Subject: Re: 2-category algebras? From: dyetter@math.ksu.edu (David Yetter) Date: Tue, 1 Jun 93 9:12:21 CDT On John Baez's question: I don't know of any linearization of 2-categories, but the "category algebra" construction gives examples of what Barry Mitchell (old work! -- dare one say classical category theory?) called "algebroids". I don't know the reference, but someone else can certainly recall it. --David Yetter +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: RE: 2-category algebras? Date: 31 May 1993 15:31:27 -0400 (EDT) From: MTHFWL@ubvms.cc.buffalo.edu Subject: The group ring of a small category Dear John Baez, see my abstract in the American Math Society Notices for 1963 Meeting Number 601, paper number 37, page 280. /A longer exposition including applications to network design, cohomology etc. was written then and I can make you a copy if you are interested. Even in 30 years not all the conclusions have been drawn. My colleague S.D. Schack also independently discovered the construction and has used it extensively in deformation theory. An important variant which applies to free categories and others was discovered by Pierre Leroux and used to construct the Moebius function of a category. Greetings, F.W. Lawvere +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: RE: 2-category algebras? Date: Wed, 2 Jun 1993 10:39:14 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) > ... what Barry Mitchell (old work! -- >dare one say classical category theory?) called "algebroids". I don't know >the reference ... ?? "Rings with several objects", Advances in Mathematics 8 (72) 1-161 Steve Vickers. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: RE: 2-category algebras Date: Thu, 3 Jun 93 9:07 GMT From: MAS010@BANGOR.AC.UK 2-category algebras: A student at Bangor, Ghaffar Mosa, did a thesis in: Higher dimensional algebroids and crossed complexes (1987). The aim was to define \omega-algebroids (following Brown-Higgins definition of \omega-groupoids) and to prove these equivalent to crossed complexes (in the context of algebroids), analogously to the Brown-Higgins equivalence between \omega-groupoids and crossed complexes of groupoids. (The former are essentially cubical.) A lot of information was found, some of which was a basis for the thesis of Al-Agl (Aspects of multiple categories, 1989), which has been subsumed in the paper of Al-Agl and Steiner in the Proc LMS. In effect, the equivalence is known up to dimension 3, for the case originally mooted, although there is presumably a version in the context of the Al-Agl/Steiner paper. An equivalence between crossed complexes of algebroids and globular infinity categories has been proved by Andy Tonks at Bangor. The point of the relation with crossed complexes is that these are part of the tradition in homological algebra from Rinehart, Frohlich, Lue, in which the notion of "chains of syzygies" starts with a resolution of an algebra in a general sense ("varieties of algebras"), leading initially to a crossed module in the appropriate context, rather than a module. In order to carry out the analogous work to that done for crossed complexes of groupoids work, it is also desirable to have a tensor product of such crossed complexes of algebras, as for the groupoid case. This was part of the aim of obtaining an equivalence between crossed complexes of algebroids and \omega-algebroids. However, even this equivalence would leave many questions open, since, as said above, crossed complexes are defined for "varieties of algebras", so one really wants a tensor product in this setting. Ronnie Brown +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: paper available From: raymond@fwi.uva.nl (Raymond Hoofman ) Date: Mon, 7 Jun 1993 11:09:02 +0200 (MET DST) The following paper is available by anonymous ftp: Information Systems as Coalgebras R. Hoofman Abstract: "In this paper we show that the category CINF of continuous information systems introduced in [1] can be constructed from the category REL of sets and relations in a systematic way: we prove that CINF is the category of coalgebras of the lower powerdomain comonad on the Karoubi envelope of REL. Informally, this means that the category of continuous dcpo's is in proportion to the Karoubi envelope of REL, like the category of sets and functions is in proportion to REL." ([1] Vickers, S., Information Systems for Continuous Posets, Theoretical Computer Science, to appear.) FTP instructions: > ftp vera.fwi.uva.nl > Name: anonymous > Password: [your email address] > cd pub/illc/raymond > binary > get coalg.dvi.Z (dvi-format) > get coalg.ps.Z (ps-format) > quit > uncompress coalg.ps > uncompress coalg.dvi With kind regards, Raymond Hoofman. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: commutative diagrams Date: Fri, 4 Jun 93 13:50:33 EDT From: barr@triples.Math.McGill.CA (Michael Barr) 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: commutative diagrams yet again From: Paul Taylor Date: Mon, 7 Jun 93 13:27:39 BST You started this, Mike, not me. ``Here is my counter-proposal'' 12.35 started typing: \begin{diagram} TT & \rTo & T \\ & \rdTo _{TT\eta'} & 1 & \rdTo\\ \dTo<{T\eta'T} & 2 & TTT' & \pile{\rTo^{\mu T'}\\\rTo_{T\sigma}} & TT'\\ && \dTo<{T\eta'TT'} & 3 & \dTo<{T\eta'T'} & \rdTo_6^{id} \\ TT'T & \rTo^{TT'T\eta'} & TT'TT' & \rTo^{TT'\sigma} & TT'T' & \rTo^{T\mu'} & TT' \\ \dTo<{\sigma T} & 4 & \dTo<{\sigma TT'} & 5 & \dTo<{\sigma T'} & 7 & \dTo>{\sigma} \\ T'T & \rTo_{TT'\eta'} & T'TT' & \rTo_{T'\sigma} & T'T' & \rTo_{\mu'} & T' \end{diagram} 12:37 finished typing the original tried LaTeX well I made rather a lot of typing mistakes because I don't think very well under "race" conditions I had "TT && \rTo && T" in the first row (thinking of the cube!). I had an extra & at the beginning of the third row (dunno why). I missed an & before the last T'. A closing } was missing from \pile. I typed || instead of \\ on one row. 12.49 I got it right. Now you can't test Mike's proposal because he hasn't implemented it, but you could try typing in his test diagram (timing yourself) and then check it using a "dry run". Maybe you could do the same with catmac or any of the other competitors. Test the other way round: without running LaTeX, read the source of the diagram and draw it on paper. Do the same with Mike's proposed syntax. Can you make sense of the same example as it appears in the catmac manual? (I know that there are plenty of my users out there who will back me up on this in private, but I'd quite appreciate it if they did so publicly.) I'm afraid I didn't understand all of the notation in Mike's example, so what I have above may not be quite what he intended. Here is the ASCII version of his example for reference. % 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' Getting back to the LaTeX 3 project, which gave rise to this, I have proposed that instead of trying to build in applications like commutative diagrams as part of a monolithic program, there should be an interface standard for autonomous programs to co-operate with LaTeX if it is present. In fact this is already my and Kris Rose's policy: unlike most of the competitors, our code is written to be compatible with LaTeX, plain TeX, AMS-TeX, etc, rather than to rely on one of them. Kris Rose has recently made a proposal for a graphics language. (It is the core of xypic, but differs conceptually quite a lot from the original manual; in particular it is no longer based on the matrix structure, and I am not quite sure how he re-implements matrices on top of the new language.) The "core" proposal is in diku/users/kris/xycore27beta.psZ at ftp.diku.dk for those who are interested. I don't believe there is such a thing as a definitive graphics package, or even a definitive package for all category theory applications, and so I think it would be better for LaTeX 3 to give access (in a structured way) to many different applications packages. As far as plain old fashioned commutative diagrams are concerned, I claim that my matrix syntax (with the "chess" rule for spanning cells between objects) is the easiest to use, though improvement is always possible. Paul +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: commutative diagrams From: Nico Verwer Date: Mon, 7 Jun 1993 14:47:28 +0100 (METDST) Here is what I think of Michael's proposal. I use XY-pic myself, and I am very satisfied by it. Therefore I am heavily biased in favour of XY-pic. | % 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 Why should I specify the size of the grid? The macros shoul do that for me. (Yes, XY-pic does that). | \object(1,1){TT} % object at pos (1,1) on the grid is TT I don't like to say \object every time. This can be inferred by the macros themselves, by letting everything be an object, except when it is a sub/superscript of an arrow. Also, I like the TeX alignment style of diagram specification, with & as a column separator, and \\ as a row separator better. This makes the code more position-independent, and less wysiwyg,ab-like. | \arrow(1,1)(2,2)_{TT\eta'} % draw an arrow with a subscript TT\eta The same goes for arrows. I like to specify a relative direction, instead of giving the exact co-ordinates. Co-ordinates are like goto's in programming languages: they make writing unreadable code very easy. The XY-pic syntax defines some standard arrows like \uto (up arrow), \drdashed (down-right dashed) or \drrto (one cell down, two to the right arrow). All these arrows can be made from more primitive commands, which specify relative, or absolute co-ordinates. This is extremely flexible, and allows the user to define her own arrows withoout much effort. | \enddiagram Yeah, I really like this. :-) | For example, sometimes you want a triangle to be | isoceles and there should be a way of specifying that. The basic Special things like triangles etc. and even whole diagrams can be made into macros in XY-pic. The relative addressing means that this is