From rrosebru@mta.ca Mon Apr 2 15:41:26 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f32Htro18580 for categories-list; Mon, 2 Apr 2001 14:55:53 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Steve Stevenson MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-ID: <15048.29190.127056.701441@merlin.cs.clemson.edu> Date: Mon, 2 Apr 2001 08:35:18 -0400 (EDT) To: categories@mta.ca Subject: categories: Probability Theory X-Mailer: VM 6.72 under 21.1 (patch 11) "Carlsbad Caverns" XEmacs Lucid Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 1 Good morning: I'm looking for papers/books on category-theoretic approaches to probability. On the AMS site, there are only two listed: one in 1980 and one in 1976. Are there more recent? best, steve From rrosebru@mta.ca Mon Apr 2 15:42:04 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f32Ht7728464 for categories-list; Mon, 2 Apr 2001 14:55:07 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Mon, 2 Apr 2001 07:16:50 -0400 (EDT) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: Categories list Subject: categories: Re: Looking for adjoints In-Reply-To: <004c01c0b98d$173d6240$218d6c18@ivideon.com> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 2 Here are a few interesting examples. One of the earliest, although not called an adjoint, was the Bohr compactification of an abelian group and the construction was similar to that of the original adjoint functor theorem. The earliest that was labelled such was Kan's Ex^\infty, which was a left adjoint to the inclusion of (what are now called) Kan simplicial sets into simplicial sets. All Galois connections are contravariant adjunctions. The contravariant power set functor is adjoint to itself on the left. Hom(A,PB) = Hom(B,PA). Valid in any topos. Also valid in toposes, for any f: A --> B, the induced inverse image function f*: Sub B --> Sub A has both a left adjoint (the familiar direct image) and a right adjoint (not familiar), considering the subobject lattices as categories. And many, many more. From rrosebru@mta.ca Mon Apr 2 21:47:03 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3305q521249 for categories-list; Mon, 2 Apr 2001 21:05:52 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 2 Apr 2001 11:42:50 -0700 (PDT) From: mjhealy@redwood.rt.cs.boeing.com (Michael Healy 425-865-3123) Message-Id: <200104021842.LAA10571@lilith.rt.cs.boeing.com> To: categories@mta.ca Subject: categories: Re: Probability Theory X-Sun-Charset: US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 3 > > I'm looking for papers/books on category-theoretic approaches to > probability. On the AMS site, there are only two listed: one in 1980 > and one in 1976. Are there more recent? > Steve and all, Here's one I've been looking at: H. T. Nguyen & N. T. Nguyen (2000). On Chu Spaces in Uncertainty Analysis, International Journal of Intelligent Systems, vol. 15, 425-440. I'm sending to categories also because I wanted to ask about this in the context of combining probabilistic analysis with domain theory, topology and logic in categories. Thanks, Mike -- =========================================================================== e Michael J. Healy A FA ----------> GA (425)865-3123 | | FAX(425)865-2964 | | Ff | | Gf c/o The Boeing Company | | PO Box 3707 MS 7L-66 \|/ \|/ Seattle, WA 98124-2207 ' ' USA FB ----------> GB -or for priority mail- e "I'm a natural man." 2760 160th Ave SE MS 7L-66 B Bellevue, WA 98008 USA michael.j.healy@boeing.com -or- mjhealy@u.washington.edu ============================================================================ From rrosebru@mta.ca Tue Apr 3 15:23:46 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f33Had607827 for categories-list; Tue, 3 Apr 2001 14:36:39 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Re: Looking for adjoints To: categories@mta.ca Date: Mon, 2 Apr 2001 22:47:04 +0100 (BST) X-Mailer: ELM [version 2.5 PL3] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: From: Tom Leinster X-Scanner: exiscan *14kCAA-0002TC-00*DMkDUwjrCXU* http://duncanthrax.net/exiscan/ Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 4 Here's an adjunction from which various basic results in category theory can be read off. (Useful, but somewhat inward-looking...) Fix a small category C, and consider the forgetful functor U: [C^op, Set] ---> [ob C, Set]. This has a left adjoint F, which can easily be written down explicitly (and whose existence is also guaranteed because it's a Kan extension). Hence U preserves limits - and this is part of what's meant by the statement that limits are computed pointwise in a presheaf category. Moreover, the adjunction is monadic, from which it follows that (a) U creates limits (which is the rest of what's meant by the "computed pointwise" slogan), and (b) every presheaf is the colimit of representables (using the fact that every algebra for a monad is a coequalizer of free algebras). Dually, U has a right adjoint, so the dual results also hold. Tom > From: Jean-Pierre Marquis > To: > > I would like to have a large pool of examples of adjoint functors in as many > different fields of mathematics as possible. I am looking for the "nicest", > in whatever sense you can think of this expression (e.g. unexpected, their > existence is equivalent to a classical theorem, etc), cases in various > fields. > > References or examples anyone? (Besides the standard ones found in Mac > Lane, etc.) > > Thank you, > Jean-Pierre Marquis > > > From rrosebru@mta.ca Tue Apr 3 15:38:02 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f33I0Yg22087 for categories-list; Tue, 3 Apr 2001 15:00:34 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <00ab01c0bc18$29692ed0$aec488c1@math.ist.utl.pt> From: "Amilcar Sernadas" To: References: <15048.29190.127056.701441@merlin.cs.clemson.edu> Subject: categories: Re: Probability Theory Date: Tue, 3 Apr 2001 09:29:06 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2314.1300 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2314.1300 X-MIME-Autoconverted: from 8bit to quoted-printable by bebop.math.ist.utl.pt id f339VPr00667 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id f338Vgb20028 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 5 ----- Original Message ----- From: Steve Stevenson To: Sent: Monday, April 02, 2001 1:35 PM Subject: categories: Probability Theory > I'm looking for papers/books on category-theoretic approaches to > probability. On the AMS site, there are only two listed: one in 1980 > and one in 1976. Are there more recent? These papers and PhD thesis should be interesting (by decreasing order of abstraction/generality): C. Hermida and P. Mateus. Internal paracategories. Preprint, Section of Computer Science, Department of Mathematics, Instituto Superior Técnico, 1049-001 Lisboa, Portugal, 2000. Submitted for publication. Get a preprint: http://www.cs.math.ist.utl.pt/ftp/pub/MateusP/00-CM-paracat.ps L. Schröder and P. Mateus. Universal aspects of probabilistic automata. Preprint, Section of Computer Science, Department of Mathematics, Instituto Superior Técnico, 1049-001 Lisboa, Portugal, 1999. Submitted for publication. Get a preprint: http://www.cs.math.ist.utl.pt/ftp/pub/MateusP/99-SM-precat2.ps P. Mateus. Interconnection of Probabilistic Systems. PhD thesis, IST, Universidade Técnica de Lisboa, 2000. Supervised by A. Sernadas and C. Sernadas. Get a preprint: http://www.cs.math.ist.utl.pt/ftp/pub/MateusP/00-M-PhDthesis.ps P. Mateus, A. Sernadas, and C. Sernadas. Precategories for combining probabilistic automata. Electronic Notes in Theoretical Computer Science, 29, 1999. Early version presented at FIREworks Meeting, Magdeburg, May 15-16, 1998. Presented at CTCS'99, Edinburgh, September 10-12, 1999. http://www.elsevier.nl/cas/tree/store/tcs/free/entcs/store/tcs29/tcs29016.ps P. Mateus, M. Cabral Morais, C. Nunes, A. Pacheco, A. Sernadas, and C. Sernadas. Randomly timed automata. Preprint, Section of Computer Science, Department of Mathematics, Instituto Superior Técnico, 1049-001 Lisboa, Portugal, March 2000. Submitted for publication. Get a preprint: http://www.cs.math.ist.utl.pt/ftp/pub/SernadasA/00-MMNPSS-rta.ps P. Mateus, A. Sernadas, and C. Sernadas. Realization of probabilistic automata: Categorical approach. In Didier Bert and Christine Choppy, editors, Recent Trends in Algebraic Development Techniques - Selected Papers, volume 1827 of Lecture Notes in Computer Science, pages 237--251. Springer-Verlag, 2000. Get a preprint: http://www.cs.math.ist.utl.pt/ftp/pub/SernadasA/99-MSS-probreal.ps ++++++++++++++++++++++++++++++++++++++++++++++++++ Amilcar Sernadas Departamento de Matematica Instituto Superior Tecnico Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL tel: 351-21-8417150 fax: 351-21-8417598 e-mail: acs@math.ist.utl.pt www: http://www.cs.math.ist.utl.pt/cs/acs.html ++++++++++++++++++++++++++++++++++++++++++++++++++ From rrosebru@mta.ca Wed Apr 4 21:42:07 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f34NxH811800 for categories-list; Wed, 4 Apr 2001 20:59:17 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Peter McBurney Reply-To: p.j.mcburney@csc.liv.ac.uk To: categories@mta.ca Subject: categories: job: @ Liverpool: Agent Communication Languages & Protocols Date: Wed, 4 Apr 2001 16:08:46 +0100 X-Mailer: KMail [version 1.0.28] Content-Type: text/plain MIME-Version: 1.0 Message-Id: <0104041610340E.12177@peter.staff.csc.liv.ac.uk> Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 6 Apologies for multiple postings. The UNIVERSITY of LIVERPOOL, UK - DEPARTMENT of COMPUTER SCIENCE RESEARCH POST in AGENT COMMUNICATION LANGUAGES & PROTOCOLS A researcher is required to work on a three year research project in the use of temporal and modal logics for the specification and verification of agent communication languages and protocols. The project will be undertaken within the Agent ART group in Department of Computer Science at the University of Liverpool (UK), under the supervision of Prof Mike Wooldridge and Dr Simon Parsons. The project will provide an opportunity to carry out research on a topic of much current interest, in a highly active and rapidly expanding group. Informal enquiries may be sent to: Prof M Wooldridge (mailto:M.J.Wooldridge@csc.liv.ac.uk). Further particulars and details of the application procedure may be requested from: Director of Personnel University of Liverpool Liverpool L69 3BX, United Kingdom tel: (+44 151) 794 2210 [24 hour answerphone] mailto: jobs@liv.ac.uk WWW: http://www.liv.ac.uk/ ** PLEASE QUOTE REFERENCE B/451 ** CLOSING DATE FOR APPLICATIONS 27 APRIL 2001 -- Mike Wooldridge http://www.csc.liv.ac.uk/~mjw/ Department of Computer Science tel (+44 151) 794 3667 University of Liverpool fax (+44 151) 794 3715 Liverpool L69 7ZF, United Kingdom mailto:M.J.Wooldridge@csc.liv.ac.uk From rrosebru@mta.ca Fri Apr 6 12:53:49 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f36F48S00218 for categories-list; Fri, 6 Apr 2001 12:04:08 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: david carlton MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-ID: <15052.54956.183609.916620@jackfruit.Stanford.EDU> Date: Thu, 5 Apr 2001 13:33:48 -0700 (PDT) To: categories@mta.ca Subject: categories: strict n-categories, equivalences X-Mailer: VM 6.72 under 21.1 "20 Minutes to Nikko" XEmacs Lucid (patch 2) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 7 I'm trying to figure out the relation between the following notions of "equivalence": We define a strict n-category to be a category enriched in strict (n-1)-categories. Then we can make a recursive definition of "equivalence" as follows: we say that an arrow f: x -> y is an equivalence if there exists an arrow g: y -> x such that there exist arrows alpha: fg -> 1_x and beta: gf -> 1_y that are also equivalences. Also, we might want to say that a (strict) functor F: C -> D of strict n-categories is an "equivalence" if: 1) For all objects d in D, there exists an object c of C and an equivalence F(c) -> d; 2) For all objects c, c' in C, the map Hom_C(c,c') -> Hom_D(F(c),F(c')) (which is a functor of (n-1)-categories) is an equivalence. (To complete these definitions, I should also say that an arrow in a 1-category is an equivalence iff it's an isomorphism and that a functor of 0-categories is an equivalence iff it's a bijection.) Then, my main question is: Q: Is it the case that an arrow f: x -> y is an equivalence if and only if, for all z, the Yoneda functor Hom(z,x) -> Hom(z,y) induced by f is an equivalence? I think and hope that the answer is true, but I'm certainly not 100% confident; it seems to me that its proof would involve a sufficiently messy (and picky) simultaneous induction that I'd be quite happy if somebody else had already worked this out. If it's not true, are there any other theorems along this line with improved definitions of "equivalence" (either on the arrow side or the functor side)? One warning: my terminology might be badly chosen, especially for functors, because I don't think that it's the case that, if one constructs the strict (n+1)-category of all n-categories, that the functors that I'm calling equivalences are all equivilances as arrows in that category. In general, are there any good references for strict n-categories? (E.g. one giving the details of the construction of nCat.) I'm not as conversant with the literature of category theory as I should be, and I'd like to avoid reinventing the wheel more than necessary. thanks, david carlton carlton@math.stanford.edu From rrosebru@mta.ca Fri Apr 6 15:43:45 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f36I5DV26188 for categories-list; Fri, 6 Apr 2001 15:05:13 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f User-Agent: Microsoft-Outlook-Express-Macintosh-Edition/5.02.2022 Date: Thu, 05 Apr 2001 12:59:11 +1000 Subject: categories: Re: Looking for adjoints From: Michael Batanin To: Message-ID: In-Reply-To: Mime-version: 1.0 Content-type: text/plain; charset="US-ASCII" Content-transfer-encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 8 This is just a small correction to Michael Barr message. The Kan's Ex^{\infty} is not a left adjoint to the inclusion of Kan simplicial sets into simplicial sets. In the original paper by Kan "On c.s.s. complexes" there is no mention about it being adjoint. Yet, the following is true Kan(Ex^{\infty}X,Y) is homotopy equivalent to Ssets(X,Y). So it is some sort of homotopy adjunction. Michael Batanin. >on 2/4/01 9:16 PM, Michael Barr at barr@barrs.org wrote: > The earliest that was labelled such was Kan's Ex^\infty, which was a left > adjoint to the inclusion of (what are now called) Kan simplicial sets into > simplicial sets. > From rrosebru@mta.ca Sat Apr 7 13:06:21 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f37FSDD22640 for categories-list; Sat, 7 Apr 2001 12:28:13 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3ACCCF90.546690F7@it-c.dk> Date: Thu, 05 Apr 2001 22:03:28 +0200 From: Mads Tofte Organization: IT University of Copenhagen X-Mailer: Mozilla 4.72 [en] (X11; U; Linux 2.2.14-5.0 i686) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: jobs: 13 positions at IT University of Copenhagen Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 9 [This job announcement is being sent to mailing lists that are read by researchers who work on aspects of information technology relevant to the IT University of Copenhagen. Feel free to circulate it! Apologies if you receive multiple copies. I'll be very happy to provide further information. Sincerely, Mads Tofte (tofte@it-c.dk) ] The IT University of Copenhagen was established by the Danish Ministry of Research in 1999. It is a graduate scool for research and research-based teaching in Information Technology. During its less that two years of existence, the IT University of Copenhagen has grown to employ 20 scientific staff at the level of assistant professor or higher, 20 ph.d.