June 1993 - Categories - categories.org.au

machines in a category
by mahler＠socs.uts.EDU.AU 30 Jun '93

30 Jun '93

Does anyone know of any work on this after Manes&Arbib74? In particular they reference their own, to be published, Kleisli Machines. Did this eventuate? thanks Daniel +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Questions about the meeting at MSRI
by cfw2＠po.CWRU.Edu 30 Jun '93

30 Jun '93

I have chased down the answers to several questions that I have been asked about the Workshop on Universal Algebra and Category Theory at the Mathematical Sciences Research Institute in Berkeley, July 12 - 23, 1993. 1. You will be able to lecture using a blackboard or using transparencies. 2. There will be talks on Saturday morning, 17 July. 3. There will be a windup discussion session at 4 PM Friday, 23 July. 4. The invited speakers should expect to speak for 50 minutes. If you have questions, you can email them at uact(a)msri.org. -- Charles Wells Department of Mathematics Case Western Reserve University

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Introductory paper on geometric logic
by sjv＠doc.ic.ac.uk 24 Jun '93

24 Jun '93

My short paper "Geometric Logic in Computer Science" is available by anonymous ftp from Imperial College. I present an introduction to geometric logic and the mathematical structures associated with it, such as categorical logic and toposes. I also describe some of its applications in computer science including its potential as a logic for specification languages. (The connectives of geometric logic are finitary conjunction, arbitrary disjunction, equality and existential quantification.) (18 pages, due to appear in - G.L. Burn, S.J. Gay and M.D. Ryan (eds) "Theory and Formal Methods 1993", Proceedings of the first Imperial College Department of Computing workshop on Theory and Formal Methods, Springer Workshops in Computer Science, 1993.) Steve Vickers Brief instructions ------------------ ftp host: theory.doc.ic.ac.uk login name: anonymous password: (type in your email address at this point) directory: papers/Vickers file: GLiCS.dvi +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Re: diagrams of all sorts
by dyetter＠math.ksu.edu 18 Jun '93

18 Jun '93

I would favor a grid-based syntax something like that illustrated in the examples below A simple commutative square might be denoted \begin{diagram}{2,2} \atnodes{A, B, C, D} %list entries at nodes in row-dominant order %would be useful in diagrams with text or %special symbols at all or most nodes \arrow [f]{1,1}{1,2} %arrow-head at second coordinate %argument \arrow [\phi ]{1,1}{2,1} \arrow [\psi ]{1,2}{2,2} \arrow [g]{2,1}{2,2} \end{diagram} A square with a 2-cell might be denoted \begin{diagram}{2,2} \atnodes{A, B, C, D} \arrow [f]{1,1}{1,2} \arrow [\phi ]{1,1}{2,1} \arrow [\psi ]{1,2}{2,2} \arrow [g]{2,1}{2,2} \among{\Swarrow x}{1,1}{1,2}{2,2}{2,1} %variable number of vertices %to define region, coordinate %arguments are placed last to %make synatax uniform, an allow %the variable number here \end{diagram} A pushout square might be denoted \begin{diagram}{2,2} \atnodes{A, B, C, D} \arrow [f]{1,1}{1,2} \arrow [\phi ]{1,1}{2,1} \arrow [\psi ]{1,2}{2,2} \arrow [g]{2,1}{2,2} \among{\rangle }{1,1}{1,2}{2,2}{2,1} %\rangle places the right-angle %symbol as a region filler near the first %listed vertex, thus a pushout would have the %same form, except {2,2} would be the first of %the coordiante arguments \end{diagram} A knot diagram for the trefoil knot as a closed braid might be denoted \begin{diagram}{4,4} \line {1,1}{1,4} %draws a line between the nodes (if node is empty %line goes to node location, if node has %symbol, line stops a small distance away) \line {4,1}{4,4} \arcplus {1,1}{1,2} %draws a semicircular arc above (or right of) line %joining nodes \arcplus {1,3}{1,4} \arcminus {4,1}{4,2} %similarly below (or left of) \arcminus {4,3}{4,4} \among{\overcrossing }{1,2}{1,3}{2,3}{2,2} \among{\overcrossing }{2,2}{2,3}{3,3}{3,2} \among{\overcrossing }{3,2}{3,3}{4,3}{4,2} %it is more convenient to %put crossings inside regions than at nodes %for certain special fillers such as %crossings, the \among command should require %a fixed number of coordinate arguments \end{diagram} One might want a command of the form \smoothing which would smooth connections at between arcs at incoming vertices. One could indicate orientation on the knot by using \arrow in place of \line. A Feynman diagram for the annihilation of an electron-positron pair might be denoted \begin{diagram}{3,3} \arrow[e^-]{1,1}{2,2} \arrow[e^+]{3,1}{2,2} \wavyarrow[\gamma ]{2,2}{2,3} \end{diagram} One would also like commands of the forms \atnode{ <text or special symbols> }{n,m} for use when most nodes do not require text or special symbols, while the syntax of \atnodes should allow a pair of commas with a space between to indicate an empty node, \arrowarcplus... \arrowarcminus... useful for oriented knot diagrams or bigonal parts of 2-categorical diagrams, \wavyline for vitural photons (and other things) and, of course, more symbols like \Swarrow for a double lined south-west pointing arrow, etc. Other symbols for use at node or in \among statements might include standard circuit design and logical gate symbols. Three cheers for a grid based syntax, so long as it has enough special symbols and the capacity to put things in regions as well as at nodes and between nodes. --David Yetter +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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diagrams
by cbj＠socs.uts.EDU.AU 17 Jun '93