completely position-independent (the origin of the diagram is local). | there should be more shapes. For example the names \ptriangle, | \Vtriangle,... that I used in my package. I think those names are It can be done easily in XY-pic. What is even better is that the arrows and the grid will be modified according to the object/arrow labels if you pass them as parameters. | Could this be implemented? It has been implemented. One of the BIG advantages of XY-pic is that it is not just a package for drawing category theoretical diagrams. In the new version (of which I am a beta-tester), diagrams are an option, which you can put on top of the XY core language. This is really a complete graphics drawing language, programmed in TeX. It is incredibly clever, and still small, and fast. There are other options for drawing flow charts, general graphs, etcetera. | One possibility | is that someone will write an extension to TeX (analogous to Michael | Ferguson's multi-lingual tex XY-pic is completely programmed in TeX, and compatible with LaTeX, AMS (La)TeX, (e)plain TeX. One of the big pro's is that it does not depend on any non-TeX features. All you need apart from TeX is MetaFont (for the beautiful arrowtips and circle-pieces), and even that is not needed if you have a standard 300dpi laser printer. At home, I use it on a slow Atari 1040ST with a Canon BJ10 inkjet printer, and it works perfectly well. In summary, I am strongly in favour of XY-pic. If you want to try it out for yourself, you can ftp it from ftp.diku.dk, or mail to the writer of this incredibly versatile, and free (!), package, Kristoffer Rose (kris@diku.dk). -- Nico Verwer | nico@cs.ruu.nl Dept. of Computer Science, University of Utrecht | phone: +31 30 533921 p.o. box 80.089, 3508 TB Utrecht, The Netherlands | fax: +31 30 513791 Date: Mon, 7 Jun 1993 19:45:09 -0300 (ADT) From: CATEGORIES@mta.ca Message-Id: <930607194509.20800540@mta.ca> Subject: Re: commutative diagrams To: rrosebrugh@macc2.mta.ca X-Vmsmail-To: MX%"rrosebrugh@macc2.mta.ca" Date: Mon, 7 Jun 1993 14:22:27 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) >From: barr@triples.Math.McGill.CA (Michael Barr) > >I am on a committee to consider what the *syntax* for commutative >diagrams ought to be in the next implementation of latex. ... Here is a proposal to replace diagrams by text in linear form. On the other hand, Jon Barwise (visiting Imperial) has just given a lecture on the formal use of diagrams instead of linear text. On the face of it, you might think it natural to combine these ideas and use a diagramatic syntax, not a textual one, for expressing diagrams - based on something like MacDraw, perhaps. But I'm not familiar with the Latex packages, so perhaps I'm thinking about the problem the wrong way. Steve Vickers. p.s. - No point in asking me about Barwise's lecture; I had to miss it. Date: Mon, 7 Jun 1993 19:38:03 -0300 (ADT) From: CATEGORIES@mta.ca Message-Id: <930607193803.2080053a@mta.ca> Subject: Re: commutative diagrams yet again To: rrosebrugh@macc2.mta.ca X-Vmsmail-To: MX%"rrosebrugh@macc2.mta.ca" Date: Mon, 7 Jun 93 15:20:31 BST From: Thorsten Altenkirch (I know that there are plenty of my users out there who will back me up on this in private, but I'd quite appreciate it if they did so publicly.) I do not use diagrams intensively, and for occasional use I found Paul's macros very useful and easy to remember. They also serve well to draw non-categorical diagrams (like Church-Rosser diagrams, Barendregt's cube , etc). I have to admit that I am a bit surprised that Michael Barr proposes something 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 First of all I don't want to calculate positions and I don't want (and can) remember the names of lots of different objects (like 'Vtriangle'). Actually I haven't used Barr's package and it may well be that these macros are better suited for a very intensive user of categorical diagrams. However, I would rather appreciate it if there is a standard for diagrams acceptable for a wider class of users. Thorsten Altenkirch Kennst du das Land, wo die Zitronen blu"hn, Laboratory for Foundations Im dunkeln Laub die Gold-Orangen glu"hn, of Computer Science Ein sanfter Wind vom blauen Himmel weht, Die Myrte still und hoch der Lorbeer steht, University of Edinburgh Kennst du es wohl? Dahin! Dahin Mo"cht ich mit dir, o mein Geliebter, ziehn. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: diagrams Date: Tue, 8 Jun 93 11:33:20 EST From: cbj@socs.uts.EDU.AU 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: commutative diagrams yet again Date: Tue, 8 Jun 1993 10:31:19 +0200 From: David Murphy For me, there are two separate issues in the debate over commuting diagrams packages; what's best for drawing simple square diagrams quickly, and what is best for achieving complex effects (non-square symmetries, lots of 2-cells, for instance). I don't like Paul's syntax for either. My objection on the first count, and the reason I like Mike's proposal, is just taste; in honesty, I think either Paul's or Mike's syntax would be reasonable for producing quick-and-easy square diagrams, even if some of us execrate the syntax. The second point is more serious; just how much of a graphics package do we want a commutative diagrams package to be? Should I be able (albeit with lots of effort) to draw pentagonal diagrams or complex interactions between 2-cells (such as in, for instance, Mike Johnson's thesis) ? Having tried to use it, I don't think Paul's package will extend cleanly in this direction. Mike's syntax might leave one with having to draw a complex diagram on graph paper first, but that is better than not being able to draw it at all, and is something that frankly ought to be done unless one is very sure of the way the results will look on the page. In summary, then, I don't believe a package that calculates its own positions for things (rather than leaving that up to the user) can achieve the highest standards for a wide class of diagrams. We must define the problem--what is the class of things we want to draw ? If it includes pentagons, heptagons,... then we may end up forcing people to do trigonometry. David Murphy +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: commutative diagrams (several posts) Date: Tue, 8 Jun 93 13:20:11 +0200 From: rosolini@athena.dima.unige.it (Pino Rosolini) It seems that diagram drawing has been well implemented in TeX, so we should think of what is still unsatisfactory. For instance, there is little choice of appropriate arrow tips and tails. Probably good ones could be added in the LaTeX fonts. Pino Rosolini PS I am one of the "satisfied users" of Paul Taylor's macros. The syntax they use is very simple and direct (I'd say, in the true spirit of TeX). +++++++++++++++++++++++++++++++++++++++++++ Date: Tue, 8 Jun 93 14:52:11 +0200 From: dybkjaer@euler.ruc.dk (Hans Dybkjaer) A number of comments: Barry Jay writes: > 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. > ... > 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. XY-pic (which I have used, and I agree much with Nico Verwer's praise) consists of core which is a general-purpose "drawing" package (entirely within TeX) in which almost everything within TeX's limited graphical capabilities may be done, and a diagram package (built on that) which makes possible specifying diagrams using the matrix analogy allowing for easy use. > To illustrate, Tex passed the music test: by adding a few macros, Music has a well-established, linear and compositional notation. Diagrams may have a harder time. > Paul Taylor proposes allowing a good interface to any package. Perhaps > by separating these issues, the need for many packages can be reduced. This is very true. The number of core macros and necessary font families should be low. David Murphy writes: > I don't like Paul's syntax for either. My objection on the first > count, and the reason I like Mike's proposal, is just taste; ... > ... > just how much of a graphics package > do we want a commutative diagrams package to be? This may be important. I don't know whether Michael Barr's initiation of the discussion was aimed at a general diagram drawing package, or just at "categorical" diagrams (then other kinds of digrams like syntax diagrams, flow charts, etc. being defined by other communities). Anyway, it is impossible to find a single (linear) notations suited for all diagram purposes, and meeting everybodys taste. I almost completely agree with Paul Taylors thoughts (though I have no strong meaning on his syntax compared to XY-pic). I would recommend: 1) The implementation of a general purpose core package based on pure TeX and compatible with Plain TeX, LaTeX and AmsTeX (at least). All packages have widespread use among mathematicians. The designers here should take a strong look at XY-pic with its possibilities of absolute, symbolic and relative adressings, uniform specifications of objects, arrows and arrow labels, easy definition of subdiagrams, and much more. 2) The addition of packages building on the core. These should at least include commutative diagrams (up to what is seen in basic category books like that of Barr and Wells, and perhaps some 2-cells and cubic drawings. I like best the notation philosophy of Paul Taylor and Kristoffer Rose, but one might include a number of common, special purpose macros a la Michael Barr). Other possibilities are syntax diagrams, flow charts, trees, and geometric drawings (note that at least here the paper-and-pencil approach is probably preferable). 3) For the core, and for each package, interfacing to the Plain, LaTeX and AmsTeX packages (possibly only a difference in the environment establishing macros; the problem is biggest for LaTeX who has tampered most with the basic TeX syntax). 