-students, some 700 students, and a teaching network of a further 90 teachers who are either part-time staff at the ITU or permanent staff of four nearby universities, namely the University of Copenhagen, Copenhagen Business School, The Technical University of Denmark (Lyngby), and Roskilde University. The four research departments of the IT University of Copenhagen currently have 13 vacant positions as assistant professor or associate professor. We welcome applications within the following areas: mobility, media and communication, software technology, software applications, theoretical computer science and virtual reality. The four research departments are: * The Department of Design and Use of IT * The Department of Digital Aesthetics and Communication * The Department of Innovation * The Theory Department The full job announcement may be found at www.it-c.dk/eng (look under Vacancies). At www.it-c.dk/eng one may also register on an electronic news service which provides information about future job announcements and scholarships. Sincerely, Mads Tofte Director of the IT University of Copenhagen From rrosebru@mta.ca Mon Apr 9 20:12:16 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f39MQWp29141 for categories-list; Mon, 9 Apr 2001 19:26:32 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3AD197A2.7C5CF426@di.epfl.ch> Date: Mon, 09 Apr 2001 13:06:10 +0200 From: Krzysztof Worytkiewicz X-Mailer: Mozilla 4.73 [en] (X11; U; Linux 2.2.16-3 i686) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Terminology Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 10 Dear Categories, Is it established terminology to call *injective* a faithful functor which is injective (in the usual sense) on objects ? Cheers, Krzysztof From rrosebru@mta.ca Thu Apr 12 16:43:43 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3CIpus11314 for categories-list; Thu, 12 Apr 2001 15:51:56 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Martin Escardo MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Tue, 10 Apr 2001 15:28:00 +0100 (BST) To: categories@mta.ca Subject: categories: A universal characterization of the unit interval X-Mailer: VM 6.43 under 20.4 "Emerald" XEmacs Lucid Message-ID: <15059.6003.377874.445195@henry.cs.bham.ac.uk> Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 11 The following extended abstract is now available on the Web. A universal characterization of the closed Euclidean interval ABSTRACT. We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. http://www.cs.bham.ac.uk/~mhe/papers/lics2001-revised.ps Best wishes, Martin Escardo & Alex Simpson -- P.s. A draft full version with proofs is available on-line at http://www.dcs.ed.ac.uk/home/als/Research/interval.ps From rrosebru@mta.ca Fri Apr 13 15:08:20 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3DHL7330961 for categories-list; Fri, 13 Apr 2001 14:21:07 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 13 Apr 2001 14:17:10 -0300 (ADT) From: Michael Barr To: categories Subject: categories: Beta version of new diagram package Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 12 I have decided to build a new diagram package as a front end to XY-pic. It is in many ways like my old one, but has much of the power of XY, with a syntax that is more familiar to people who have used the old one. This doubtless contains a number of errors and I would appreciate hearing about them. Here is the macro file, diagxy and a document package diaxydoc follows - it will compile under latex 2e. I have not tested this package in plain tex, but I believe it will work. As for AMS tex, well good luck. It has been tried with both the old and new latex. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input xy \xyoption{arrow} \xyoption{curve} \newdir{ >}{{ }*!/-9pt/@{>}} \newdir{< }{!/9pt/@{<}*{ }} \newbox\label \newdimen\high \newdimen\deep \newdimen\ul \newcount\deltax \newcount\deltay \newcount\xend \newcount\yend \newcount\xpos \newcount\ypos \newcount\default \default=500 \newif\ifat \newcount\atcode \atcode=\catcode`\@ \catcode`\@=11 \ul=.01em \X@xbase=\ul \Y@ybase=\ul \let\ifnextchar\@ifnextchar \catcode`\@=12 \let\bfig\xy \let\efig\endxy \def\car#1#2\nil{#1} \def\morphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]{% \xend#1\advance \xend by #5% \yend#2\advance \yend by #6% \def\next{#4}% \ifx\next\empty \atfalse% \else \def\next{\car#4\nil}% \if\next@\attrue \else \atfalse \fi% \fi% \ifat \def\next{#4}% \ifx\next\empty% \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}% \else \if#3l% \ifnum #6>0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3m% \setbox\label=\hbox{\kern .5pt $\labelstyle #9$\kern .5pt}% \high=\ht\label \advance\high by 1pt \ht\label=\high% \deep=\dp\label \advance\deep by 1pt \dp\label=\deep% \POS(#1,#2)*+{#7}\ar#4|-{\box\label} (\xend,\yend)*+{#8}% \else \if#3r% \ifnum #6<0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3a% \ifnum #5>0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3b% \ifnum #5<0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \fi\fi\fi\fi\fi\fi% \else% \def\next{#4}% \ifx\next\empty% \POS(#1,#2)*+{#7}; (\xend,\yend)*+{#8}% \else \def\next{#4}% \ifx\next\empty% \POS(#1,#2)*+{#7}\ar (\xend,\yend)*+{#8}% \else \if#3l% \ifnum #6>0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3m% \setbox\label=\hbox{\kern .5pt $\labelstyle #9$\kern .5pt}% \high=\ht\label \advance\high by 1pt \ht\label=\high% \deep=\dp\label \advance\deep by 1pt \dp\label=\deep% \POS(#1,#2)*+{#7}\ar@{#4}|-{\box\label} (\xend,\yend)*+{#8}% \else \if#3r% \ifnum #6<0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3a% \ifnum #5>0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3b% \ifnum #5<0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \fi\fi\fi\fi\fi\fi\fi% \fi\ignorespaces}% \def\location(#1,#2){\xpos=#1\ypos=#2\ignorespaces} \def\squareplacements|#1`#2`#3`#4|{% \def\xa{#1}\def\xb{#2}\def\xc{#3}\def\xd{#4}\ignorespaces} \def\sizes<#1,#2>{\deltax=#1\deltay=#2\ignorespaces} \def\squaredata[#1`#2`#3`#4;#5`#6`#7`#8]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}% \def\labela{#5}\def\labelb{#6}\def\labelc{#7}\def\labeld{#8}\ignorespaces} \def\squarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{% \location(#1)% \squareplacements|#2|% \sizes<#7>% \squaredata[#8]% \morphism(\xpos,\ypos)|\xd|/#6/<\deltax,0>[\nodec`\noded`\labeld] \advance \ypos by \deltay \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb] \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela] \advance \xpos by \deltax \morphism(\xpos,\ypos)|\xc|/#5/<0,-\deltay>[\nodeb`\noded`\labelc] \ignorespaces} \def\square{\ifnextchar({\squarep}{\squarep(0,0)}} \def\squarep(#1){\ifnextchar|{\squarepp(#1)}{\squarepp(#1)|a`l`r`b|}} \def\squarepp(#1)|#2|{\ifnextchar/{\squareppp(#1)|#2|}% {\squareppp(#1)|#2|/>`>`>`>/}} \def\squareppp(#1)|#2|/#3`#4`#5`#6/{% \ifnextchar<{\squarepppp(#1)|#2|/#3`#4`#5`#6/}% {\squarepppp(#1)|#2|/#3`#4`#5`#6/<\default,\default>}} \def\triplacements|#1`#2`#3|{\def\xa{#1}\def\xb{#2}\def\xc{#3}} \def\tridata[#1`#2`#3;#4`#5`#6]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}% \def\labela{#4}\def\labelb{#5}\def\labelc{#6}} \def\ptrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela] \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb] \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc] \ignorespaces} \def\ptriangle{\ifnextchar({\ptrianglep}{\ptrianglep(0,0)}} \def\ptrianglep(#1){\ifnextchar|{\ptrianglepp(#1)}{\ptrianglepp(#1)|a`l`r|}} \def\ptrianglepp(#1)|#2|{\ifnextchar/{\ptriangleppp(#1)|#2|}% {\ptriangleppp(#1)|#2|/>`>`>/}} \def\ptriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\ptrianglepppp(#1)|#2|/#3`#4`#5/}% {\ptrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\qtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela] \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb] \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xc|/#5/<0,-\deltay>[\nodeb`\nodec`\labelc] \ignorespaces} \def\qtriangle{\ifnextchar({\qtrianglep}{\qtrianglep(0,0)}} \def\qtrianglep(#1){\ifnextchar|{\qtrianglepp(#1)}{\qtrianglepp(#1)|a`l`r|}} \def\qtrianglepp(#1)|#2|{\ifnextchar/{\qtriangleppp(#1)|#2|}% {\qtriangleppp(#1)|#2|/>`>`>/}} \def\qtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\qtrianglepppp(#1)|#2|/#3`#4`#5/}% {\qtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\dtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc] \advance\ypos by \deltay\advance \xpos by \deltax \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela] \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb] \ignorespaces} \def\dtriangle{\ifnextchar({\dtrianglep}{\dtrianglep(0,0)}} \def\dtrianglep(#1){\ifnextchar|{\dtrianglepp(#1)}{\dtrianglepp(#1)|l`r`b|}} \def\dtrianglepp(#1)|#2|{\ifnextchar/{\dtriangleppp(#1)|#2|}% {\dtriangleppp(#1)|#2|/>`>`>/}} \def\dtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\dtrianglepppp(#1)|#2|/#3`#4`#5/}% {\dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\btrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc] \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodeb`\labela] \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb] \ignorespaces} \def\btriangle{\ifnextchar({\btrianglep}{\btrianglep(0,0)}} \def\btrianglep(#1){\ifnextchar|{\btrianglepp(#1)}{\btrianglepp(#1)|l`r`b|}} \def\btrianglepp(#1)|#2|{\ifnextchar/{\btriangleppp(#1)|#2|}% {\btriangleppp(#1)|#2|/>`>`>/}} \def\btriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\btrianglepppp(#1)|#2|/#3`#4`#5/}% {\btrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\Atrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% {\multiply\deltax by 2% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]}% \advance\ypos by \deltay\advance\xpos by \deltax \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela] \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb] \ignorespaces} \def\Atriangle{\ifnextchar({\Atrianglep}{\Atrianglep(0,0)}} \def\Atrianglep(#1){\ifnextchar|{\Atrianglepp(#1)}{\Atrianglepp(#1)|l`r`b|}} \def\Atrianglepp(#1)|#2|{\ifnextchar/{\Atriangleppp(#1)|#2|}% {\Atriangleppp(#1)|#2|/>`>`>/}} \def\Atriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Atrianglepppp(#1)|#2|/#3`#4`#5/}% {\Atrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\Vtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb] {\multiply\deltax by 2% \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \global\advance\xpos by \deltax}% \morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc] \ignorespaces} \def\Vtriangle{\ifnextchar({\Vtrianglep}{\Vtrianglep(0,0)}} \def\Vtrianglep(#1){\ifnextchar|{\Vtrianglepp(#1)}{\Vtrianglepp(#1)|a`l`b|}} \def\Vtrianglepp(#1)|#2|{\ifnextchar/{\Vtriangleppp(#1)|#2|}% {\Vtriangleppp(#1)|#2|/>`>`>/}} \def\Vtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Vtrianglepppp(#1)|#2|/#3`#4`#5/}% {\Vtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\Ctrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance \ypos by \deltay \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodeb`\nodec`\labelc] \advance\ypos by \deltay \advance \xpos by \deltax \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela] \multiply\deltay by 2% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces} \def\Ctriangle{\ifnextchar({\Ctrianglep}{\Ctrianglep(0,0)}} \def\Ctrianglep(#1){\ifnextchar|{\Ctrianglepp(#1)}{\Ctrianglepp(#1)|a`r`b|}} \def\Ctrianglepp(#1)|#2|{\ifnextchar/{\Ctriangleppp(#1)|#2|}% {\Ctriangleppp(#1)|#2|/>`>`>/}} \def\Ctriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Ctrianglepppp(#1)|#2|/#3`#4`#5/}% {\Ctrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\Dtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\xpos by \deltax \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc] \advance\xpos by -\deltax \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodeb`\labelb]% \multiply \deltay by 2 \morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodec`\labela]% \ignorespaces} \def\Dtriangle{\ifnextchar({\Dtrianglep}{\Dtrianglep(0,0)}} \def\Dtrianglep(#1){\ifnextchar|{\Dtrianglepp(#1)}{\Dtrianglepp(#1)|a`l`b|}} \def\Dtrianglepp(#1)|#2|{\ifnextchar/{\Dtriangleppp(#1)|#2|}% {\Dtriangleppp(#1)|#2|/>`>`>/}} \def\Dtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Dtrianglepppp(#1)|#2|/#3`#4`#5/}% {\Dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}} \def\tripairplacements|#1`#2`#3`#4`#5|{\def\xa{#1}\def\xb{#2}\def\xc{#3}% \def\xd{#4}\def\xe{#5}} \def\tripairdata[#1`#2`#3`#4;#5`#6`#7`#8`#9]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}% \def\labela{#5}\def\labelb{#6}\def\labelc{#7}\def\labeld{#8}\def\labele{#9}} \def\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \morphism(\xpos,\ypos)|\xd|/#6/<\deltax,0>[\nodeb`\nodec`\labeld]% \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xe|/#7/<\deltax,0>[\nodec`\noded`\labele]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodea`\noded`\labelc]% \ignorespaces} \def\Atrianglepair{\ifnextchar({\Atrianglepairp}{\Atrianglepairp(0,0)}}% \def\Atrianglepairp(#1){\ifnextchar|{\Atrianglepairpp(#1)}% {\Atrianglepairpp(#1)|l`m`r`b`b|}}% \def\Atrianglepairpp(#1)|#2|{\ifnextchar/{\Atrianglepairppp(#1)|#2|}% {\Atrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Atrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodea`\noded`\labelc]% \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,0>[\nodeb`\nodec`\labelb]% \morphism(\xpos,\ypos)|\xd|/#6/<0,-\deltay>[\nodeb`\noded`\labeld]% \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xe|/#7/<-\deltax,-\deltay>[\nodec`\noded`\labele]% \ignorespaces} \def\Vtrianglepair{\ifnextchar({\Vtrianglepairp}{\Vtrianglepairp(0,0)}}% \def\Vtrianglepairp(#1){\ifnextchar|{\Vtrianglepairpp(#1)}% {\Vtrianglepairpp(#1)|a`a`l`m`r|}}% \def\Vtrianglepairpp(#1)|#2|{\ifnextchar/{\Vtrianglepairppp(#1)|#2|}% {\Vtrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Vtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xe|/#7/<0,-\deltay>[\nodec`\noded`\labele]% \advance\xpos by -\deltax \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \morphism(\xpos,\ypos)|\xd|/#6/<\deltax,-\deltay>[\nodeb`\noded`\labeld]% \advance\ypos by \deltay \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces} \def\Ctrianglepair{\ifnextchar({\Ctrianglepairp}{\Ctrianglepairp(0,0)}}% \def\Ctrianglepairp(#1){\ifnextchar|{\Ctrianglepairpp(#1)}% {\Ctrianglepairpp(#1)|l`r`m`l`r|}}% \def\Ctrianglepairpp(#1)|#2|{\ifnextchar/{\Ctrianglepairppp(#1)|#2|}% {\Ctrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Ctrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \morphism(\xpos,\ypos)|\xd|/#6/<0,-\deltay>[\nodeb`\noded`\labeld]% \advance\ypos by \deltay \morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]% \advance\ypos by -\deltay \advance\xpos by \deltax \morphism(\xpos,\ypos)|\xe|/#7/<-\deltax,-\deltay>[\nodec`\noded`\labele]% \ignorespaces} \def\Dtrianglepair{\ifnextchar({\Dtrianglepairp}{\Dtrianglepairp(0,0)}}% \def\Dtrianglepairp(#1){\ifnextchar|{\Dtrianglepairpp(#1)}% {\Dtrianglepairpp(#1)|l`r`m`l`r|}}% \def\Dtrianglepairpp(#1)|#2|{\ifnextchar/{\Dtrianglepairppp(#1)|#2|}% {\Dtrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Dtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \catcode`\@=\atcode \def\place(#1,#2)[#3]{\POS(#1,#2)*+{#3}} \def\pullback#1]#2]{\square#1]\trident#2]} \def\tridentdata[#1;#2`#3`#4]{\def\nodee{#1}\def\labele{#2}% \def\labelf{#3}\def\labelg{#4}} \def\trident{\ifnextchar|{\tridentp}{\tridentp|a`m`b|}} \def\tridentp|#1|{\ifnextchar/{\tridentpp|#1|}{\tridentpp|#1|/{>}`{>}`{>}/}} \def\tridentpp|#1|/#2/{\ifnextchar<{\tridentppp|#1|/#2/}% {\tridentppp|#1|/#2/<500,500>}} \def\tridentppp|#1`#2`#3|/#4`#5`#6/<#7,#8>[#9]{% \tridentdata[#9]% \advance \xpos by -\deltax \advance \xpos by -#7\advance \ypos by #8% {\advance\deltax by #7% \morphism(\xpos,\ypos)|#1|/#4/<\deltax,-#8>[\nodee`\nodeb`\labele]}% \morphism(\xpos,\ypos)|#2|/#5/<#7,-#8>[\nodee`\nodea`\labelf]% \advance\deltay by #8% \morphism(\xpos,\ypos)|#3|/#6/<#7,-\deltay>[\nodee`\nodec`\labelg]} From rrosebru@mta.ca Fri Apr 13 15:34:00 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3DHleX22459 for categories-list; Fri, 13 Apr 2001 14:47:40 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 13 Apr 2001 14:24:09 -0300 (ADT) From: Michael Barr To: categories Subject: categories: documentation for new diagram package Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 13 %% %% documentation for diagxy \documentclass{article} \input diagxy \xyoption{curve} \begin{document} \def\xypic{\hbox{\rm\Xy-pic}} \title{A new diagram package} \author{Michael Barr} \maketitle \section*{Why a new diagram package?