17 Jun '93

Two comments: 1) Nico Verwer suggests using a general drawing package XY-pic. Drawings and diagrams differ fundamentally on the issue of who decides the position of picture elements. In a drawing package the user decides, in a diagram the program decides. Obviously there are many situations where control is shared in a limited way, but I believe that this difference is sufficiently fundamental to warrant separate packages and syntax for each purpose. In particular the syntax for a diagram can be much terser, since less information is required for processing. Of course, one would want to be able to incorporate a picture in a diagram, or a diagram in a picture, but their natures really are different. Paul Taylor proposes allowing a good interface to any package. Perhaps by separating these issues, the need for many packages can be reduced. 2) The value of having both Tex and Latex is universally acknowledged. The former is a fundamental package, giving full power to the user, while the latter is convenient for the average user. Presumably our diagram tools should be designed in the same way. We need a diagram language based on general principles, and a convenience package built on top for the average user. To illustrate, Tex passed the music test: by adding a few macros, musical scores could be typeset in Tex without any fundamental problems, even though it was not designed for that purpose. Since musical scores are diagrams, perhaps the music test should apply to any diagram package. Barry Jay +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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contribution to the drawing diagrams debate
by F.J.de.Vries＠cwi.nl 17 Jun '93

17 Jun '93

_____________ The discussion on a drawing module for LaTeX at present is only about the syntax in which one has to "program" one drawings. Of course that involves quite a lot of personal taste and routine. But shouldn't we first somehow decide how we want to make our drawings? As I see it there are two sort of drawings I make. Routine ones of a type that I make often and new ones. Simple commutative diagrams e.g. belong to the first class, and I am a happy user of Paul Taylor's package to express them quick and concisely in LaTeX. Drawings of the second type I typically make with pencil and paper. If I decide to include such a drawing in my LaTeX text, I have a problem. How to to represent this drawing in LaTeX? I don't want to experiment with many different packages to discover that they can or cannot represent my drawing in LaTeX. So I remember my old MacIntosh and its MacDraw program. Yes, I can make the drawing, but for the problem (1) that I cannot use LaTeX for the various bits of text which are included in my drawing. Anyway suppose I overcome this somehow, then I have a second problem (2) to include such a drawing in LaTeX. Yes, I can make a postscript version and include it, but this process is slow and clumsy especially if I want to edit the drawing at some stage. Suppose I just want to rename some labels, I am forced to go through the whole procedure... What I would like is the following: the drawing in my document is made by a special editor, part of the tex system. If I want to make a picture, I would like to be able to open a window, as on the Mac, and make the picture in this window, being able to use LaTeX text fragments (which depending whether I click on it with the mouse is either in editable LaTeX form or in "print" form). And have all the wysiwyg drawing freedom that such MacDraw package provides me (moving around bits and pieces, turning them, drawing lines, arrows with definable heads and tails etc): including fancy stuff as stretching and bending lines, hiding objects behind others, shading them etc etc. I don't think I want to see the "TeX" code of the picture, This bit of code should automatically be part of my document (say as an appendix produced by the TeX system), so that I can send it to other people as one file. When I LateX the document, the picture should appear in the text, at the specified place. It this a feasible dream? Would it be useful? Does it exist already? Isn't this easier then decide upon some abstract syntax? Is such a dream heresy from the TeX point of view? Fer-Jan de Vries, CWI, Amsterdam. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Diagrams-briefly
by Paul Taylor 16 Jun '93