4) One set of font families. Hans Dybkj{\ae}r +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Date: Tue, 8 Jun 1993 10:48 AST From: RDAWSON@HUSKY1.STMARYS.CA What would be really useful for quick work, etc, would be a sort of 'CAD' in which the 'D' stands for diagram. This would permit such things as: -constructing cones and prisms over arbitrary parts of a diagram; -adjusting highly nonplanar diagrams to avoid accidental co- lineations; -handling such things as curved arrows, etc, which could be very difficult to specify using coordinates. What has this got to do with LaTeX? Ideally, such a program would put out, not a bitmap, but a LaTeX or similar file that can be included in a paper. [In a perfect world, the diagram editor should pop up during the TeX editing session as required. Most nonDos OS's permit this, and even the DOS world is beginning to be able to do such wild & crazy things]. Now, a graphic interface may make various symbols (such as curved arrows) practical that are not easy to use by drawing a sketch on paper and typing in eyeballed coordinates. Perhaps any such diagram element that would be nice to use, even if not practical for 'hand entry', shpuld be considered for inclusion. A similar situation occurs in some ray-tracing programs, which have some primitive shapes (spheres, ellipsoids, cones...) intended for hand entry, and some (Bezier patches, Phong triangles) with many parameters that are in- tended to be entered via software tools. -Robert Dawson +++++++++++++++++++++++++++++++++++++++++++++++++++++ Date: Tue, 8 Jun 93 14:59:03 BST From: jrk@information-systems.east-anglia.ac.uk (Richard Kennaway) Barry Jay writes: >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. A sufficiently expressive drawing package would allow a diagram package to be implemented on top of it, rather than implementing both independently in TeX. (Cf. the implementation of LaTeX on top of TeX.) Personally, while I use category-diagram-like pictures, I also use pictures that are beyond the capabilities of any TeX package I know of. What I want (if I'm going to use TeX at all, but that's a separate theological issue) is a TeX-compatible drawing package that can do, say, at least everything that a program like MacDraw can do. For category diagrams, I find Paul Taylor's package sufficient. It's the only one I've used, so that isn't a comment on the relative merits of the others. -- ____ Richard Kennaway __\_ / School of Information Systems Internet: jrk@sys.uea.ac.uk \ X/ University of East Anglia uucp: ...mcsun!ukc!uea-sys!jrk \/ Norwich NR4 7TJ, U.K. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: commutative diagrams yet again From: dyetter@math.ksu.edu (David Yetter) Date: Tue, 8 Jun 93 14:19:03 CDT Since the debate has begun anew, I again want to appeal to the designers of LaTeX 3's diagram package to be mindful of the needs of *ALL* likely users who need diagrammatic algebra. One should certainly be able to handle 2-categorical diagrams as in Karponov and Voevodsky's recent work, as well as Joyal/Street "string diagrams" (and with them at no extra cost, except allowing the user to specify straight, dotted, double, or wavy lines with or without arrows, Feynman diagrams, knot and braid diagrams, the "Chinese character" diagrams arising in the theory of Vassiliev invariants, and linear logic proof nets). Algebra loosed from the constraints of living in strings of symbols was once the exclusive province of categorists, but no more, we are now joined by low-dimensional topologists, theoretical physicists, and a host of others. Let's not build a tool for the needs of the early 1980's when by 2010 half of mathematics will need what some in this debate had derided as extravigances. --David Yetter +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: XYpic examples Date: Wed, 9 Jun 93 20:28:06 +1000 From: ross@macadam.mpce.mq.edu.au (Ross Moore) 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!! :-) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Diagrams-briefly From: Paul Taylor Date: Wed, 9 Jun 1993 15:51:55 +0100 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* +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Braids and categories Date: Sat, 12 Jun 93 15:36:19 PDT From: baez@ucrmath.ucr.edu (john baez) 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: XYpic examples From: ross@mpce.mq.edu.au Date: Fri, 11 Jun 93 16:21:40 -0500 In my earlier posting, I gave an incorrect directory path to the XY-pic examples. At the ftp site: ftp.mpce.mq.edu.au the directory pub/maths/Quantum should have been pub/maths/TeX/Quantum This has been fixed so that BOTH paths now work. Sorry to anyone who may have been inconvenienced by my Oops :-) Ross Moore +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Diagrams-briefly From: dyetter@math.ksu.edu (David Yetter) Date: Mon, 14 Jun 93 9:55:16 CDT Paul Taylor has consistantly refused to reply to my comments regarding the design of a diagrams package for LaTeX 3. I can only suspect that this is because to acknowledge the use of diagrammatic algebra in such roles as Feynman diagrams, proof-nets, knot and braid