} This started when a user of my old package, diagram, wrote to ask me if dashed lines were possible. The old package had dashed lines for horizontal and vertical arrows, but not any other direction. The reason for this was that \LaTeX\ used rules for horizontal and vertical arrows, but had its own fonts for other directions. While rules could be made any size, the smallest lines in other directions were too long for decent looking dashes. Presumably, Lamport was worried about compile time and file size if the lines were too short, considerations that have diminished over the years since. Also arrows could be drawn in only 48 different directions, which is limiting. My macros were not very well implemented for slopes like 4 and 1/4, since I never used such lines. There was certainly an alternative, \xypic, for those who wanted something better. But it was hard to learn and I was not entirely happy with the results. The basic interface used an \verb+\halign+. This meant that no extra space was allotted to an arrow that had a large label. In addition, the different slots could have different horizontal size, which could result in a misshapen diagram. I used it in a paper that had a `W' shaped diagram whose nodes had different widths and the result was quite obviously misshapen. On the other hand, the graphics engine underlying \xypic\ is really quite remarkable and it occurred to me that I could try to redraft diagram as a front end. The result is the current package. It has been tested only with version 3.7, so there is no guarantee it would work with any earlier version (or, for that matter, later). The syntax described below is not compatible with the original diagram package. Every front end of this sort represents a trade-off between simplicity and utility. A package that simply upgraded the syntax to allow more dashed (and dotted) lines would have been just as easy to implement, but would have made poor use of the wonderful possibilities of the underlying \xypic\ package. Also there would have been too many different arrow specifications (they would have had to go at least in the range $[-9,9]$) for easy memory) and still would not have included things like inclusion arrows. To those who would have liked a simpler syntax, I apologize. To those who would want more flexibility, I remind that the entire \xypic\ package is there for use. There is one point that cannot be made too strongly. {\em Watch for errant spaces.} Unlike the diagram package, which was carried out in math mode so that spaces were ignored. \xypic\ is not in math mode and spaces will mess up your diagrams. Unlike diagram, it should not be necessary to put any objects inside \verb+\phantom+. In diagram, for reasons that I now understand, objects did not always register properly. This was a flaw built in to the very heart of the package and is not worth correcting, although it could have been done better in the first place. If you see an object in double vision, then the virtually certain cause is that a space has gotten in there somewhere. I have attempted to prevent this by liberal use of \verb+\ignorespaces+ in the definitions, but one cannot always be sure and while testing, I found a number seemingly innocuous spaces that messed up diagrams. \section*{The basic syntax} The basic syntax is built around an operation \verb+\morphism+ that is used as \begin{verbatim} \morphism(x,y)|label pos|/{shape}/[N`N`A] \end{verbatim} The parameters \verb+x+ and \verb+y+ give the location of the source node within a fixed local coordinate system and \verb-x+dx- and \verb-y+dy- and locate the target. To be precise, the first coordinate is the horizontal center of the node and the second is that of the base line. These distances are given in terms of \verb+\ul+'s, (for unitlength), which is user assignable. I have, for the time being, set it to .01em, which is probably too small. In the original version of diagram, it was .1em. This was reasonable for diagrams, but since the multipliers had to be integers, it turned out to inconveniently large for certain internal computations. (Strictly speaking, the multipliers could be any decimal number, but \TeX\ does not permit arithmetic with any but whole numbers.) Since all the internal computations are carried out by \xypic, this is no longer a consideration and in the production version, I may go back to the original .1em. Note that by basing distances on ems, I make things automatically scale when changing point size. The parameter \verb+label pos+ is one of \verb+a,b,l,r,m+ that stand for above, below, left, right, or mid and describe the positioning of the arrow label on the arrow. The label \verb+m+ stands for a label positioned in the middle of a split arrow. Although \verb+a+ and \verb+b+ can be used in any direction, they give uncertain results when used on vertical arrows and similarly \verb+l+ and \verb+r+ should not be used on horizontal arrows. The \verb+shape+ describes the shape of the arrow. An arrow is thought of as consisting of three parts, the tail, the shaft and the head. You may specify just the head, in which case the shaft will be an ordinary line, or all three. However, since the tail can be (and usually is) empty, in practice you can also describe the shaft and tail. In addition, it is possible to modify the arrow in various ways. Although the parameter is shown withing braces, the braces can be omitted unless one of the modifier characters is \verb+/+, in which case, {\em the entire paramater} must be put in braces. It is important to note that it will not work just to put the \verb+/+ inside the braces, since this will interfere with the internal parsing of \xypic. The head and tail shapes are basically one of \verb+>+, \verb+>>+, \verb+(+, \verb+)+, \verb*+ >+, and \verb*+< +. Each of these may also be preceded by \verb+^+ or \verb+_+ and others are user definable. For details, see the \xypic\ reference manual. The first of these is an ordinary head, while the second is for an epic arrow. The third is not much used, but the superscripted version makes and inclusion tail, as will be illustrated below. The reverse ones give reversed arrowheads. The sign \verb*+ + stands for an obligatory space and it leaves extra space for a tailed (monic) arrow, which otherwise runs into the source node. Although there are many possibilities for shafts, including alphanumeric characters, there are basically three that interest us: \verb+-+, which is an ordinary shaft, \verb+--+, which produces a dashed arrow, \verb+=+, which gives a double arrow (although with only one arrowhead), and \verb+.+, which makes a dotted arrow. Thus \verb+>+ or \verb+->+ will produce an ordinary arrow, \verb+->>+ an epic arrow, \verb*+ >->+ makes a monic arrow, and \verb+-->>+ would make a dashed epic arrow. The descriptions \verb+<-, <<-,+ \verb*+<-< +, and \verb+<<--+ give the reversed versions. Note that \verb+<+ does not give a reversed arrow, since \xypic\ interprets that as a reversed head, not a tail. If the shape parameter begins with an \verb+@+, it is interpeted differently. In that case, it has the form \verb+@{shape}@+ modifier, where the modifier is as described in the \xypic\ reference guide. I just mention a couple of them. The paramater \verb+@{->}@<3pt>+, for example, would give an ordinary arrow moved three points in the direction perpendicular to that of the arrow. If you give \verb+{@{-->}@/^5pt/}+, you will get an epic arrow that is curved in the direction perpenducular to the direction of the arrow by five points. (As far as I can tell, there is no difference between \verb+^-5pt+ and \verb+_5pt+ so the syntax could be simplified and made consistent with that of \verb++ described above.) It is imperative that a specification such as \verb+@{>}@/5pt/+ be enclosed in braces because of the \verb+/+s. Finally, the \verb+N+s are the source and target of the arrow and \verb+L+ is the label. Currently, the \verb+L+ is set in \verb+\scriptstyle+ size, but this can easily be changed by putting \verb+\let\labelstyle=\textstyle+ in your file. \subsection*{A word about parameters} I have already mentioned the necessity of enclosing certain arrow shape specifications in braces. Owing to certain built-in limitations of \TeX, I have used a number of different delimiters: \verb+(, ), |, /, [, ], `, ;+. Any of these that appear inside an argument could conceivably cause problems. They were chosen as the least likely to appear inside mathematics (except for \verb+(, )+ which appear in positions that are unlikely to cause problems). However, be warned that this is a possible cause of mysterious error messages. If it is indeed then enclosing the offending parameter in braces should cure the problem. In most, but not all, cases, enclosing the offending character will solve the problem. The exceptions come when the braces interfere with \xypic's somewhat arcane parsing mechanism. There will be no problem if it is just part of a mathematical expression. \section*{Shapes} Using the basic \verb+\morphism+ macro, I have a defined a number of shapes, squares (really rectangles) and variously oriented triangles. I will probably do a few other shapes as I had done for the old package, such as \verb+\cube+ and a $3 \times 3$ diagram. The basic shapes are exactly the same, but the options are done entirely differently. Here is the syntax of the \verb+\square+ macro: \begin{verbatim} \square(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L`L] \end{verbatim} Each of the first four sets of parameters is optional and any subset of them can be omitted. (Note that only the sets can be omitted, once you give \verb+dx+, you also have to give \verb+dy+ and so on.) The first two describe the horizontal and vertical size of the square, the next four give the label placements using the same five characters as above, the next four give the shapes of the arrows using the same syntax as discussed above. The last group is the location. The meaning of this varies a bit with the shape. For a square, it is the midpoint of the baseline of the lower left corner node. For a triangle that has a lower left corner, it is the same. Otherwise, \verb+x+ is the midpoint of the leftmost node and \verb+y+ is the baseline of the lowest node. As with the old macros, different shapes should fit together accurately, in fact more accurately than with the old ones. Here is a listing of the shapes, together with the groups of parameters. In all cases, the first four groups are optional and any subset of them will work. However, they must come in order. Note that the names of the triangles are related to the shape as the shape that best approximates the shape of the letter. For example, a \verb+\ptriangle+ is a right triangle that has its hypotenuse going from upper right to lower left. Triangles with lower case names have their legs horizontal and vertical, while those with capitalized names have their hypotenuse horizontal or vertical. In those cases, one of \verb+dx+ or \verb+dy+ is the length of a leg and the other is {\em half} the length of the hypotenuse. In all cases, the order of the nodes and of the arrows is linguistic, first moving from left to right and then down. The defaults are reasonable, but with triangles, there is not always a natural direction for arrows. I always made mistakes in the order with my macros and this is certainly a defect. But the order is the same. In every case the braces around the shape specification can be removed unless it includes the following delimiter (that is, \verb+`+ or \verb+/+, as the case may be.) \begin{verbatim} \square(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L] \ptriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \qtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \dtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \btriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Atriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Vtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Ctriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Dtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Atrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \Vtrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \Ctrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \Dtrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \end{verbatim} (Of course, the \verb+%+ signs are required if you break the macro at such points.) To make a diagram, you have to enclose it inside \verb+\xy ... \endxy+. You will almost always want it displayed, for which the simplest is to enclose it in \verb+$$\xy ... \endxy$$+. For old times sake, I have also let \verb+\bfig+ and \verb+\efig+ be synonyms for \verb+\xy+ and \verb+\endxy+, resp. (In case you wonder, these go all the back to a main frame formatter running at McGill when Charles Wells and I were first writing TTT, where \verb+.BFIG+ and \verb+.EFIG+ were used to make displays, at least as far as that primitive formatter was capable of.) You could instead make definitions as follows: \begin{verbatim} \def\bfig{$$\xy} \def\efig{\endxy$$} \end{verbatim} in which case you will automatically get a display. I do not do this because my spelling checker is programmed to ignore stuff in math or display mode and the \verb+$$+ marks the display mode. Besides which, I have defined the \verb+Ctl-F+ key on my editor so that pressing it gives \begin{verbatim} $$\bfig _ \efig$$ \end{verbatim} with the cursor at the position marked by the underline. \section{Examples} Many people---including me--- learn mainly by example and I will give a number of examples here. The formal syntax can be learned in the \xypic\ reference manual. We begin with \begin{verbatim} $$\bfig \square[A`B`C`D;e`f`g`m] \efig$$ \end{verbatim} produces $$\bfig \square[A`B`C`D;e`f`g`m] \efig$$ This can be modified, for example \begin{verbatim} $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \efig$$ \end{verbatim} produces $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \efig$$ This can be put together with a morphism as follows: \begin{verbatim} $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \morphism(500,500)|m|/.>/<-500,-500>[B`C`h] \efig$$ \end{verbatim} which makes a familiar diagram: $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \morphism(500,500)|m|/.>/<-500,-500>[B`C`h] \efig$$ The same diagram could have been made by \begin{verbatim} $$\bfig \ptriangle|a`l`m|/>>`>`.