16 Jun '93

As "task coordinator" for commutative diagrams in the LaTeX 3 project, I would be grateful if you would turn your minds to the question of the idiom in which you think it is best to express the majority of diagrams in the main stream of category theory and other parts of algebra. The extremes are not a good test of this: (1) for diagrams which are just a square or just a triangle, it makes no difference what package you use, because you can always add \square and \triangle macros on top of it. Mike did that for LaTeX pictures, and anyone moderately competent in writing TeX macros could do it for any of the graphics packages. I don't think there's any utility in it but others may do. (2) if you are writing about the foundations of the theory of braids, by definition you are doing something which is novel, peculiar and not main-stream, and necessarily this will involve ad hoc methods of creating your graphics. The low-level ad-hoc-ery needed for this is a BURDEN to the use and development of tools for idiomatic uses. The matrix syntax has been used by several macro designers, including Kris Rose (who, as he acknowledges, took it from me) for XY-PIC, Francis Borceux, Mike Spivak (lamstex) and me. In one form or another I think this has proved to be very useful. Mike Barr refuses to say anything that might be interpreted as approval for my package, but everyone else who has actually used TeX for commutative diagrams seems to agree. In my report to the LaTeX 3 project may I say that that is the consensus of the category theory community? Paul PS The current version of my package emulates AMSTEX (not lamstex). That is, you can take your existing amstex document, add \input diagrams \diagramstyle[amstex] and it will replace the amateurish mess by some pretty diagrams. anonymous FTP theory.doc.ic.ac.uk /tex/contrib/Taylor/tex/diagram* +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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commutative diagrams
by barr＠triples.Math.McGill.CA 16 Jun '93