theory, the representation theory of quantum groups, etc. would completely undermine his position, which seems limited to those sorts of diagrams used by categorists interested in *logical* applications of 1-category theory only. I take strong exception to his remark > (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. > which shows a peculiar notion of the main-stream. The portion of category theory which has had the most fruitful interactions with the main-stream of mathematics *as a whole* has of late been the part which uses braid diagrams. In answer to Paul's question: > In my report to the LaTeX 3 project may I say that that is the consensus > of the category theory community? *NO* not if you ask me. Not to be wholely negative, I want to point out that the suggestion of a syntax allowing one to specify a size of matrix, locations of text at matrix nodes (usually objects), starting and endings of arrows (and labelling text), and (for 2-categorists) labels (including short arrows and text) for regions, would in fact permit one to specify knot diagrams, Feynman diagrams, proof-nets, etc. if one had the options of a. specifying (in place of text) various sorts of nodes (trivalent vertex, overcrossing, undercrossing, box with text, empty circle, etc.) b. specifying various types of connections (arrow, line, wavy line, semicircular arc on either side of the line, etc.) Similarly such syntax would be perfectly adequate for pentagonal, hexagonal, etc. diagrams unless one demands regular polygons. Personally, I a quite happy with hexagons with two right angle and four of 3\pi/4. --David Yetter +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: commutative diagrams (several posts) Note from moderator: Several posts on the topic of the moment follow. I regret that Charles Wells' was delayed until today. I would like to thank Michelle Boers for helping with sending out posts during my absence from telnet facilities. The posts which follow have been slightly edited for heat. If anyone wishes the changes restored, let me know. Bob Rosebrugh ++++++++++++++++++++++++++++++++ Date: Wed, 9 Jun 93 11:49:35 -0400 From: cfw2@po.CWRU.Edu (Charles F. Wells) Subject: Syntax for Diagrams Comments concerning diagram drawing packages for TeX: 1. As some have already said, TeX needs a general drawing module in the spirit of commercial drawing programs. A package for implementing diagrams as used by categorists could be added on top of that. Mathematicians communicate with structured drawings of all sorts, not just category diagrams. *************************************************************** * A system for producing printed mathematical texts that does * * not provide for sophisticated drawing as well as * * sophisticated setting of text is _only_half_a_system_ * * because mathematics is the marriage of geometry and logic. * *************************************************************** Such a drawing module, of course, is not really part of the LaTeX-s project. 2. xypic and Paul Taylor's package use a grid metaphor with syntax based on TeX's array notation. Mike Barr has suggested another grid based notation in which you refer to the grid points by coordinates. I have not used either system but Mike's looked easier to use. I am a mathematician. I THINK the respondents who liked the array notation systems were computer scientists. I suspect there may be a connection. Most mathematicians are in math departments and have to teach calculus regularly, so are used to locating points by coordinates. Perhaps computer scientists are less used to such things and more used to learning new syntax. What is amenable syntax is a very personal thing. I think I could learn to use xypic's & notation easily; I KNOW I could use the name-the-coordinates system because it is just like what I do when I teach calculus. Mike Barr's original syntax is still good for common simple squares and triangles, and his underlying \putmorphism macro works pretty well for very complicated diagrams that are not obviously grid-based. It has two flaws: you have to put in \phantoms on overlapping nodes, which is a royal pain, and you are restricted as to the angles you can draw arrows. Both flaws are the result of flaws in the underlying system. I repeat: We need a full-fledged drawing module. I would expect a grid-based system to work better than Mike's for complicated diagrams in which imposing a grid structure does not do too much violence to the original diagram, since Mike's shapes are a pain to paste together. I've become rather proficient at using Mike's system, but it's like programming in assembly language whereas the grid based systems look like programming in a higher level language. Charles Wells Department of Mathematics Case Western Reserve University 10900 Euclid Avenue Cleveland, OH 44106-7058 216-368-2893 cfw2@po.cwru.edu +++++++++++++++++++++++++++++++++++++ Date: Mon, 14 Jun 1993 12:48:32 +0200 From: David Murphy Subject: Re: Diagrams-briefly