>/[A`B`C;e`f`h] \dtriangle/`>` >->/[B`C`D;`g`m] \efig$$ \end{verbatim} You can fit two squares together, horizontally: \begin{verbatim} $$\bfig \square|a`l`m`b|[A`B`C`D;f`g`h`k] \square(500,0)/>``>`>/[B`E`D`F;l``m`n] \efig$$ \end{verbatim} $$\bfig \square|a`l`m`b|[A`B`C`D;f`g`h`k] \square(500,0)/>``>`>/[B`E`D`F;l``m`n] \efig$$ or vertically \begin{verbatim} $$\bfig \square(0,500)|a`l`r`m|[A`B`C`D;f`g`h`k] \square/`>`>`>/[C`D`E`F;`l`m`n] \efig$$ \end{verbatim} $$\bfig \square(0,500)|a`l`r`m|[A`B`C`D;f`g`h`k] \square/`>`>`>/[C`D`E`F;`l`m`n] \efig$$ or a square and a triangle \begin{verbatim} $$\bfig \Ctriangle/<-`>`>/<400,400>[\hbox{\rm rec}(A,B)`B`X;r_0(A,B)`f`t_0] \square(400,0)/<-``>`<-/<1000,800>[\hbox{\rm rec}(A,B)`A\times\hbox{\rm rec}(A,B)`X`A\times X;r(A,B)``\hbox{\rm id}_A\times f`t] \efig$$ \end{verbatim} gives the diagram $$\bfig \Ctriangle/<-`>`>/<400,400>[\hbox{\rm rec}(A,B)`B`X;r_0(A,B)`f`t_0] \square(400,0)/<-``>`<-/<1000,800>[\hbox{\rm rec}(A,B)`A\times\hbox{\rm rec}(A,B)`X`A\times X;r(A,B)``\hbox{\rm id}_A\times f`t] \efig$$ This diagram is on page 361 of the third edition of Category Theory for Computing Science to describe recursion. Here is an example using the procedure for sliding an arrow sideways. This one could even be used in a text, \xy \morphism(0,0)|a|/@{>}@<3pt>/<400,0>[A`B`d] \morphism(0,0)|b|/@{>}@<-3pt>/<400,0>[A`B`e]\endxy which was made using \begin{verbatim} \xy \morphism(0,0)|a|/@{>}@<3pt>/<400,0>[A`B`d] \morphism(0,0)|b|/@{>}@<-3pt>/<400,0>[A`B`e]\endxy \end{verbatim} Something a bit different that illustrates the use of another shaft \verb+=+ that gives a 2-arrow, as well as curved arrows: \begin{verbatim} $$\bfig \morphism(0,0)|a|/{@{>}@/^1em/}/<500,0>[A`B`f] \morphism(0,0)|b|/{@{>}@/_1em/}/<500,0>[A`B`g] \morphism(250,50)|a|/=>/<0,-100>[``] \efig$$ \end{verbatim} $$\bfig \morphism(0,0)|a|/{@{>}@/^1em/}/<500,0>[A`B`f] \morphism(0,0)|b|/{@{>}@/_1em/}/<500,0>[A`B`g] \morphism(250,50)|a|/=>/<0,-100>[``] \efig$$ In order to use curved arrows, you must insert \verb+\xyoption{curve}+ into your file. There is a macro \verb+\place(x,y)[N]+ where \verb+N+ is any object that places that object anywhere. I have changed the name in order to avoid clashing with the \LaTeX\ picture mode's \verb+\put+. Here is an example that uses a construction that is undocmented here, but uses an \Xy\ construction: \begin{verbatim} \newbox\anglebox \setbox\anglebox=\hbox{\xy \POS(75,0)\ar@{-} (0,0) \ar@{-} (75,75)\endxy} \def\angle{\copy\anglebox} $$\bfig \square[A`B`C`D;f`g`h`k] \place(100,400)[\angle] \efig$$ \end{verbatim} \newbox\anglebox \setbox\anglebox=\hbox{\xy \POS(75,0)\ar@{-} (0,0) \ar@{-} (75,75)\endxy} \def\angle{\copy\anglebox} $$\bfig \square[A`B`C`D;f`g`h`k] \place(100,400)[\angle] \efig$$ Notice that you get a headless arrow by using \verb+\ar@{-}+. Here is a special code installed at the request of Jonathon Funk: \begin{verbatim} $$\bfig \pullback|b`r`r`b|<800,800>[P`X`Y`Z;t`u`v`w]% |a`m`b|/>`-->`>/<500,500>[A;f`g`h] \efig$$ \end{verbatim} $$\bfig \pullback|b`r`r`b|<800,800>[P`X`Y`Z;t`u`v`w]% |a`m`b|/>`-->`>/<500,500>[A;f`g`h] \efig$$ The full syntax for this is \begin{verbatim} \pullback(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L]% |lp`lp`lp|/{sh}`{sh}`{sh}/[N;L`L`L] \end{verbatim} Of these only the nodes placed inside brackets are obligatory. The first sets of parameters are exactly as for \verb+\square+ and the remaining parameters are for the nodes and labels of the outer arrows. There is no positioning paramters for them; rather you set the horizontal and vertical separations of the outer node from the square. The last example, is a complicated diagram from TTT: \begin{verbatim} $$\bfig \morphism(0,2100)|l|/>/<0,-1400>[TT`TT'T`T\eta'T] \morphism(0,2100)|a|/>/<700,0>[TT`T`\mu] \morphism(0,2100)|l|/>/<700,-700>[TT`TTT'`TT\eta'] \morphism(700,2100)|r|/>/<700,-700>[`TT'`T\eta] \place(700,1750)[1] \morphism(700,1400)|b|/@<-20\ul>/<700,0>[TTT'`TT'`T\sigma] \square(700,700)|a`m`m`a|/@<20\ul>`>``>/<700,700>[TTT'`TT'`TT'TT'`TT'T';\mu T' `T\eta'TT'``] \morphism(700,700)|a|/>/<700,0>[TT'TT'`TT'T'`TT'\sigma] \place(300,1400)[2] \place(1050,1050)[3] \btriangle(1400,700)|m`r`a|/>`>`>/<700,700>[TT'`TT'T'`TT';T\eta'T'`\hbox{\rm id}`] \place(1600,1050)[6] \square|a`l`m`b|<700,700>[TT'T`TT'TT'`T'T`T'TT';TT'T\eta'`\sigma T`\sigma TT'`T'T\eta'] \square(1400,0)|a`m`r`b|<700,700>[TT'T'`TT'`T'T'`T';T\mu'`\sigma T' `\sigma`\mu'] \morphism(700,0)|b|/>/<700,0>[T'TT'`T'T'`T'\sigma T'] \place(300,350)[4] \place(1050,350)[5] \place(1750,350)[7] \efig$$ \end{verbatim} $$\bfig \morphism(0,2100)|l|/>/<0,-1400>[TT`TT'T`T\eta'T] \morphism(0,2100)|a|/>/<700,0>[TT`T`\mu] \morphism(0,2100)|l|/>/<700,-700>[TT`TTT'`TT\eta'] \morphism(700,2100)|r|/>/<700,-700>[`TT'`T\eta] \place(700,1750)[1] \morphism(700,1400)|b|/@<-20\ul>/<700,0>[TTT'`TT'`T\sigma] \square(700,700)|a`m`m`a|/@<20\ul>`>``>/<700,700>[TTT'`TT'`TT'TT'`TT'T';\mu T' `T\eta'TT'``] \morphism(700,700)|a|/>/<700,0>[TT'TT'`TT'T'`TT'\sigma] \place(300,1400)[2] \place(1050,1050)[3] \btriangle(1400,700)|m`r`a|/>`>`>/<700,700>[TT'`TT'T'`TT';T\eta'T'`\hbox{\rm id}`] \place(1600,1050)[6] \square|a`l`m`b|<700,700>[TT'T`TT'TT'`T'T`T'TT';TT'T\eta'`\sigma T`\sigma TT'`T'T\eta'] \square(1400,0)|a`m`r`b|<700,700>[TT'T'`TT'`T'T'`T';T\mu'`\sigma T' `\sigma`\mu'] \morphism(700,0)|b|/>/<700,0>[T'TT'`T'T'`T'\sigma T'] \place(300,350)[4] \place(1050,350)[5] \place(1750,350)[7] \efig$$ \end{document} From rrosebru@mta.ca Mon Apr 16 20:29:45 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3GMEDx27420 for categories-list; Mon, 16 Apr 2001 19:14:13 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 16 Apr 2001 14:27:04 -0700 Message-Id: <200104162127.OAA24479@kamiak.eecs.wsu.edu> From: "David Benson" To: categories@mta.ca Subject: categories: Foundational Methods in Computer Science Workshop (FMCS'01) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 14 PRELIMINARY ANNOUNCEMENT Workshop on Foundational Methods in Computer Science -- a series of informal workshops on categories and logic in computer science FMCS'01 : May 31 -- June 3, 2001 Spokane, Washington May 31: Arrival day -- Reception in the evening June 1: Tutorial Talks June 2: Research Talks Conference Banquet June 3: Research Talks in the morning Noon -- end of workshop A subsequent email will describe the venue, a hotel directly on the Spokane River, with nearby parks and a view of the Spokane Falls, as well as being just across the river from downtown. The necessaries about registration and obtaining hotel accomodations (around $US73/night) will be included in these subsequent messages. If you plan to attend, please reply as soon as may be to dbenson@eecs.wsu.edu and an indicate whether you would like to give a tutorial or a research talk. Thank you! -- Professor David B. Benson (509) 335-2706 School of EE and Computer Science (EME 102A) (509) 335-3818 fax PO Box 642752, Washington State University dbenson@eecs.wsu.edu Pullman WA 99164-2752 U.S.A. From rrosebru@mta.ca Tue Apr 17 21:24:32 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3HNgm614663 for categories-list; Tue, 17 Apr 2001 20:42:48 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3ADC4B5C.9BB45075@mcs.le.ac.uk> Date: Tue, 17 Apr 2001 14:55:40 +0100 From: "V. Schmitt" X-Mailer: Mozilla 4.76 [en] (X11; U; Linux 2.2.18 i686) X-Accept-Language: en MIME-Version: 1.0 To: "categories@mta.ca" Subject: categories: ECT and computer science Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 15 Dear all, I am interested in knowing relevant applications of enriched categories to computer science. Would you please send me references if any? Thanks, Best regards, Vincent Schmitt. From rrosebru@mta.ca Thu Apr 19 10:06:45 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3JCRn509670 for categories-list; Thu, 19 Apr 2001 09:27:49 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <20010418235052.95337.qmail@web12205.mail.yahoo.com> Date: Wed, 18 Apr 2001 16:50:52 -0700 (PDT) From: Galchin Vasili Subject: categories: Category Theory and Hereditarily-Finite Sets To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 16 Hello Cat Theory Community, Hereditarily-finite sets are becoming increasingly more popular in computer science research. 1) What kind of interesting categories exist where an "object" is a hereditarily-finite set plus some structure on the set and a "morphism" would be a structure-preserving function. (I can think of the obvious subcategory of SET and also the category where an object is a hereditarily-finite set together with a "SET" endomorphism on that set, but neither of these categories would have interesting or useful properties, in my opinion) 2) What kind of papers can I read on this subject? Regards, Bill Halchin From rrosebru@mta.ca Thu Apr 19 17:49:51 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3JKHPU13443 for categories-list; Thu, 19 Apr 2001 17:17:25 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 19 Apr 2001 19:29:13 +0100 From: Paul Taylor Message-Id: <200104191829.TAA04249@koi-pc.dcs.qmw.ac.uk> To: categories@mta.ca Subject: categories: Re: Category Theory and Hereditarily-Finite Sets Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 > Hereditarily-finite sets are becoming increasingly more popular > in computer science research. Why? Because some ill-advised first year maths lecturer told you that the element relation was the foundation of mathematics, maybe? > "object" is a hereditarily-finite set plus some structure on the set > and a "morphism" would be a structure-preserving function. If you're really interested in heredity, so the "structure" is the element relation, this is a well-founded coalgebra for the covariant powerset functor. Coalgebras for the powerset functor were first studied by Gerhard Osius in JPAA in 1974, although he considered recursion rather than induction. Well founded coalgebras for general functors (but with some emphasis on the powerset) are defined in Section 6.3 of my book "Practical Foundations of Mathematics" (Cambridge University Press, 1999). The exercises for that chapter show how various ideas with recursive programs may be expressed in these terms. In particular, unary recursion (with at most one recursive call at each level) is reduced to tail recursion (equivalent to while programs) together with an accumulator monoid. Paul From rrosebru@mta.ca Fri Apr 20 12:31:05 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3KEn0N17529 for categories-list; Fri, 20 Apr 2001 11:49:00 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 20 Apr 2001 14:13:10 +0200 (MET DST) Message-Id: <200104201213.OAA03519@volterra.math.kth.se> From: Dan Laksov To: categories@mta.ca Subject: categories: groupoids Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 18 Could anyone help me with a good reference to groupoid schemes, or sheaves on groupoid schemes? Sincerely Dan Laksov From rrosebru@mta.ca Fri Apr 20 12:33:52 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3KEmXJ26265 for categories-list; Fri, 20 Apr 2001 11:48:33 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3ADF5672.F53C9735@essex.ac.uk> Date: Thu, 19 Apr 2001 22:19:46 +0100 From: Wilkins E B Organization: University of Essex X-Mailer: Mozilla 4.6 [en] (X11; I; Linux 2.2.14 i686) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Category Theory and Hereditarily-Finite Sets References: <200104191829.TAA04249@koi-pc.dcs.qmw.ac.uk> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 19 Paul Taylor wrote: > > Hereditarily-finite sets are becoming increasingly more popular > > in computer science research. > > Why? Because some ill-advised first year maths lecturer told you that > the element relation was the foundation of mathematics, maybe? Maybe because someone read papers by Friedman (1977) which gave the set theory B which is [Beeson] "the theory of hereditary extensional sets of finite rank" and which is strong enough to model Bishop-style constructive mathematics. This seems to be a good reason. Elwood -- Dr Elwood Wilkins tel: (+44) (0)1206 872771 Senior Research Officer fax: (+44) (0)1206 872788 Department of Computer Science University of Essex, Colchester, Essex, UK From rrosebru@mta.ca Fri Apr 20 12:33:57 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3KEoa217353 for categories-list; Fri, 20 Apr 2001 11:50:36 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Mime-Version: 1.0 X-Sender: nielson@cstpop.it.dtu.dk (Unverified) Message-Id: Date: Tue, 17 Apr 2001 10:50:39 +0200 To: categories@mta.ca From: Flemming Nielson Subject: categories: job: positions at the Technical University of Denmark Content-Type: text/plain; charset="us-ascii" ; format="flowed" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 20 The Technical University of Denmark is happy to announce a number of associate professorships (tenured) and assistant professorships (untenured) http://www.imm.dtu.dk/cst/jobs.htm covering data security, algorithmics, distributed systems, software engineering, embedded systems etc. The Technical University of Denmark was created in 1829 by the discoverer of electromagnetism, H.C.Oersted, and became the first private Danish University in 2001, thanks to a donation by the Danish Government, in order to increase its impact on research and teaching in engineering and science. The Department of Informatics & Mathematical Modelling has been given the mission to strengthen Computer Science and Technology and will be announcing a substantial number of positions within selected areas in the coming years. -- ---------------------------------------------------------------------- Professor dr.scient. Flemming Nielson direct: (+45) 4525 3735 Informatics & Mathematical Modelling mailto:nielson@imm.dtu.dk The Technical University of Denmark http://www.imm.dtu.dk/~nielson ---------------------------------------------------------------------- From rrosebru@mta.ca Sun Apr 22 20:21:02 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3MMjfY22317 for categories-list; Sun, 22 Apr 2001 19:45:41 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Sun, 22 Apr 2001 08:38:25 -0400 (EDT) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: Categories list Subject: categories: diaxydoc.tex Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 \documentclass{article} \input diagxy \xyoption{curve} \begin{document} \def\xypic{\hbox{\rm\Xy-pic}} \title{A new diagram package (Version 2)} \author{Michael Barr\\Dept of Math and Stats\\McGill University\\805 Sherbrooke St. W\\Montreal, QC Canada H3A 2K6\\[5pt] barr@barrs.org} \maketitle \section*{Why a new diagram package?} This started when a user of my old package, diagram, wrote to ask me if dashed lines were possible. The old package had dashed lines for horizontal and vertical arrows, but not any other direction. The reason for this was that \LaTeX\ used rules for horizontal and vertical arrows, but had its own fonts for other directions. While rules could be made any size, the smallest lines in other directions were too long for decent looking dashes. Presumably, Lamport was worried about compile time and file size if the lines were too short, considerations that have diminished over the years since. Also arrows could be drawn in only 48 different directions, which is limiting. My macros were not very well implemented for slopes like 4 and 1/4, since I never used such lines. There was certainly an alternative, \xypic, for those who wanted something better. But it was hard to learn and I was not entirely happy with the results. The basic interface used an \verb+\halign+. This meant that no extra space was allotted to an arrow that had a large label. In addition, the different slots could have different horizontal size, which could result in a misshapen diagram. I used it in a paper that had a `W' shaped diagram whose nodes had different widths and the result was quite obviously misshapen. On the other hand, the graphics engine underlying \xypic\ is really quite remarkable and it occurred to me that I could try to redraft diagram as a front end. The result is the current package. It has been tested only with version 3.7, so there is no guarantee it would work with any earlier version (or, for that matter, later). The syntax described below is not compatible with the original diagram package. Every front end of this sort represents a trade-off between simplicity and utility. A package that simply upgraded the syntax to allow more dashed (and dotted) lines would have been just as easy to implement, but would have made poor use of the wonderful possibilities of the underlying \xypic\ package. Also there would have been too many different arrow specifications (they would have had to go at least in the range $[-9,9]$) for easy memory) and still would not have included things like inclusion arrows. To those who would have liked a simpler syntax, I apologize. To those who would want more flexibility, I remind that the entire \xypic\ package is there for use. There is one point that cannot be made too strongly. {\em Watch for errant spaces.