16 Jun '93

I am on a committee to consider what the *syntax* for commutative diagrams ought to be in the next implementation of latex. Thinking about I realized that this cannot and should not be based on just what I need, but should be based broadly on the needs of the whole community, meaning the whole categorical community, since we are clearly the heaviest consumers of this technology. Now I am going to make a preliminary proposal. I emphasize that this is purely mine, has not been approved of, or even seen by the committee and I want your reactions. I am interested either in a reaction like, ``Here is my counter-proposal'' or ``This aspect could be improved in this way''. ``I think it stinks'' is not helpful (even if true). I should say that this is not a complete proposal, just a fragment and would need lots of filling out to become a complete proposal. Consider the following diagram, which is a correction of something from TTT. % TT-------------\>T % |\ \ % | \ \ % | \ \ % | \ \ % | \ \ % | \ \ % | \ \ % | \ 1 \ % | TT\eta'\ T\eta'\ % | \ \ % | \ \ % | \ \ % | \ \ % | \lr \mu%T' \lr % | TTT'===========\>TT' % T\eta'T| 2 | T\sigma | \ % | | | \ % | | | \ % | | | \ % | T\eta'TT'| 3 T\eta'T'| 6 \\$id$\l % (6) | | | \ % | | | \ % \v TT'T\eta' \v TT'\sigma \v T\mu' \lr\l % TT'T---------\>TT'TT'-------\>TT'T'----\>TT' % | | | | % | | | | % | | | | % | | | | % \sigma%T| 4 \sigma%TT'| 5 \sigma%T'| 7 |\sigma\l % | | | | % | | | | % \v \v \v \v % T'T---------\>T'TT'---------\>T'T'-----\>T' % T'T\eta' T'\sigma \mu' % \efig What I am thinking of would have the following syntax: \diagram(4,4) % make a diagram on a 4 x 4 grid \object(1,1){TT} % object at pos (1,1) on the grid is TT \object(2,1)T % object at pos (2,1) is T \object(2,2){TTT'} % and so on ..... \arrow(1,1)(2,2)_{TT\eta'} % draw an arrow with a subscript TT\eta % from position (1,1) to (2,2). Subscript means that it would be a % subscript if it were rotated to pointing along the x axis. \arrow(1,1)(2,1)_\mu % Similarly \twoarrow(2,2)(3,2)_{T\sigma}^{\mu T} % draw a double arrow, with a % subscript T\sigma on the bottom one and a superscript \mu T on top. ... % continue in this way to build a diagram, piece by piece \enddiagram Without the comments, this looks like: \diagram(4,4) \object(1,1){TT} \object(2,1)T \object(2,2){TTT'} ..... \arrow(1,1)(2,2)_{TT\eta'} \arrow(1,1)(2,1)_\mu \twoarrow(2,2)(3,2)_{T\sigma}^{\mu T} ... \enddiagram all in all, 12 objects and 15 arrows. Of course, abbreviations \ob and \ar can obviously be used. Would you prefer to be able to say, e.g. \object11{TT} Or even \object11 TT (I don't know if it is possible to use line ends as an argument delimiter, but if it is then that could be done.) More is needed. For example, sometimes you want a triangle to be isoceles and there should be a way of specifying that. The basic idea of this syntax is that TeX should work out the arrangements and place the labels (centred on the arrows, needless to say) and grow the arrows to encompass the labels and so on. In addition I think there should be more shapes. For example the names \ptriangle, \Vtriangle,... that I used in my package. I think those names are one of the best things about my package. I never have trouble recalling which is which. There is also the question of those numbers that are placed in the diagrams. I really don't know how to handle them, but they are quite unusual. Here is one kind of problem diagram I can forsee. If there is an shape \Vtriangle it will, of course, be isoceles. But consider a diagram of the shape . -----------> . | | | | | | | | | | | | v v . ------------> . \ / \ / \ / \ / \ / \ / v v . I want to be able to say \subdiagram(1,1)(3,2){\square...} \subdiagram(1,2)(3,3){\Vtriangle...} whose semantics is, I hope, evident. Many other things come to mind. Of course, sometimes there are three arrows, sometimes they go in different directions, sometimes they should be dashed and so on. Could this be implemented? Of course it can; TeX is Turing complete. The real question is, is it feasible? I don't know and it is not really my job to find out. Well, it is in a sense, since an unfeasible syntax will just not be implemented. One possibility is that someone will write an extension to TeX (analogous to Michael Ferguson's multi-lingual tex that allows accented words to be hyphenated) that are non-standard, but are an initialization option at least in some versions and that do it quickly, while high level code is written that takes a long time to function. I await your responses, if not with bated breath, then perhaps with my arms covering my face. Michael +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Braids and categories
by baez＠ucrmath.ucr.edu 15 Jun '93

15 Jun '93

Paul Taylor writes: (2) if you are writing about the foundations of the theory of braids, by definition you are doing something which is novel, peculiar and not main-stream, I take exception to this - one has only to look at (e.g.) New Developments in the Theory of Knots, a 900-page reprint volume, to see that while this work may be novel (it took off in '85) and may be peculiar (that's a matter of taste), it is very much mainstream. One might argue that this is not category theory, but in fact the most significant recent applications of category theory to mathematical physics (my field) are connected with braids and the like. Whether LaTeX 3 should attempt to take this fact into account is of course another question; I would *hope* it would, but it might be too much of a bother. John Baez +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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XYpic examples
by ross＠macadam.mpce.mq.edu.au 12 Jun '93

12 Jun '93

With all this discussion on commutative diagrams, I think it is about time some of you saw what XY-pic can do. Take a look at some of the files on: ftp.mpce.mq.edu.au in the directory: /pub/maths/Quantum -rw-rw-r-- 1 ross ftpmaths 87743 Jun 4 17:43 Section12.ps.Z -rw-rw-r-- 1 ross ftpmaths 66125 Jun 4 17:43 Section12.psfonts.Z -rw-rw-r-- 1 ross ftpmaths 35953 Jun 9 19:42 examples.ps.Z (Actually the June 4 has now been updated to June 9) These are PostScript files, created using dvips If you want a .dvi file instead, then I will put it there upon request. The file Section12.psfonts.Z requires your printer to have access to PostScript versions of the TeX CM fonts, and AMS fonts. (This may also be the case with examples.ps.Z so if anyone has trouble printing this, just let me know and I'll provide an easier version.) Enjoy, Ross Moore. PS: these files require XY-pic 2.7, not yet released. This gives a taste of what is to come, real soon now!! :-) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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