says. Illegitimi non carborundum, Vaughn. One theme in his messages (as I perceive it) is that category theory appears unnecessarily arcane to the rest of the technical world. Let us take heed. -- Charles Wells Department of Mathematics, Case Western Reserve University University Circle, Cleveland, OH 44106-7058, USA 216 368 2893 ==============================================================================
Thanks to Vaughan for putting a literary slant on this which I can't match. Please excuse me if *I* adopt a conservative policy and only provide macros which I believe I can support, with orthogonal design, in the long term. If Mike Barr or Kris Rose or anyone else wants to fill the niche markets with their non-orthogonal methods, that's fine by me. Just remember the "general advice" I gave earlier. Paul ==============================================================================
This message has nothing to do with categories, but that's the point. Paul Taylor writes:
LaTeX 3 is a general and extensive re-write of LaTeX. I don't fully understand the terms of reference, though they started as making amstex work with LaTeX. Some very amateurish CDs were in amstex, so we may take it that the present objectives are to do (ie make compatible and recommend a package to do) that much, but more professionally. There were CDs in amstex because (presumably) whoever at AMS specified it considered that they were already an important mode of expression IN MATHEMATICS AS A WHOLE.
Why the emphasis on category diagrams? Obviously, we want to be able to do them easily in LaTeX, because it's a topic we're interested in, but I would rather that LaTeX was extended with an entire language for drawing pictures, instead of just a few special cases like category diagrams, braids, inverted ampersands, and so on. (I don't consider embedded postscript to be practical for this purpose.) Has anything like this been considered by whoever is designing LaTeX 3, or by anyone else? -- Richard Kennaway SYS, University of East Anglia, Norwich NR4 7TJ, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk ==============================================================================
One theme in his messages (as I perceive it) is that category theory appears unnecessarily arcane to the rest of the technical world. Let us take heed. I understand that in response to public pressure Mattel will be changing Barbie's complaint to "Category theory is tough," with the expectation that any remaining protests will be unintelligible to the media. -Vaughan ==============================================================================
participants (5)
-
Charles Wells -
jrk＠information-systems.east-anglia.ac.uk -
Paul Taylor -
pratt＠cs.stanford.edu -
unknown＠categories.mta.ca