} Unlike the diagram package, which was carried out in math mode so that spaces were ignored. \xypic\ is not in math mode and spaces will mess up your diagrams. Unlike diagram, it should not be necessary to put any objects inside \verb+\phantom+. In diagram, for reasons that I now understand, objects did not always register properly. This was a flaw built in to the very heart of the package and is not worth correcting, although it could have been done better in the first place. If you see an object in double vision, then the virtually certain cause is that a space has gotten in there somewhere. I have attempted to prevent this by liberal use of \verb+\ignorespaces+ in the definitions, but one cannot always be sure and while testing, I found a number seemingly innocuous spaces that messed up diagrams. \subsection*{Differences from the previous version} \begin{enumerate} \item \verb+\scalefactor{decimal number}+ changes all dimensions by the factor. \item New macros \verb+\Square, \hSquares, \vSquares+ that calculate their own widths. \item New macro \verb+\cube+. \item Internal macro \verb+\label+ that conflicted with \LaTeX\ renamed. \item Added certain procedures designed for inline and displayed equations that are not full-fledged diagrams. These also calculate the length of the arrow to contain the label. \item Corrected certain spacing anomolies. \item Now does work with plain \TeX. \end{enumerate} \section*{The basic syntax} The basic syntax is built around an operation \verb+\morphism+ that is used as \begin{verbatim} \morphism(x,y)|label pos|/{shape}/[N`N`A] \end{verbatim} The parameters \verb+x+ and \verb+y+ give the location of the source node within a fixed local coordinate system and \verb-x+dx- and \verb-y+dy- and locate the target. To be precise, the first coordinate is the horizontal center of the node and the second is that of the base line. These distances are given in terms of \verb+\ul+'s, (for unitlength), which is user assignable. I have, for the time being, set it to .01em, which is probably too small. In the original version of diagram, it was .1em. This was reasonable for diagrams, but since the multipliers had to be integers, it turned out to inconveniently large for certain internal computations. (Strictly speaking, the multipliers could be any decimal number, but \TeX\ does not permit arithmetic with any but whole numbers.) Since all the internal computations are carried out by \xypic, this is no longer a consideration and in the production version, I may go back to the original .1em. Note that by basing distances on ems, I make things automatically scale when changing point size. The parameter \verb+label pos+ is one of \verb+a,b,l,r,m+ that stand for above, below, left, right, or mid and describe the positioning of the arrow label on the arrow. The label \verb+m+ stands for a label positioned in the middle of a split arrow. Although \verb+a+ and \verb+b+ can be used in any direction, they give uncertain results when used on vertical arrows and similarly \verb+l+ and \verb+r+ should not be used on horizontal arrows. The \verb+shape+ describes the shape of the arrow. An arrow is thought of as consisting of three parts, the tail, the shaft and the head. You may specify just the head, in which case the shaft will be an ordinary line, or all three. However, since the tail can be (and usually is) empty, in practice you can also describe the shaft and tail. In addition, it is possible to modify the arrow in various ways. Although the parameter is shown within braces, the braces can be omitted unless one of the modifier characters is \verb+/+, in which case, {\em the entire parameter} must be put in braces. It is important to note that it will not work just to put the \verb+/+ inside the braces, since this will interfere with the internal parsing of \xypic. The head and tail shapes are basically one of \verb+>+, \verb+>>+, \verb+(+, \verb+)+, \verb*+ >+, and \verb*+< +. Each of these may also be preceded by \verb+^+ or \verb+_+ and others are user definable. For details, see the \xypic\ reference manual. The first of these is an ordinary head, while the second is for an epic arrow. The third is not much used, but the superscripted version makes and inclusion tail, as will be illustrated below. The reverse ones give reversed arrowheads. The sign \verb*+ + stands for an obligatory space and it leaves extra space for a tailed (monic) arrow, which otherwise runs into the source node. Although there are many possibilities for shafts, including alphanumeric characters, there are basically three that interest us: \verb+-+, which is an ordinary shaft, \verb+--+, which produces a dashed arrow, \verb+=+, which gives a double arrow (although with only one arrowhead), and \verb+.+, which makes a dotted arrow. Thus \verb+>+ or \verb+->+ will produce an ordinary arrow, \verb+->>+ an epic arrow, \verb*+ >->+ makes a monic arrow, and \verb+-->>+ would make a dashed epic arrow. The descriptions \verb+<-, <<-,+ \verb*+<-< +, and \verb+<<--+ give the reversed versions. Note that \verb+<+ does not give a reversed arrow, since \xypic\ interprets that as a reversed head, not a tail. If the shape parameter begins with an \verb+@+, it is interpreted differently. In that case, it has the form \verb+@{shape}@+ modifier, where the modifier is as described in the \xypic\ reference guide. I just mention a couple of them. The parameter \verb+@{->}@<3pt>+, for example, would give an ordinary arrow moved three points in the direction perpendicular to that of the arrow. If you give \verb+{@{-->}@/^5pt/}+, you will get an epic arrow that is curved in the direction perpendicular to the direction of the arrow by five points. (As far as I can tell, there is no difference between \verb+^-5pt+ and \verb+_5pt+ so the syntax could be simplified and made consistent with that of \verb++ described above.) It is imperative that a specification such as \verb+@{>}@/5pt/+ be enclosed in braces because of the \verb+/+s. Finally, the \verb+N+s are the source and target of the arrow and \verb+L+ is the label. Currently, the \verb+L+ is set in \verb+\scriptstyle+ size, but this can easily be changed by putting \verb+\let\labelstyle=\textstyle+ in your file. \subsection*{A word about parameters} I have already mentioned the necessity of enclosing certain arrow shape specifications in braces. Owing to certain built-in limitations of \TeX, I have used a number of different delimiters: \verb+(, ), |, /, [, ], `, ;+. Any of these that appear inside an argument could conceivably cause problems. They were chosen as the least likely to appear inside mathematics (except for \verb+(, )+ which appear in positions that are unlikely to cause problems). However, be warned that this is a possible cause of mysterious error messages. If it is indeed then enclosing the offending parameter in braces should cure the problem. In most, but not all, cases, enclosing the offending character will solve the problem. The exceptions come when the braces interfere with \xypic's somewhat arcane parsing mechanism. There will be no problem if it is just part of a mathematical expression. \section*{Shapes} Using the basic \verb+\morphism+ macro, I have a defined a number of shapes, squares (really rectangles) and variously oriented triangles. I will probably do a few other shapes as I had done for the old package, such as \verb+\cube+ and a $3 \times 3$ diagram. The basic shapes are exactly the same, but the options are done entirely differently. Here is the syntax of the \verb+\square+ macro: \begin{verbatim} \square(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L`L] \end{verbatim} Each of the first four sets of parameters is optional and any subset of them can be omitted. (Note that only the sets can be omitted, once you give \verb+dx+, you also have to give \verb+dy+ and so on.) The first two describe the horizontal and vertical size of the square, the next four give the label placements using the same five characters as above, the next four give the shapes of the arrows using the same syntax as discussed above. The last group is the location. The meaning of this varies a bit with the shape. For a square, it is the midpoint of the baseline of the lower left corner node. For a triangle that has a lower left corner, it is the same. Otherwise, \verb+x+ is the midpoint of the leftmost node and \verb+y+ is the baseline of the lowest node. As with the old macros, different shapes should fit together accurately, in fact more accurately than with the old ones. Here is a listing of the shapes, together with the groups of parameters. In all cases, the first four groups are optional and any subset of them will work. However, they must come in order. Note that the names of the triangles are related to the shape as the shape that best approximates the shape of the letter. For example, a \verb+\ptriangle+ is a right triangle that has its hypotenuse going from upper right to lower left. Triangles with lower case names have their legs horizontal and vertical, while those with capitalized names have their hypotenuse horizontal or vertical. In those cases, one of \verb+dx+ or \verb+dy+ is the length of a leg and the other is {\em half} the length of the hypotenuse. In all cases, the order of the nodes and of the arrows is linguistic, first moving from left to right and then down. The defaults are reasonable, but with triangles, there is not always a natural direction for arrows. I always made mistakes in the order with my macros and this is certainly a defect. But the order is the same. In every case the braces around the shape specification can be removed unless it includes the following delimiter (that is, \verb+`+ or \verb+/+, as the case may be.) \begin{verbatim} \square(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L] \ptriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \qtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \dtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \btriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Atriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Vtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Ctriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Dtriangle(x,y)|lp`lp`lp|/{sh}`{sh}`{sh}/[N`N`N;L`L`L] \Atrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \Vtrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \Ctrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \Dtrianglepair(x,y)|lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N;L`L`L`L`L] \end{verbatim} (Of course, the \verb+%+ signs are required if you break the macro at such points.) To make a diagram, you have to enclose it inside \verb+\xy ... \endxy+. You will almost always want it displayed, for which the simplest is to enclose it in \verb+$$\xy ... \endxy$$+. For old times sake, I have also let \verb+\bfig+ and \verb+\efig+ be synonyms for \verb+\xy+ and \verb+\endxy+, resp. (In case you wonder, these go all the back to a main frame formatter running at McGill when Charles Wells and I were first writing TTT, where \verb+.BFIG+ and \verb+.EFIG+ were used to make displays, at least as far as that primitive formatter was capable of.) You could instead make definitions as follows: \begin{verbatim} \def\bfig{$$\xy} \def\efig{\endxy$$} \end{verbatim} in which case you will automatically get a display. I do not do this because my spelling checker is programmed to ignore stuff in math or display mode and the \verb+$$+ marks the display mode. Besides which, I have defined the \verb+Ctl-F+ key on my editor so that pressing it gives \begin{verbatim} $$\bfig _ \efig$$ \end{verbatim} with the cursor at the position marked by the underline. \section{Examples} Many people---including me--- learn mainly by example and I will give a number of examples here. The formal syntax can be learned in the \xypic\ reference manual. We begin with \begin{verbatim} $$\bfig \square[A`B`C`D;e`f`g`m] \efig$$ \end{verbatim} produces $$\bfig \square[A`B`C`D;e`f`g`m] \efig$$ This can be modified, for example \begin{verbatim} $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \efig$$ \end{verbatim} produces $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \efig$$ This can be put together with a morphism as follows: \begin{verbatim} $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \morphism(500,500)|m|/.>/<-500,-500>[B`C`h] \efig$$ \end{verbatim} which makes a familiar diagram: $$\bfig \square/>>`>`>` >->/[A`B`C`D;e`f`g`m] \morphism(500,500)|m|/.>/<-500,-500>[B`C`h] \efig$$ The same diagram could have been made by \begin{verbatim} $$\bfig \ptriangle|a`l`m|/>>`>`.>/[A`B`C;e`f`h] \dtriangle/`>` >->/[B`C`D;`g`m] \efig$$ \end{verbatim} You can fit two squares together, horizontally: \begin{verbatim} $$\bfig \square|a`l`m`b|[A`B`C`D;f`g`h`k] \square(500,0)/>``>`>/[B`E`D`F;l``m`n] \efig$$ \end{verbatim} $$\bfig \square|a`l`m`b|[A`B`C`D;f`g`h`k] \square(500,0)/>``>`>/[B`E`D`F;l``m`n] \efig$$ or vertically \begin{verbatim} $$\bfig \square(0,500)|a`l`r`m|[A`B`C`D;f`g`h`k] \square/`>`>`>/[C`D`E`F;`l`m`n] \efig$$ \end{verbatim} $$\bfig \square(0,500)|a`l`r`m|[A`B`C`D;f`g`h`k] \square/`>`>`>/[C`D`E`F;`l`m`n] \efig$$ or a square and a triangle \begin{verbatim} $$\bfig \Ctriangle/<-`>`>/<400,400>[\hbox{\rm rec}(A,B)`B`X;r_0(A,B)`f`t_0] \square(400,0)/<-``>`<-/<1000,800>[\hbox{\rm rec}(A,B)`A\times\hbox{\rm rec}(A,B)`X`A\times X;r(A,B)``\hbox{\rm id}_A\times f`t] \efig$$ \end{verbatim} gives the diagram $$\bfig \Ctriangle/<-`>`>/<400,400>[\hbox{\rm rec}(A,B)`B`X;r_0(A,B)`f`t_0] \square(400,0)/<-``>`<-/<1000,800>[\hbox{\rm rec}(A,B)`A\times\hbox{\rm rec}(A,B)`X`A\times X;r(A,B)``\hbox{\rm id}_A\times f`t] \efig$$ This diagram is on page 361 of the third edition of Category Theory for Computing Science to describe recursion. Here is an example using the procedure for sliding an arrow sideways. This one could even be used in a text, \xy \morphism(0,0)|a|/@{>}@<3pt>/<400,0>[A`B`d] \morphism(0,0)|b|/@{>}@<-3pt>/<400,0>[A`B`e]\endxy which was made using \begin{verbatim} \xy \morphism(0,0)|a|/@{>}@<3pt>/<400,0>[A`B`d] \morphism(0,0)|b|/@{>}@<-3pt>/<400,0>[A`B`e]\endxy \end{verbatim} Later we will introduce a number of inline procedures. Something a bit different that illustrates the use of another shaft \verb+=+ that gives a 2-arrow, as well as curved arrows: \begin{verbatim} $$\bfig \morphism(0,0)|a|/{@{>}@/^1em/}/<500,0>[A`B`f] \morphism(0,0)|b|/{@{>}@/_1em/}/<500,0>[A`B`g] \morphism(250,50)|a|/=>/<0,-100>[``] \efig$$ \end{verbatim} $$\bfig \morphism(0,0)|a|/{@{>}@/^1em/}/<500,0>[A`B`f] \morphism(0,0)|b|/{@{>}@/_1em/}/<500,0>[A`B`g] \morphism(250,50)|a|/=>/<0,-100>[``] \efig$$ In order to use curved arrows, you must insert \verb+\xyoption{curve}+ into your file. Here are two ways of doing three arrows between two objects, depending on what you like: \begin{verbatim} $$\bfig \morphism(0,0)|a|/@{>}@<5pt>/<500,0>[A`B`f] \morphism(0,0)|m|/@{>}/<500,0>[A`B`g] \morphism(0,0)|b|/@{>}@<-5pt>/<500,0>[A`B`h] \efig$$ \end{verbatim} which gives $$\bfig \morphism(0,0)|a|/@{>}@<5pt>/<500,0>[A`B`f] \morphism(0,0)|m|/@{>}/<500,0>[A`B`g] \morphism(0,0)|b|/@{>}@<-5pt>/<500,0>[A`B`h] \efig$$ and \begin{verbatim} $$\bfig \morphism(0,0)|a|/{@{>}@/^5pt/}/<500,0>[A`B`f] \morphism(0,0)|m|/@{>}/<500,0>[A`B`g] \morphism(0,0)|b|/{@{>}@/^-5pt/}/<500,0>[A`B`h] \efig$$ \end{verbatim} which gives $$\bfig \morphism(0,0)|a|/{@{>}@/^5pt/}/<500,0>[A`B`f] \morphism(0,0)|m|/@{>}/<500,0>[A`B`g] \morphism(0,0)|b|/{@{>}@/^-5pt/}/<500,0>[A`B`h] \efig$$ Either of these could also be used inline. There is a macro \verb+\place(x,y)[N]+ where \verb+N+ is any object that places that object anywhere. I have changed the name in order to avoid clashing with the \LaTeX\ picture mode's \verb+\put+. Here is an example that uses a construction that is undocumented here, but uses a documented \Xy\ construction: \begin{verbatim} \newbox\anglebox \setbox\anglebox=\hbox{\xy \POS(75,0)\ar@{-} (0,0) \ar@{-} (75,75)\endxy} \def\angle{\copy\anglebox} $$\bfig \square[A`B`C`D;f`g`h`k] \place(100,400)[\angle] \efig$$ \end{verbatim} \newbox\anglebox \setbox\anglebox=\hbox{\xy \POS(75,0)\ar@{-} (0,0) \ar@{-} (75,75)\endxy} \def\angle{\copy\anglebox} $$\bfig \square[A`B`C`D;f`g`h`k] \place(100,400)[\angle] \efig$$ Notice that you get a headless arrow by using \verb+\ar@{-}+. Here is a special code installed at the request of Jonathon Funk: \begin{verbatim} $$\bfig \pullback|b`r`r`b|<800,800>[P`X`Y`Z;t`u`v`w]% |a`m`b|/>`-->`>/<500,500>[A;f`g`h] \efig$$ \end{verbatim} $$\bfig \pullback|b`r`r`b|<800,800>[P`X`Y`Z;t`u`v`w]% |a`m`b|/>`-->`>/<500,500>[A;f`g`h] \efig$$ The full syntax for this is \begin{verbatim} \pullback(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L]% |lp`lp`lp|/{sh}`{sh}`{sh}/[N;L`L`L] \end{verbatim} Of these only the nodes placed inside brackets are obligatory. The first sets of parameters are exactly as for \verb+\square+ and the remaining parameters are for the nodes and labels of the outer arrows. There is no positioning parameters for them; rather you set the horizontal and vertical separations of the outer node from the square. Here are some more special constructions. In general, if you are doing a square, you should use \verb+\Square+ instead of \verb+\square+ because if figures its own width. The syntax is almost the same, except that \verb+dx+ is omitted. For example, \begin{verbatim} $$\bfig \Square/^{ (}->`>`>`^{ (}->/<350>[{\rm Hom}(A,2^B)`{\rm Sub}(A\times B)` {\rm Hom}(A',2^{B'})`{\rm Sub}(A'\times B');\alpha(A,B)```\alpha(A',B')] \efig$$ \end{verbatim} will produce the square $$\bfig \Square/^{ (}->`>`>`^{ (}->/<350>[{\rm Hom}(A,2^B)`{\rm Sub}(A\times B)` {\rm Hom}(A',2^{B'})`{\rm Sub}(A'\times B');\alpha(A,B)```\alpha(A',B')] \efig$$ There are a couple of points to note here. Note the use of the argument \verb+^{ (}->+ to get the inclusion arrow. The complication is created by the necessity of adding a bit of extra space before the hook. You get pretty much the same effect by putting a bit of extra space after the node: \begin{verbatim} $$\bfig \Square/^(->`>`>`^(->/<350>[{\rm Hom}(A,2^B)\,`{\rm Sub}(A\times B)` {\rm Hom}(A',2^{B'})\,`{\rm Sub}(A'\times B');\alpha(A,B)```\alpha(A',B')] \efig$$ \end{verbatim} The full syntax is \begin{verbatim} \Square(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L`L] \end{verbatim} There are also macros for placing two \verb+\Square+s together horizontally or vertically. The first is \verb+\hSquares+ with the syntax \begin{verbatim} \hSquares(x,y)|lp`lp`lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N`N`N;L`L`L`L`L`L`L] \end{verbatim} The second is \verb+\vSquares+ with a similar syntex except that there are two \verb+dy+ parameters, one for each square: \begin{verbatim} \hSquares(x,y)|lp`lp`lp`lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}`{sh}`{sh}`{sh}/% [N`N`N`N`N`N;L`L`L`L`L`L`L] \end{verbatim} There is a macro for making cubes. The syntax is \begin{verbatim} \cube(x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L`L]% (x,y)|lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[N`N`N`N;L`L`L`L]% |lp`lp`lp`lp|/{sh}`{sh}`{sh}`{sh}/[L`L`L`L] \end{verbatim} The first line of parameters is for the outer square and the second for the inner square, while the remaining parameters are for the arrows between the squares. Only the parameters in square brackets are required; there are defaults for the others. Here is an example: \begin{verbatim} $$\bfig \cube(0,0)|a`r`l`b|/ >->` >->`>`>/<1500,1500>[A`B`C`D;f`g`h`k]% (300,300)|a`r`l`b|/>`>`>`>/<400,400>[A'`B'`C'`D';f'`g'`h'`k']% |m`m`m`m|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta] \efig$$ \end{verbatim} gives the somewhat misshapen diagram $$\bfig \cube(0,0)|a`r`l`b|/ >->` >->`>`>/<1500,1500>[A`B`C`D;f`g`h`k]% (300,300)|a`r`l`b|/>`>`>`>/<400,400>[A'`B'`C'`D';f'`g'`h'`k']% |m`m`m`m|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta] \efig$$ because the parameters were oddly chosen. The defaults center the squares. I discovered accidently, while debugging the cube that what I though was an out-of-range choice of parameters would produce an offset cube: \begin{verbatim} $$\bfig \cube|a`r`l`b|/ >->` >->`>`>/<1000,1000>[A`B`C`D;f`g`h`k]% (400,400)|a`r`l`b|/>`>`>`>/<900,900>[A'`B'`C'`D';f'`g'`h'`k']% |r`r`r`r|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta] \efig$$ \end{verbatim} gives $$\bfig \cube|a`r`l`b|/ >->` >->`>`>/<1000,1000>[A`B`C`D;f`g`h`k]% (400,400)|a`r`l`b|/>`>`>`>/<900,900>[A'`B'`C'`D';f'`g'`h'`k']% |r`r`r`r|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta] \efig$$ \subsection*{Empty placement and moving labels} The label placements within \verb+|x|+ is valid only for \verb+x = a,b,r,l,m+. If you use any other value, or leave it empty, the label entry is ignored, but you can use any valid \xypic\ label, as described in Figure 13 of the reference manual. One place you might want to use this is for the placement of the labels along an arrow. In \xypic\ the default placement of the label is midway between the midpoints of the nodes. If the two nodes are of widely different sizes, this can result in strange placements; therefore I always place them midway along the arrow. However, as the following illustrates, this can be changed. \begin{verbatim} $$\bfig \morphism(0,900)||/@{->}^<>(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \morphism(0,600)||/@{->}^<(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \morphism(0,300)||/@{->}^>(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \morphism(0,0)||/@{->}^(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \efig$$ \end{verbatim} which produces $$\bfig \morphism(0,900)||/@{->}^<>(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \morphism(0,600)||/@{->}^<(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \morphism(0,300)||/@{->}^>(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \morphism(0,0)||/@{->}^(0.7){f}/<800,0>[A^B\times B^C\times C`C`] \efig$$ Here is the explanation. The label placement argument is empty (it cannot be omitted) and the arrow entry is empty. However, placing \verb+^(0.7){f}+ inside the arrow shape places the label \verb+f+ 7/10 of the way between the nodes. Unmodified, this places it 7/10 of the way between the centers of the nodes. This may be modified by \verb+<+, which moves the first (here the left) reference point to the beginning of the arrow, \verb+>+ which moves the second reference point to the end of the arrow, or by both, which moves both reference points. In most cases, you will want both. Incidentally, \verb+-+ is a synonym for the sequence \verb+<>(.5)+ and that is the default placement in my package. Here are some more examples that illustrates the special sequence \verb+\hole+ used in conjunction with \verb+|+ that implements \verb+m+ as well as the fact that these things can be stacked. For more details, I must refer you to the reference manual. \begin{verbatim} $$\bfig \morphism(0,600)||/@{->}|-\hole/<800,0>[A^B\times B^C\times C`C`] \morphism(0,300)||/@{->}|-\hole^<>(.7)f/<800,0>[A^B\times B^C\times C`C`] \morphism(0,0)||/@{->}|-\hole^<>(.7)f_<>(.3)g/<800,0>[A^B\times B^C\times C`C`] \efig$$ \end{verbatim} produces $$\bfig \morphism(0,600)||/@{->}|-\hole/<800,0>[A^B\times B^C\times C`C`] \morphism(0,300)||/@{->}|-\hole^<>(.7)f/<800,0>[A^B\times B^C\times C`C`] \morphism(0,0)||/@{->}|-\hole^<>(.7)f_<>(.3)g/<800,0>[A^B\times B^C\times C`C`] \efig$$ Here is another version of the cube we looked at above, using these special placements and \verb+\hole+'s to break some lines and make it neater. \begin{verbatim} $$\bfig \cube|a`r`l`b|/@{ >->}^<>(.6){f}` >->`@{>}_<>(.4){h}`>/% <1000,1000>[A`B`C`D;`g``k]% (400,400)|a```b|/>`@{>}|(.33)\hole^<>(.6){g'}`>`@{>}% |(.67)\hole_<>(.4){k'}/<900,900>[A'`B'`C'`D';f'``h'`]% |r`r`r`r|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta] \efig$$ \end{verbatim} $$\bfig \cube|a`r`l`b|/@{ >->}^<>(.6){f}` >->`@{>}_<>(.4){h}`>/% <1000,1000>[A`B`C`D;`g``k]% (400,400)|a```b|/>`@{>}|(.33)\hole^<>(.6){g'}`>`% @{>}|(.67)\hole_<>(.4){k'}/<900,900>[A'`B'`C'`D';f'``h'`]% |r`r`r`r|/<-`<-`<-`<-/[\alpha`\beta`\gamma`\delta] \efig$$ This one is probably worth saving as a template. Note that the \verb+\hole+'s by reference to the center of the nodes. That way they will always miss hitting the arrows no matter what the size of the node. If the nodes are unusually large, the cube may be magnified using \verb+\scalefactor+. \subsection*{Inline macros} Here we illustrate a few of the macros for inline---or displayed---equations the package contains. In each case, the macro may have a superscript or subscript or both (in which case the superscript must come first) and the arrow(s) grow long enough to hold the super- or subscript. If you type \verb+$A\to B\to^f C\to_g D\to^h_{{\rm Hom}(X,Y)} E$+, you get $A\to B\to^f C\to_g D\to^h_{{\rm Hom}(X,Y)} E$. Similarly, the macro \verb+\toleft+ reverses the arrows. The remaining macros of this sort are \verb+\mon+ which gives a monic arrow, \verb+\epi+ which gives an epic arrow, \verb+\two+ that gives a pair of arrows, as well as leftwards pointing versions, \verb+\monleft+, \verb+\epileft+, and \verb+twoleft+ of each of them. Here is one more example: \begin{verbatim} $A\twoleft B\twoleft^f C\twoleft_g D\twoleft^h_{{\rm Hom}(X,Y)} E$ \end{verbatim} gives $A\twoleft B\twoleft^f C\twoleft_g D\twoleft^h_{{\rm Hom}(X,Y)} E$. There is an almost unlimited variety of such procedures possible. The ones that are provided can be used as templates to define new ones with, say, curved arrows or three arrows or whatever a user might have need of. \subsection*{2-arrows} There is a macro for making 2-arrows of a fixed size, but varying orientation. They should be put at the appropriate position in a diagram. The size actually varies with the direction by about 10\% because there is no simple way of implementing the sin$\circ$arctan and cos$\circ$arctan functions and I have used rational approximating functions, but the variation seems acceptable. The worst cases are those of slopes $\pm3$ and $\pm1/3$, which are about 5\% too short, and $\pm1$, which are about 5\% too long. These two cases are illustrated below. The two parameters are two integers \verb+dx+ and \verb+dy+ whose ratio is the slope of the arrow. They need not be relatively prime, but arithmetic overflow could occur if they were too large. Here is a sampler \begin{verbatim} $$\bfig \place(0,0)[\twoar(1,0)] \place(200,0)[\twoar(0,1)] \place(400,0)[\twoar(1,1)] \place(600,0)[\twoar(0,-1)] \place(800,0)[\twoar(1,2)] \place(1000,0)[\twoar(1,3)] \place(1200,0)[\twoar(1,-3)] \place(1400,0)[\twoar(-3,1)] \place(1600,0)[\twoar(-1,-3)] \place(1800,0)[\twoar(255,77)] \efig$$ \end{verbatim} $$\bfig \place(0,0)[\twoar(1,0)] \place(200,0)[\twoar(0,1)] \place(400,0)[\twoar(1,1)] \place(600,0)[\twoar(0,-1)] \place(800,0)[\twoar(1,2)] \place(1000,0)[\twoar(1,3)] \place(1200,0)[\twoar(1,-3)] \place(1400,0)[\twoar(-3,1)] \place(1600,0)[\twoar(-1,-3)] \place(1800,0)[\twoar(255,77)] \efig$$ \subsection*{Diagram from TTT} The last example is a complicated diagram from TTT: \begin{verbatim} $$\bfig \morphism(0,2100)|l|/>/<0,-1400>[TT`TT'T`T\eta'T] \morphism(0,2100)|a|/>/<700,0>[TT`T`\mu] \morphism(0,2100)|l|/>/<700,-700>[TT`TTT'`TT\eta'] \morphism(700,2100)|r|/>/<700,-700>[`TT'`T\eta] \place(700,1750)[1] \morphism(700,1400)|b|/@<-20\ul>/<700,0>[TTT'`TT'`T\sigma] \square(700,700)|a`m`m`a|/@<20\ul>`>``>/<700,700>[TTT'` TT'`TT'TT'`TT'T';\mu T' `T\eta'TT'``] \morphism(700,700)|a|/>/<700,0>[TT'TT'`TT'T'`TT'\sigma] \place(300,1400)[2] \place(1050,1050)[3] \btriangle(1400,700)|m`r`a|/>`>`>/<700,700>[TT'`TT'T'`TT'; T\eta'T'`\hbox{\rm id}`] \place(1600,1050)[6] \square|a`l`m`b|<700,700>[TT'T`TT'TT'`T'T`T'TT'; TT'T\eta'`\sigma T`\sigma TT'`T'T\eta'] \square(1400,0)|a`m`r`b|<700,700>[TT'T'`TT'`T'T'`T'; T\mu'`\sigma T' `\sigma`\mu'] \morphism(700,0)|b|/>/<700,0>[T'TT'`T'T'`T'\sigma T'] \place(300,350)[4] \place(1050,350)[5] \place(1750,350)[7] \efig$$ \end{verbatim} $$\bfig \morphism(0,2100)|l|/>/<0,-1400>[TT`TT'T`T\eta'T] \morphism(0,2100)|a|/>/<700,0>[TT`T`\mu] \morphism(0,2100)|l|/>/<700,-700>[TT`TTT'`TT\eta'] \morphism(700,2100)|r|/>/<700,-700>[`TT'`T\eta] \place(700,1750)[1] \morphism(700,1400)|b|/@<-20\ul>/<700,0>[TTT'`TT'`T\sigma] \square(700,700)|a`m`m`a|/@<20\ul>`>``>/<700,700>[TTT'` TT'`TT'TT'`TT'T';\mu T' `T\eta'TT'``] \morphism(700,700)|a|/>/<700,0>[TT'TT'`TT'T'`TT'\sigma] \place(300,1400)[2] \place(1050,1050)[3] \btriangle(1400,700)|m`r`a|/>`>`>/<700,700>[TT'`TT'T'`TT'; T\eta'T'`\hbox{\rm id}`] \place(1600,1050)[6] \square|a`l`m`b|<700,700>[TT'T`TT'TT'`T'T`T'TT'; TT'T\eta'`\sigma T`\sigma TT'`T'T\eta'] \square(1400,0)|a`m`r`b|<700,700>[TT'T'`TT'`T'T'`T'; T\mu'`\sigma T' `\sigma`\mu'] \morphism(700,0)|b|/>/<700,0>[T'TT'`T'T'`T'\sigma T'] \place(300,350)[4] \place(1050,350)[5] \place(1750,350)[7] \efig$$ \end{document} From rrosebru@mta.ca Sun Apr 22 20:30:36 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3MMjMP01649 for categories-list; Sun, 22 Apr 2001 19:45:22 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Sun, 22 Apr 2001 08:37:44 -0400 (EDT) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: Categories list Subject: categories: diagxy.tex Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 22 Appended is the latest version. It is, I hope, very close to the final version. I have gotten almost no feedback from the first version. I encourage you to try it and let me know any problems or questions. The next email will be the revised version of the documentation. %%%%%%%%%%%%%%%%%%%%%%%%%% diagxy.tex %%%%%%%%%%%%%%% % This is a package of commutative diagram macros built on top of Xy-pic % by Michael Barr (email: barr@barrs.org). This may be freely % distributed, unchanged, for non-commercial use. If changed, it must be % renamed. Inclusion in a commercial software package is also % permitted, provided I receive one free copy of the software package for % my personal use. There are no guarantees that this package is good for % anything. I have tested it with LaTeX 2e, LaTeX 2.09 and Plain TeX. % Although I know of no reason it will not work with AMSTeX, I have not % tested it. \input xy \xyoption{arrow} \newdir{ >}{{ }*!/-9pt/@{>}} \newdir{ (}{{ }*!/-5pt/@{(}} \newdir^{ (}{{ }*!/-5pt/@^{(}} \newdir{< }{!