Paul writes: > 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. ...about Paul's package... I have used it. I don't find his syntax helpful, intuitive, or sufficiently extensible. I do not feel that the best interest of the community would be served by Paul's syntax becoming the standard, and will communicate this (as a minority view) to the LaTeX 3 project. David. ++++++++++++++++++++++++++++++++++++ From: Paul Taylor Date: Mon, 14 Jun 1993 21:20:18 +0100 Subject: Re: Braids and categories John Baez takes exception to my excluding braids from asking about commutative diagrams syntax. Why? Would you take exception if I put out a general enquiry about topology and said I was specifically interested in applications to computer science rather than analysis? You know perfectly well what kind of diagrams we are talking about, and I am asking you (those of you who draw such diagrams, which, I guess, is more or less everyone on this list) how to express them in ascii. Moreover I defend my use of the words "novel" and "peculiar". I have seen thousands of "commutative" diagrams and know what the idiom is, and have developed a TeX macro package which has an input and output graphical language matching this idiom. I know conceptually what braids are, but I have not seen enough diagrams of them to know what the idiom is. Previously when we got on to this topic and I had no response besides counterexamples I made a public invitation to propose an ascii idiom but had no useful response. I have no wish to denigrate knot theory, but I would like members of the community to apply their minds to the question which has been asked ... Paul ++++++++++++++++++++++++++++++++++++++ Date: Tue, 15 Jun 1993 02:17:28 -0400 From: Jon Berrick Subject: Re: Diagrams-briefly A small side-issue relating to commutative squares concerns introduction of a standard notation for pull-backs and push-outs. In my Pitman book (Research Notes #56) I introduced the following symbols in the middle of the square. For a pull-back, use a small square with upper and left-hand edge removed but top left vertex intact. "Dually" for push-out. TEXperts will know how to encode this. These symbols have found some following among topologists (perhaps by default). No doubt your more ardent correspondents will wish to comment. Jon Berrick. ++++++++++++++++++++++++++++++++++++ Date: Tue, 15 Jun 93 10:16:19 -0400 From: jds@math.upenn.edu Subject: Re: commutative diagrams being minimally competent, I will opt for for using more keystrokes but keeping things as readable as possible i.e. (12,3) for a position is fine standard use of _{ } for subscripting arrows names for special configs being as mnemonic as possible get the wrinkles out of 2-D diagrams before worrying about 3-D don't forget the need for curved arrows more later thanks jim +++++++++++++++++++++++++++++++++++ From: Paul Taylor Date: Tue, 15 Jun 1993 15:16:22 +0100 Subject: Re: Diagrams-briefly To reply to David Yetter's "not being entirely negative", I myself suspected that the matrix notation with special nodes for over/undercrossings (such as \HonV and \VonH) would extend to braids and the like. Maybe those who consider them to be so important conceptually would spend a few minutes thinking out how to express braids in a matrix notation ... BTW I can't readily quote you or edit my own text, because I'm working on a dumb terminal 500km from home. The over and undercrossings of horizontal and vertical lines have been in my package for ages. There's also some kind of "break" feature in xypic. If you care to look in the international TeX archive, you will find macro packages intended for drawing Feynmann diagrams and trees. There are also prototypes of mine for proof trees and natural deduction proof boxes, and many other things. No one package is ever going to do all of the jobs, and nor should it: if you want wysiwyg, use a pen an paper. Paul +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: contribution to the drawing diagrams debate Date: Wed, 16 Jun 1993 17:20:07 +0200 From: F.J.de.Vries@cwi.nl _____________ 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. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: diagrams of all sorts From: dyetter@math.ksu.edu (David Yetter) Date: Thu, 17 Jun 93 9:43:06 CDT 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{ }{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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: diagrams From: Nico Verwer Date: Thu, 17 Jun 1993 11:20:20 +0100 (METDST) I would like to contribute to the diagram discussion again, addressing some points that were brought up by others. One of the things that I think we should keep in mind, is that a drawing package must be more general than just a category theoretical diagram drawing package. Preferably, it would provide a well-chosen set of drawing primitives, on top of which a specific `front-end' can be built. Thus one could have the category theoretical diagram front-end, the electronic circuit front-end, the Feynman diagram front-end, the tree drawing front-end, etcetera... A highly specialized set of macros, like Michael Barr's, is not interesting for 99% of the LaTeX users. Of course his package could be a front-end on top of the drawing-kernel. Paul Taylor could make his macros into a front-end as well, and they would both be happy. I prefer the XY-pic syntax for diagrams, so I am already happy. :-) The rest of this posting seems to be a plug for XY-pic, but this is only because it solves many of the problems that people have brought up. From: cbj@socs.uts.EDU.AU (Barry Jay) | 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. In the core of XY-pic, the user decides the position of the picture elements. The diagram-drawing option is a `user' from the XY-core point of view, and it translates the alignment-like grid into co-ordinates. The syntax of the diagram option differs radically from that of the core, so you can have it both ways. | 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. This is exactly what XY-pic provides. From: dyetter@math.ksu.edu (David Yetter) | a. specifying (in place of text) various sorts of nodes (trivalent | vertex, overcrossing, undercrossing, box with text, empty circle, etc.) | b. specifying various types of connections (arrow, line, wavy line, | semicircular arc on either side of the line, etc.) | Similarly such syntax would be perfectly adequate for pentagonal, | hexagonal, etc. diagrams unless one demands regular polygons. It is all present in XY-pic. From: cfw2@po.CWRU.Edu (Charles F. Wells) | 2. xypic and Paul Taylor's package use a grid metaphor with | syntax based on TeX's array notation. From XY-pic 2.7 onwards this is definitely _not_ the case. The grid is added on top of the XY-core. In fact, this has alwasy been the case, but the lower levels were not really accessible to the user. That has now changed, and Kris has already written some excellent documentation on this. | I have not used either system but Mike's | looked easier to use. I am a mathematician. I THINK the | respondents who liked the array notation systems were computer | scientists. I suspect there may be a connection. I am a computer scientist, and for me specifying absolute co-ordinates is like the GOTO statement in programming languages, which has been obsolete since the late sixties. A grid notation is easier to design (for simple diagrams I don't even make a sketch on paper first, I just type them in), easier to read (sometimes other people have to read and work with my LaTeX source), and easier to modify (you can easily change layout, add rows or columns, etc.). Running the risk of being a cs-chauvinist, I suspect that computer scientists are the people who make mathematics usable for the rest of the world, that is, _users_. LaTeX is used by all kinds of people, not just mathematicians. | What is amenable syntax is a very personal thing. I think I | could learn to use xypic's & notation easily; I KNOW I could use | the name-the-coordinates system because it is just like what I | do when I teach calculus. I think that once you learn to use the & and // notation, you will never want to use co-ordinates again. XY-pic can draw diagrams that are not grid-based easily, and has lines in many (that is, very many) directions. It automatically removes overlap with nodes at the end of lines. From: David Murphy | ...about Paul's package... I have used it. I don't find his syntax helpful, | intuitive, or sufficiently extensible. I do not feel that the best interest | of the community would be served by Paul's syntax becoming the standard, and | will communicate this (as a minority view) to the LaTeX 3 project. Is this whole discussion communicated to the LaTeX-3 project. Are we going to take a vote? Is there a place to send suggestions for LaTeX-3 to? How will the LaTeX-3 group decide on a drawing module? From: jds@math.upenn.edu | don't forget the need for curved arrows Once again: they exist in XY-pic already. From: F.J.de.Vries@cwi.nl | 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 There are several such systems, like TeXCAD (PC) and TeXdraw (Atari ST). TeXdraw can generate MetaFont output, epic, or a LaTeX picture environment. It would be feasible to make a drawing program generating XY-pic core commands. This would meet all the demands that you put upon it. -- Nico Verwer | nico@cs.ruu.nl Dept. of Computer Science, University of Utrecht | phone: +31 30 533921 p.o. box 80.089, 3508 TB Utrecht, The Netherlands | fax: +31 30 513791 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Introductory paper on geometric logic Date: Thu, 24 Jun 1993 12:57:10 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Questions about the meeting at MSRI Date: Tue, 29 Jun 93 11:28:43 -0400 From: cfw2@po.CWRU.Edu (Charles F. Wells) 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@msri.org. -- Charles Wells Department of Mathematics Case Western Reserve University