/9pt/@{<}*{ }} \newbox\Label% \newdimen\high% \newdimen\deep% \newdimen\ul% \newcount\deltax% \newcount\deltay% \newcount\deltaX% \newcount\deltaY% \newdimen\wdth \newcount\xend% \newcount\yend% \newcount\Xend \newcount\Yend \newcount\xpos% \newcount\ypos% \newcount\default \default=500% \newcount\defaultmargin \defaultmargin=200% \newif\ifat% \newcount\topw% \newcount\botw% \newcount\Xpos% \newcount\Ypos% \def\ratchet#1#2{\ifnum#1<#2\global #1=#2\fi}% \newcount\atcode \atcode=\catcode`\@% \catcode`\@=11 \expandafter\ifx\csname @ifnextchar\endcsname\relax \def\ifnextchar#1#2#3{\let\@tempe #1\def\@tempa{#2}\def\@tempb{#3}\futurelet \@tempc\@ifnch} \def\@ifnch{\ifx \@tempc \@sptoken \let\@tempd\@xifnch \else \ifx \@tempc \@tempe\let\@tempd\@tempa\else\let\@tempd\@tempb\fi \fi \@tempd} \def\:{\let\@sptoken= } \: % this makes \@sptoken a space token \def\:{\@xifnch} \expandafter\def\: {\futurelet\@tempc\@ifnch} \else \let\ifnextchar\@ifnextchar \fi \ul=.01em% \X@xbase =.01em% \Y@ybase =.01em% \def\scalefactor#1{\ul=#1\ul \X@xbase=#1\X@xbase \Y@ybase=#1\Y@ybase}% \catcode`\@=12% \let\bfig\xy% \let\efig\endxy% \def\car#1#2\nil{#1}% \def\morphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]{% \xend#1\advance \xend by #5% \yend#2\advance \yend by #6% \domorphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]} \def\domorphism(#1,#2)|#3|/#4/<#5,#6>[#7`#8`#9]{% \edef\next{#4}% \ifx\next\empty \atfalse \else \def\next{\car#4\nil}% \if\next@\relax\attrue \else \atfalse \fi \fi% \ifat% \edef\next{#4}% \ifx\next\empty% \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}% \else \if#3l% \ifnum #6>0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3m% \edef\next{#9}% \ifx\next\empty \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}% \else \setbox\Label=\hbox{\kern .5pt $\labelstyle #9$\kern .5pt}% \high=\ht\Label \advance\high by 1pt \ht\Label=\high% \deep=\dp\Label \advance\deep by 1pt \dp\Label=\deep% \POS(#1,#2)*+{#7}\ar#4|-{\box\Label} (\xend,\yend)*+{#8}% \fi \else \if#3r% \ifnum #6<0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3a% \ifnum #5>0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3b% \ifnum #5<0% \POS(#1,#2)*+{#7}\ar#4^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar#4_-{#9} (\xend,\yend)*+{#8}% \fi% \else \POS(#1,#2)*+{#7}\ar#4 (\xend,\yend)*+{#8}% \fi\fi\fi\fi\fi\fi% \else% \edef\next{#4}% \ifx\next\empty% \POS(#1,#2)*+{#7}; (\xend,\yend)*+{#8}% \else \edef\next{#4}% \ifx\next\empty% \POS(#1,#2)*+{#7}\ar (\xend,\yend)*+{#8}% \else \if#3l% \ifnum #6>0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3m% \edef\next{#9}% \ifx\next\empty \POS(#1,#2)*+{#7}\ar@{#4} (\xend,\yend)*+{#8}% \else \setbox\Label=\hbox{\kern .5pt $\labelstyle #9$\kern .5pt}% \high=\ht\Label \advance\high by 1pt \ht\Label=\high% \deep=\dp\Label \advance\deep by 1pt \dp\Label=\deep% \POS(#1,#2)*+{#7}\ar@{#4}|-{\box\Label} (\xend,\yend)*+{#8}% \fi \else \if#3r% \ifnum #6<0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3a% \ifnum #5>0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \if#3b% \ifnum #5<0% \POS(#1,#2)*+{#7}\ar@{#4}^-{#9} (\xend,\yend)*+{#8}% \else% \POS(#1,#2)*+{#7}\ar@{#4}_-{#9} (\xend,\yend)*+{#8}% \fi% \else \POS(#1,#2)*+{#7}\ar@{#4} (\xend,\yend)*+{#8}% \fi\fi\fi\fi\fi\fi\fi% \fi\ignorespaces}% \def\location(#1,#2){\xpos=#1\ypos=#2\ignorespaces}% \def\squareplacements|#1`#2`#3`#4|{% \def\xa{#1}\def\xb{#2}\def\xc{#3}\def\xd{#4}\ignorespaces}% \def\sizes<#1,#2>{\deltax=#1\deltay=#2\ignorespaces}% \def\squaredata[#1`#2`#3`#4;#5`#6`#7`#8]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}% \def\labela{#5}\def\labelb{#6}\def\labelc{#7}\def\labeld{#8}\ignorespaces}% \def\squaredata[#1`#2`#3`#4;#5]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}% \edef\next{#5}% \ifx\next\empty \def\labela{}\def\labelb{}\def\labelc{}\def\labeld{}% \else\gdef\next[##1`##2`##3`##4]{% \def\labela{##1}\def\labelb{##2}\def\labelc{##3}\def\labeld{##4}}% \next[#5]% \fi \ignorespaces}% \def\squarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{% \location(#1)% \squareplacements|#2|% \sizes<#7>% \squaredata[#8]% \morphism(\xpos,\ypos)|\xd|/#6/<\deltax,0>[\nodec`\noded`\labeld]% \advance \ypos by \deltay% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \advance \xpos by \deltax% \morphism(\xpos,\ypos)|\xc|/#5/<0,-\deltay>[\nodeb`\noded`\labelc]% \ignorespaces}% \def\square{\ifnextchar({\squarep}{\squarep(0,0)}}% \def\squarep(#1){\ifnextchar|{\squarepp(#1)}{\squarepp(#1)|a`l`r`b|}}% \def\squarepp(#1)|#2|{\ifnextchar/{\squareppp(#1)|#2|}% {\squareppp(#1)|#2|/>`>`>`>/}}% \def\squareppp(#1)|#2|/#3`#4`#5`#6/{% \ifnextchar<{\squarepppp(#1)|#2|/#3`#4`#5`#6/}% {\squarepppp(#1)|#2|/#3`#4`#5`#6/<\default,\default>}}% \def\triplacements|#1`#2`#3|{\def\xa{#1}\def\xb{#2}\def\xc{#3}}% \def\tridata[#1`#2`#3;#4`#5`#6]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}% \def\labela{#4}\def\labelb{#5}\def\labelc{#6}}% \def\ptrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc]% \ignorespaces}% \def\ptriangle{\ifnextchar({\ptrianglep}{\ptrianglep(0,0)}}% \def\ptrianglep(#1){\ifnextchar|{\ptrianglepp(#1)}{\ptrianglepp(#1)|a`l`r|}}% \def\ptrianglepp(#1)|#2|{\ifnextchar/{\ptriangleppp(#1)|#2|}% {\ptriangleppp(#1)|#2|/>`>`>/}}% \def\ptriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\ptrianglepppp(#1)|#2|/#3`#4`#5/}% {\ptrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\qtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xc|/#5/<0,-\deltay>[\nodeb`\nodec`\labelc]% \ignorespaces}% \def\qtriangle{\ifnextchar({\qtrianglep}{\qtrianglep(0,0)}}% \def\qtrianglep(#1){\ifnextchar|{\qtrianglepp(#1)}{\qtrianglepp(#1)|a`l`r|}}% \def\qtrianglepp(#1)|#2|{\ifnextchar/{\qtriangleppp(#1)|#2|}% {\qtriangleppp(#1)|#2|/>`>`>/}}% \def\qtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\qtrianglepppp(#1)|#2|/#3`#4`#5/}% {\qtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\dtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \advance\ypos by \deltay\advance \xpos by \deltax% \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces}% \def\dtriangle{\ifnextchar({\dtrianglep}{\dtrianglep(0,0)}}% \def\dtrianglep(#1){\ifnextchar|{\dtrianglepp(#1)}{\dtrianglepp(#1)|l`r`b|}}% \def\dtrianglepp(#1)|#2|{\ifnextchar/{\dtriangleppp(#1)|#2|}% {\dtriangleppp(#1)|#2|/>`>`>/}}% \def\dtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\dtrianglepppp(#1)|#2|/#3`#4`#5/}% {\dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\btrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces}% \def\btriangle{\ifnextchar({\btrianglep}{\btrianglep(0,0)}}% \def\btrianglep(#1){\ifnextchar|{\btrianglepp(#1)}{\btrianglepp(#1)|l`r`b|}}% \def\btrianglepp(#1)|#2|{\ifnextchar/{\btriangleppp(#1)|#2|}% {\btriangleppp(#1)|#2|/>`>`>/}}% \def\btriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\btrianglepppp(#1)|#2|/#3`#4`#5/}% {\btrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\Atrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \multiply\deltax by 2% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \divide\deltax by 2 \advance\ypos by \deltay\advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces}% \def\Atriangle{\ifnextchar({\Atrianglep}{\Atrianglep(0,0)}}% \def\Atrianglep(#1){\ifnextchar|{\Atrianglepp(#1)}{\Atrianglepp(#1)|l`r`b|}}% \def\Atrianglepp(#1)|#2|{\ifnextchar/{\Atriangleppp(#1)|#2|}% {\Atriangleppp(#1)|#2|/>`>`>/}}% \def\Atriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Atrianglepppp(#1)|#2|/#3`#4`#5/}% {\Atrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\Vtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]% \multiply\deltax by 2% \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \advance\xpos by \deltax \divide \deltax by 2 \morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc]% \ignorespaces}% \def\Vtriangle{\ifnextchar({\Vtrianglep}{\Vtrianglep(0,0)}}% \def\Vtrianglep(#1){\ifnextchar|{\Vtrianglepp(#1)}{\Vtrianglepp(#1)|a`l`b|}}% \def\Vtrianglepp(#1)|#2|{\ifnextchar/{\Vtriangleppp(#1)|#2|}% {\Vtriangleppp(#1)|#2|/>`>`>/}}% \def\Vtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Vtrianglepppp(#1)|#2|/#3`#4`#5/}% {\Vtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\Ctrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance \ypos by \deltay% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodeb`\nodec`\labelc]% \advance\ypos by \deltay \advance \xpos by \deltax% \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \multiply\deltay by 2% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces}% \def\Ctriangle{\ifnextchar({\Ctrianglep}{\Ctrianglep(0,0)}}% \def\Ctrianglep(#1){\ifnextchar|{\Ctrianglepp(#1)}{\Ctrianglepp(#1)|a`r`b|}}% \def\Ctrianglepp(#1)|#2|{\ifnextchar/{\Ctriangleppp(#1)|#2|}% {\Ctriangleppp(#1)|#2|/>`>`>/}}% \def\Ctriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Ctrianglepppp(#1)|#2|/#3`#4`#5/}% {\Ctrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\Dtrianglepppp(#1)|#2|/#3`#4`#5/<#6>[#7]{% \location(#1)% \triplacements|#2|% \sizes<#6>% \tridata[#7]% \advance\xpos by \deltax \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xc|/#5/<-\deltax,-\deltay>[\nodeb`\nodec`\labelc]% \advance\xpos by -\deltax \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodeb`\labelb]% \multiply \deltay by 2% \morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodec`\labela]% \ignorespaces}% \def\Dtriangle{\ifnextchar({\Dtrianglep}{\Dtrianglep(0,0)}}% \def\Dtrianglep(#1){\ifnextchar|{\Dtrianglepp(#1)}{\Dtrianglepp(#1)|a`l`b|}}% \def\Dtrianglepp(#1)|#2|{\ifnextchar/{\Dtriangleppp(#1)|#2|}% {\Dtriangleppp(#1)|#2|/>`>`>/}}% \def\Dtriangleppp(#1)|#2|/#3`#4`#5/{% \ifnextchar<{\Dtrianglepppp(#1)|#2|/#3`#4`#5/}% {\Dtrianglepppp(#1)|#2|/#3`#4`#5/<\default,\default>}}% \def\tripairplacements|#1`#2`#3`#4`#5|{\def\xa{#1}\def\xb{#2}\def\xc{#3}% \def\xd{#4}\def\xe{#5}}% \def\tripairdata[#1`#2`#3`#4;#5`#6`#7`#8`#9]{% \def\nodea{#1}\def\nodeb{#2}\def\nodec{#3}\def\noded{#4}% \def\labela{#5}\def\labelb{#6}\def\labelc{#7}\def\labeld{#8}\def\labele{#9}}% \def\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \morphism(\xpos,\ypos)|\xd|/#6/<\deltax,0>[\nodeb`\nodec`\labeld]% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xe|/#7/<\deltax,0>[\nodec`\noded`\labele]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodea`\noded`\labelc]% \ignorespaces}% \def\Atrianglepair{\ifnextchar({\Atrianglepairp}{\Atrianglepairp(0,0)}}% \def\Atrianglepairp(#1){\ifnextchar|{\Atrianglepairpp(#1)}% {\Atrianglepairpp(#1)|l`m`r`b`b|}}% \def\Atrianglepairpp(#1)|#2|{\ifnextchar/{\Atrianglepairppp(#1)|#2|}% {\Atrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Atrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Atrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xa|/#3/<\deltax,0>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,-\deltay>[\nodea`\noded`\labelc]% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,0>[\nodeb`\nodec`\labelb]% \morphism(\xpos,\ypos)|\xd|/#6/<0,-\deltay>[\nodeb`\noded`\labeld]% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xe|/#7/<-\deltax,-\deltay>[\nodec`\noded`\labele]% \ignorespaces}% \def\Vtrianglepair{\ifnextchar({\Vtrianglepairp}{\Vtrianglepairp(0,0)}}% \def\Vtrianglepairp(#1){\ifnextchar|{\Vtrianglepairpp(#1)}% {\Vtrianglepairpp(#1)|a`a`l`m`r|}}% \def\Vtrianglepairpp(#1)|#2|{\ifnextchar/{\Vtrianglepairppp(#1)|#2|}% {\Vtrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Vtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Vtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xe|/#7/<0,-\deltay>[\nodec`\noded`\labele]% \advance\xpos by -\deltax% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \morphism(\xpos,\ypos)|\xd|/#6/<\deltax,-\deltay>[\nodeb`\noded`\labeld]% \advance\ypos by \deltay% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xa|/#3/<-\deltax,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<0,-\deltay>[\nodea`\nodec`\labelb]% \ignorespaces}% \def\Ctrianglepair{\ifnextchar({\Ctrianglepairp}{\Ctrianglepairp(0,0)}}% \def\Ctrianglepairp(#1){\ifnextchar|{\Ctrianglepairpp(#1)}% {\Ctrianglepairpp(#1)|l`r`m`l`r|}}% \def\Ctrianglepairpp(#1)|#2|{\ifnextchar/{\Ctrianglepairppp(#1)|#2|}% {\Ctrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Ctrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Ctrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<#8>[#9]{% \location(#1)% \tripairplacements|#2|% \sizes<#8>% \tripairdata[#9]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xc|/#5/<\deltax,0>[\nodeb`\nodec`\labelc]% \morphism(\xpos,\ypos)|\xd|/#6/<0,-\deltay>[\nodeb`\noded`\labeld]% \advance\ypos by \deltay% \morphism(\xpos,\ypos)|\xa|/#3/<0,-\deltay>[\nodea`\nodeb`\labela]% \morphism(\xpos,\ypos)|\xb|/#4/<\deltax,-\deltay>[\nodea`\nodec`\labelb]% \advance\ypos by -\deltay% \advance\xpos by \deltax% \morphism(\xpos,\ypos)|\xe|/#7/<-\deltax,-\deltay>[\nodec`\noded`\labele]% \ignorespaces}% \def\Dtrianglepair{\ifnextchar({\Dtrianglepairp}{\Dtrianglepairp(0,0)}}% \def\Dtrianglepairp(#1){\ifnextchar|{\Dtrianglepairpp(#1)}% {\Dtrianglepairpp(#1)|l`r`m`l`r|}}% \def\Dtrianglepairpp(#1)|#2|{\ifnextchar/{\Dtrianglepairppp(#1)|#2|}% {\Dtrianglepairppp(#1)|#2|/>`>`>`>`>/}}% \def\Dtrianglepairppp(#1)|#2|/#3`#4`#5`#6`#7/{% \ifnextchar<{\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/}% {\Dtrianglepairpppp(#1)|#2|/#3`#4`#5`#6`#7/<\default,\default>}}% \def\place(#1,#2)[#3]{\POS(#1,#2)*+{#3}}% \def\pullback#1]#2]{\square#1]\trident#2]}% \def\tridentdata[#1;#2`#3`#4]{\def\nodee{#1}\def\labele{#2}% \def\labelf{#3}\def\labelg{#4}}% \def\trident{\ifnextchar|{\tridentp}{\tridentp|a`m`b|}}% \def\tridentp|#1|{\ifnextchar/{\tridentpp|#1|}{\tridentpp|#1|/{>}`{>}`{>}/}}% \def\tridentpp|#1|/#2/{\ifnextchar<{\tridentppp|#1|/#2/}% {\tridentppp|#1|/#2/<500,500>}}% \def\tridentppp|#1`#2`#3|/#4`#5`#6/<#7,#8>[#9]{% \tridentdata[#9]% \advance \xpos by -\deltax% \advance \xpos by -#7\advance \ypos by #8% \advance\deltax by #7% \morphism(\xpos,\ypos)|#1|/#4/<\deltax,-#8>[\nodee`\nodeb`\labele]% \advance\deltax by -#7% \morphism(\xpos,\ypos)|#2|/#5/<#7,-#8>[\nodee`\nodea`\labelf]% \advance\deltay by #8% \morphism(\xpos,\ypos)|#3|/#6/<#7,-\deltay>[\nodee`\nodec`\labelg]% \ignorespaces}% \def\setmorphismwidth#1#2#3#4{% \setbox0=\hbox{$#1#3#3#2$}#4=\wd0% \divide #4 by 2 \divide #4 by \ul% \advance #4 by \defaultmargin \ratchet{#4}{500}% }% \def\setSquarewidth[#1`#2`#3`#4;#5`#6`#7`#8]{% \setmorphismwidth{#1}{#2}{#5}{\topw}% \setmorphismwidth{#3}{#4}{#8}{\botw}% \ratchet{\topw}{\botw}}% \def\Squarepppp(#1)|#2|/#3/<#4>[#5]{% \setSquarewidth[#5]% \squarepppp(#1)|#2|/#3/<\topw,#4>[#5]% \ignorespaces}% \def\Square{\ifnextchar({\Squarep}{\Squarep(0,0)}}% \def\Squarep(#1){\ifnextchar|{\Squarepp(#1)}{\Squarepp(#1)|a`l`r`b|}}% \def\Squarepp(#1)|#2|{\ifnextchar/{\Squareppp(#1)|#2|}% {\Squareppp(#1)|#2|/>`>`>`>/}}% \def\Squareppp(#1)|#2|/#3`#4`#5`#6/{% \ifnextchar<{\Squarepppp(#1)|#2|/#3`#4`#5`#6/}% {\Squarepppp(#1)|#2|/#3`#4`#5`#6/<\default>}}% \def\Placements|#1`#2`#3`#4`#5`#6`#7|{% \def\Xa{#1}% \def\Xb{#2}% \def\Xc{#3}% \def\Xd{#4}% \def\Xe{#5}% \def\Xf{#6}% \def\Xg{#7}% }% \def\Nodes[#1`#2`#3`#4`#5`#6]{% \def\Nodea{#1}% \def\Nodeb{#2}% \def\Nodec{#3}% \def\Noded{#4}% \def\Nodee{#5}% \def\Nodef{#6}% }% \def\Labels[#1`#2`#3`#4`#5`#6`#7]{% \def\Labela{#1}% \def\Labelb{#2}% \def\Labelc{#3}% \def\Labeld{#4}% \def\Labele{#5}% \def\Labelf{#6}% \def\Labelg{#7}% }% \def\hSquarespppp(#1,#2)|#3|/#4/<#5>[#6;#7]{% \Xpos=#1\Ypos=#2% \Placements|#3|% \deltaY=#5% \Nodes[#6]% \Labels[#7]% \dohSquares/#4/}% \def\dohSquares/#1`#2`#3`#4`#5`#6`#7/{% \Squarepppp(\Xpos,\Ypos)|\Xa`\Xc`\Xd`\Xf|/#1`#3`#4`#6/<\deltaY>% [\Nodea`\Nodeb`\Noded`\Nodee;\Labela`\Labelc`\Labeld`\Labelf]% \advance \Xpos by \topw \Squarepppp(\Xpos,\Ypos)|\Xb`\Xd`\Xe`\Xg|/#2``#5`#7/<\deltaY>% [\Nodeb`\Nodec`\Nodee`\Nodef;\Labelb``\Labele`\Labelg]% \ignorespaces}% \def\hSquares{\ifnextchar({\hSquaresp}{\hSquaresp(0,0)}}% \def\hSquaresp(#1){\ifnextchar|{\hSquarespp(#1)}{\hSquarespp% (#1)|a`a`l`m`r`b`b|}}% \def\hSquarespp(#1)|#2|{\ifnextchar/{\hSquaresppp(#1)|#2|}% {\hSquaresppp(#1)|#2|/>`>`>`>`>`>`>/}}% \def\hSquaresppp(#1)|#2|/#3/{% \ifnextchar<{\hSquarespppp(#1)|#2|/#3/}% {\hSquarespppp(#1)|#2|/#3/<\default>}}% \def\vSquarespppp(#1,#2)|#3|/#4/<#5,#6>[#7;#8]{% \Xpos=#1\Ypos=#2% \Placements|#3|% \deltaX=#5% \deltaY=#6% \Nodes[#7]% \Labels[#8]% \dovSquares/#4/\ignorespaces}% \def\dovSquares/#1`#2`#3`#4`#5`#6`#7/{% \setmorphismwidth{\Nodea}{\Nodeb}{\Labela}{\topw}% \setmorphismwidth{\Nodec}{\Noded}{\Labeld}{\botw}% \ratchet{\topw}{\botw}% \setmorphismwidth{\Nodee}{\Nodef}{\Labelg}{\botw}% \ratchet{\topw}{\botw}% \square(\Xpos,\Ypos)|\Xd`\Xe`\Xf`\Xg|/`#5`#6`#7/<\topw,\deltaX>% [\Nodec`\Noded`\Nodee`\Nodef;`\Labele`\Labelf`\Labelg]% \advance \Ypos by \deltaX% \square(\Xpos,\Ypos)|\Xa`\Xb`\Xc`\Xd|/#1`#2`#3`#4/<\topw,\deltaY>% [\Nodea`\Nodeb`\Nodec`\Noded;\Labela`\Labelb`\Labelc`\Labeld]% }% \def\vSquares{\ifnextchar({\vSquaresp}{\vSquaresp(0,0)}}% \def\vSquaresp(#1){\ifnextchar|{\vSquarespp(#1)}{\vSquarespp% (#1)|a`l`r`m`l`r`b|}}% \def\vSquarespp(#1)|#2|{\ifnextchar/{\vSquaresppp(#1)|#2|}% {\vSquaresppp(#1)|#2|/>`>`>`>`>`>`>/}}% \def\vSquaresppp(#1)|#2|/#3/{% \ifnextchar<{\vSquarespppp(#1)|#2|/#3/}% {\vSquarespppp(#1)|#2|/#3/<\default>}}% \def\osquarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{\squarepppp% (#1)|#2|/#3`#4`#5`#6/<#7>[#8]% \let\Nodea\nodea\let\Nodeb\nodeb% \let\Nodec\nodec\let\Noded\noded\Xpos=\xpos\Ypos=\ypos% \deltaX=\deltax \deltaY=\deltay \isquare} \def\cube{\ifnextchar({\osquarep}{\osquarep(0,0)}}% \def\osquarep(#1){\ifnextchar|{\osquarepp(#1)}{\osquarepp(#1)|a`l`r`b|}}% \def\osquarepp(#1)|#2|{\ifnextchar/{\osquareppp(#1)|#2|}% {\osquareppp(#1)|#2|/>`>`>`>/}}% \def\osquareppp(#1)|#2|/#3`#4`#5`#6/{% \ifnextchar<{\osquarepppp(#1)|#2|/#3`#4`#5`#6/}% {\osquarepppp(#1)|#2|/#3`#4`#5`#6/<1500,1500>}}% \def\isquarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]{% \squarepppp(#1)|#2|/#3`#4`#5`#6/<#7>[#8]% \ifnextchar|{\cubep}{\cubep|m`m`m`m|}}% \def\cubep|#1|{\ifnextchar/{\cubepp|#1|}{\cubepp|#1|/>`>`>`>/}}% \def\isquare{\ifnextchar({\isquarep}{\isquarep(\default,\default)}}% \def\isquarep(#1){\ifnextchar|{\isquarepp(#1)}{\isquarepp(#1)|a`l`r`b|}} \def\isquarepp(#1)|#2|{\ifnextchar/{\isquareppp(#1)|#2|}% {\isquareppp(#1)|#2|/>`>`>`>/}}% \def\isquareppp(#1)|#2|/#3`#4`#5`#6/{% \ifnextchar<{\isquarepppp(#1)|#2|/#3`#4`#5`#6/}% {\isquarepppp(#1)|#2|/#3`#4`#5`#6/<500,500>}}% \def\cubepp|#1`#2`#3`#4|/#5`#6`#7`#8/[#9]{% \def\Labels[##1`##2`##3`##4]{\gdef\Labela{##1}% \gdef\Labelb{##2}\gdef\Labelc{##3}\gdef\Labeld{##4}}\Labels[#9]% \xend\xpos \yend\ypos \Xend\xend\advance\Xend by -\Xpos \Yend\yend\advance\Yend by -\Ypos \domorphism(\Xpos,\Ypos)|#2|/#6/<\Xend,\Yend>[\Nodeb`\nodeb`\Labelb]% \advance\Xpos by-\deltaX \advance\xend by-\deltax \Xend\xend\advance\Xend by -\Xpos \domorphism(\Xpos,\Ypos)|#1|/#5/<\Xend,\Yend>[\Nodea`\nodea`\Labela]% \advance\Ypos by-\deltaY \advance\yend by-\deltay \Yend\yend\advance\Yend by -\Ypos \domorphism(\Xpos,\Ypos)|#3|/#7/<\Xend,\Yend>[\Nodec`\nodec`\Labelc]% \advance\Xpos by\deltaX \advance\xend by\deltax \Xend\xend\advance\Xend by -\Xpos \domorphism(\Xpos,\Ypos)|#4|/#8/<\Xend,\Yend>[\Noded`\noded`\Labeld]% \ignorespaces} \def\setwdth#1#2{\setbox0\hbox{$#1$}\wdth=\wd0 \setbox0\hbox{$#2$}\ifnum\wdth<\wd0 \wdth=\wd0 \fi} \def\tx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 100 \ratchet{\deltax}{200} \ifx\next\empty \ifx\nextt\empty \>\xy \ar(\deltax,0)\endxy\>% \else \>\xy \ar_{#2}(\deltax,0)\endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar^{#1}(\deltax,0)\endxy\>% \else \>\xy \ar^{#1}_{#2}(\deltax,0)\endxy\>% \fi \fi} \def\txx^#1{\ifnextchar_{\tx^{#1}}{\tx^{#1}_{}}} \def\to{\ifnextchar^{\txx}{\txx^{}}} \def\txleft^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 100 \ratchet{\deltax}{200} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@{<-}(\deltax,0)\endxy\>% \else \>\xy \ar@{<-}_{#2}(\deltax,0)\endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@{<-}^{#1}(\deltax,0)\endxy\>% \else \>\xy \ar@{<-}^{#1}_{#2}(\deltax,0)\endxy\>% \fi \fi} \def\txxleft^#1{\ifnextchar_{\txleft^{#1}}{\txleft^{#1}_{}}} \def\toleft{\ifnextchar^{\txxleft}{\txxleft^{}}} \def\monx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 125 \ratchet{\deltax}{225} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@{ >->}(\deltax,0)\endxy\>% \else \>\xy \ar@{ >->}_{#2}(\deltax,0)\endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@{ >->}^{#1}(\deltax,0)\endxy\>% \else \>\xy \ar@{ >->}^{#1}_{#2}(\deltax,0)\endxy\>% \fi \fi} \def\monxx^#1{\ifnextchar_{\monx^{#1}}{\monx^{#1}_{}}} \def\mon{\ifnextchar^{\monxx}{\monxx^{}}} \def\monleftx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 125 \ratchet{\deltax}{225} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@{<-< }(\deltax,0)\endxy\>% \else \>\xy \ar@{<-< }_{#2}(\deltax,0)\endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@{<-< }^{#1}(\deltax,0)\endxy\>% \else \>\xy \ar@{<-< }^{#1}_{#2}(\deltax,0)\endxy\>% \fi \fi} \def\monleftxx^#1{\ifnextchar_{\monleftx^{#1}}{\monleftx^{#1}_{}}} \def\monleft{\ifnextchar^{\monleftxx}{\monleftxx^{}}} \def\\epileftx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 125 \ratchet{\deltax}{225} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@{<<-}(\deltax,0)\endxy\>% \else \>\xy \ar@{<<-}_{#2}(\deltax,0)\endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@{<<-}^{#1}(\deltax,0)\endxy\>% \else \>\xy \ar@{<<-}^{#1}_{#2}(\deltax,0)\endxy\>% \fi \fi} \def\\epileftxx^#1{\ifnextchar_{\\epileftx^{#1}}{\\epileftx^{#1}_{}}} \def\\epileft{\ifnextchar^{\\epileftxx}{\\epileftxx^{}}} \def\epix^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 125 \ratchet{\deltax}{225} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@{->>}(\deltax,0)\endxy\>% \else \>\xy \ar@{->>}_{#2}(\deltax,0)\endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@{->>}^{#1}(\deltax,0)\endxy\>% \else \>\xy \ar@{->>}^{#1}_{#2}(\deltax,0)\endxy\>% \fi \fi} \def\epixx^#1{\ifnextchar_{\epix^{#1}}{\epix^{#1}_{}}} \def\epi{\ifnextchar^{\epixx}{\epixx^{}}} \def\twx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 100 \ratchet{\deltax}{200} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@<2.5pt>(\deltax,0) \ar@<-2.5pt>(\deltax,0) \endxy\>% \else \>\xy \ar@<2.5pt>(\deltax,0) \ar@<-2.5pt>_{#2}(\deltax,0) \endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@<2.5pt>^{#1}(\deltax,0) \ar@<-2.5pt>(\deltax,0) \endxy\>% \else \>\xy \ar@<2.5pt>^{#1}(\deltax,0) \ar@<-2.5pt>_{#2}(\deltax,0) \endxy\>% \fi \fi} \def\twxx^#1{\ifnextchar_{\twx^{#1}}{\twx^{#1}_{}}} \def\two{\ifnextchar^{\twxx}{\twxx^{}}} \def\twleftx^#1_#2{\allowbreak\edef\next{#1}\edef\nextt{#2} \setwdth{#1}{#2}\deltax=\wdth \divide \deltax by \ul \advance \deltax by 100 \ratchet{\deltax}{200} \ifx\next\empty \ifx\nextt\empty \>\xy \ar@{<-}@<2.5pt>(\deltax,0) \ar@{<-}@<-2.5pt>(\deltax,0) \endxy\>% \else \>\xy \ar@{<-}@<2.5pt>(\deltax,0) \ar@{<-}@<-2.5pt>_{#2}(\deltax,0) \endxy\>% \fi \else \ifx\nextt\empty \>\xy \ar@{<-}@<2.5pt>^{#1}(\deltax,0) \ar@{<-}@<-2.5pt>(\deltax,0) \endxy\>% \else \>\xy \ar@{<-}@<2.5pt>^{#1}(\deltax,0) \ar@{<-}@<-2.5pt>_{#2}(\deltax,0) \endxy\>% \fi \fi} \def\twleftxx^#1{\ifnextchar_{\twleftx^{#1}}{\twleftx^{#1}_{}}} \def\twoleft{\ifnextchar^{\twleftxx}{\twleftxx^{}}} \def\twoar(#1,#2){% \deltax#1\deltay#2% \deltaX=\ifnum\deltax<0-\fi\deltax \deltaY=\ifnum\deltay<0-\fi\deltay \multiply\deltax by 300 \multiply\deltay by 300 \ifnum \deltaX > \deltaY \multiply \deltaX by 3 \advance \deltaX by \deltaY \else \multiply \deltaY by 3 \advance \deltaX by \deltaY \fi \divide \deltax by \deltaX \divide \deltay by \deltaX \xy \ar@{=>}(\deltax,\deltay) \endxy } \catcode`\@=\atcode% From rrosebru@mta.ca Mon Apr 23 12:50:06 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3NEvGl10049 for categories-list; Mon, 23 Apr 2001 11:57:16 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 23 Apr 2001 15:10:13 +0100 From: Paul Taylor Message-Id: <200104231410.PAA30611@koi-pc.dcs.qmw.ac.uk> To: categories@mta.ca Subject: categories: Hausdorff's book Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 23 Does anyone know of an English translation or summary of the chapters (7-9) on topology in the first edition of Hausdorff's "Grundz\"uge der Mengenlehre"? This material was omitted from the later editions, on which the published English translations were based. Problem is, I don't read German, but I could cope with a French version, if such exists but there is no English one. Hausdorff's axiomatisation is based on the family of open neighbourhoods of each point; the usual "finite intersections and arbitrary unions" of open subsets are due, I believe, to Bourbaki. Paul From rrosebru@mta.ca Wed Apr 25 09:22:01 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3PBhG523348 for categories-list; Wed, 25 Apr 2001 08:43:16 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Gonzalo Reyes Message-Id: <200104242126.f3OLQvQ00576@telemaque.dms.umontreal.ca> Content-Type: text/plain MIME-Version: 1.0 (NeXT Mail 4.2mach_patches v148.2) Date: Tue, 24 Apr 2001 17:26:57 -0400 To: categories@mta.ca Subject: categories: preprint: Some differential equations in Synthetic Differential Geometry Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 24 The following paper is available at the address given below Some differential equations in Synthetic Differential Geometry by Anders Kock and Gonzalo E. Reyes Abstract: Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces is exploited. In particular, we utilize distribution spaces in our study of wave and heat equations. Address: http://xxx.lanl.gov/list/math.CT/recent#2 From rrosebru@mta.ca Wed Apr 25 12:10:00 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3PEe3u29244 for categories-list; Wed, 25 Apr 2001 11:40:03 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Kleisli and colimits To: categories@mta.ca Date: Wed, 25 Apr 2001 15:31:08 +0100 (BST) X-Mailer: ELM [version 2.5 PL3] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: From: Tom Leinster X-Scanner: exiscan *14sQJm-0001ed-00*j1LRnZ0fCe2* http://duncanthrax.net/exiscan/ Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 25 Does anyone know if the Kleisli construction (sending a monad to its Kleisli category) behaves in any decent way with respect to colimits? E.g. does it in any sense preserve or reflect them? The actual situation that I have is a fixed category C, and a certain coequalizer diagram in the category of monads on C. The resulting fork in Cat is also a coequalizer, and the proofs that both diagrams are coequalizers have some ingredients in common, but I can't at present see how to deduce one from the other. Thanks, Tom Leinster From rrosebru@mta.ca Wed Apr 25 14:13:19 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3PGdRr01165 for categories-list; Wed, 25 Apr 2001 13:39:27 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3AE6F592.C9298FD8@cs.bham.ac.uk> Date: Wed, 25 Apr 2001 17:04:34 +0100 From: Carsten Fuhrmann Organization: School of Computer Science, The University of Birmingham, U.K. X-Mailer: Mozilla 4.77 [en] (X11; U; SunOS 5.7 sun4u) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Kleisli and colimits References: Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 26 The construction of the Kleisli category can be turned into a left adjoint functor whose domain is the category of Mnd monads (on arbitrary categories) and monad morphisms. However, the codomain of this functor is not Cat, but a category AbsKl whose objects I call "abstract Kleisli categories". An abstract Kleisli category is a category K together with a functor L:K->K, a natural transformation \epsilon: L->Id, and a (not generally natural) transformation \theta:Id->L, such that (1) \theta_L is a natural transformation (2) L\theta o \theta = \theta_L o \theta (3) \epsilon o \theta = id (3) L\epsilon o \theta_L = id A morphism K->K' of AbsKl is a functor that preserves the solutions of the non-naturality square \theta o f = Lf o \theta (I) The Kleisli construction forms a functor Mnd->AbsKl. Its right adjoint sends an abstract Kleisli category K to the evident monad on the subcategory given by the solutions of Equation (I). The counit of this adjunction is in fact an iso, so AbsKl is a full reflective subcategory of Mnd. The full subcategory of Mnd which is equivalent to Abskl is given by those monads for which every component of the unit is a regular mono. Cheers, Carsten From rrosebru@mta.ca Fri Apr 27 09:54:22 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3RBrPs09426 for categories-list; Fri, 27 Apr 2001 08:53:25 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3AE9162E.7030606@math.uit.no> Date: Fri, 27 Apr 2001 08:48:14 +0200 From: Per User-Agent: Mozilla/5.0 (Windows; U; Win98; en-US; m18) Gecko/20001108 Netscape6/6.0 X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Extension of natural transformations Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 27 Is there anything known about the following type of problem. Let F,G : C -> D be functors and let E be a subcategory of C and let F' G' be the restirctions of F and G to E. Let a' : F' -> G' be a natural transformation. When does there exist a natural transformation alpha : F -> G that extends alpha' so that alpha restricted to E is equal to alpha'? What if E is a set of generators for C? Thanks Per K. Jakobsen From rrosebru@mta.ca Sun Apr 29 11:51:37 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3TEA9o12092 for categories-list; Sun, 29 Apr 2001 11:10:09 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs Date: Sat, 28 Apr 2001 20:26:23 -0400 (EDT) From: Michael Barr X-Sender: barr@triples.math.mcgill.ca To: Categories list Subject: categories: Syntax question on diagxy Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 28 I have a question for all you who are interested in diagxy. There is an inconsistency in the current version that is caused by attempting to keep it similar to the old diagram. In all the procedures except \morphism, the basic syntax includes a string [N`..`N;L`..`L] where the N's represent nodes and the L's labels. The exception is \morphism, where it is [N`N`L]. I must have done it that way 15 years ago because there were only three arguments and it didn't seem necessary to have different separators. There are still only those three arguments, but the inconsistency bothers me. Now is the time to change it, since once it is in general release it will be impossible. Right now, it is a trivial change to make it [N`N;L] and I am getting ready for general release. I would like it to be in the next version of the CTAN disks. If I don't hear from anyone, I am inclined to make the change. Michael From rrosebru@mta.ca Sun Apr 29 11:51:52 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3TECEq18232 for categories-list; Sun, 29 Apr 2001 11:12:14 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <200104270727.JAA05523@irmast2.u-strasbg.fr> Date: Fri, 27 Apr 2001 09:27:21 +0200 (MET DST) From: Philippe Gaucher Reply-To: Philippe Gaucher Subject: categories: GETCO 2001 : 2nd Call for papers To: categories@mta.ca MIME-Version: 1.0 Content-Type: TEXT/plain; charset=ISO-8859-1 Content-MD5: 95PWRRyDEJA6qAY6rHDlmw== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.3.5 SunOS 5.7 sun4u sparc Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from QUOTED-PRINTABLE to 8bit by mailserv.mta.ca id f3R7Rcb13886 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 29 3rd workshop on Geometric and Topological Methods in Concurrency GETCO 2001 hosted by Concur 2001, and Aalborg University Venue: Aalborg University, Denmark Conference dates: CONCUR: August 21-24, 2001 GETCO: August 25, 2001 2nd Call for papers Scope of the workshop The main mathematical disciplines that have been used in computer science are discrete mathematics (especially, graph theory and ordered structures), logics (mostly proof theory for all kinds of logics, classical, intuitionistic, modal etc.) and category theory (cartesian closed categories, topoi etc.). General Topology has also been used for instance in denotational semantics, with relations to ordered structures in particular. Recently, ideas and notions from mainstream "geometric" topology and algebraic topology have entered the scene in Concurrency Theory and Distributed Systems Theory (some of them based on older ideas). They have been applied in particular to problems dealing with coordination of multi-processor and distributed systems (see the historical note ). Among those are techniques borrowed from algebraic and geometric topology: Simplicial techniques have led to new theoretical bounds for coordination problems. Higher dimensional automata have been modelled as cubical complexes with a partial order reflecting the time flows, and their homotopy properties allow to reason about a system's global behaviour. The first workshop on the subject Geometric and Topological Methods in Concurrency Theory has been held in Aalborg, Denmark, in June 1999. GETCO 2000 was organised as a workshop affiliated with CONCUR 2000 at Penn State University. The recent volumes 10 of Math. Struct. in Comp. Science and 39, issue 2, of Electr. Notes Theor. Comp. Science have been devoted to this area. The 3rd workshop in this series will be organised as a satellite to CONCUR 2001 at Aalborg University (Aug.21 - Aug.24). It aims at bringing together researchers from both the mathematical (geometry, topology, algebraic topology etc.) and computer scientific side (concurrency theorists, semanticians, researchers in distributed systems etc.) with an active interest in these or related developments. Topics include (but are not limited to) Semantics, Concurrency Theory, Model-checking, Abstract Interpretation, Fault-tolerant Protocols for Distributed Systems, Geometrical/Topological models, Applications of algebraic topology, Category theory etc. Paper submission The deadline for submission to the workshop is May 20, 2001 (midnight). Submissions may be of two forms: Short abstracts: up to 4 pages, in format A4, typeset 11 points Full papers: up to 12 pages, in format A4, typeset 11 points (excluding bibliography and technical appendices) Both forms of submission should include a separate page with the following informations: title, author(s), corresponding author, contact information and a 12-15 lines summary. Simultaneous submission to other conferences or journals is only allowed for short abstracts. Electronic submission is strongly encouraged. The paper or abstract should be sent by e-mail in the form of a postscript file to both the addresses raussen@math.auc.dk and goubault@aigle.saclay.cea.fr. The accompanying page should be sent in a separate email message. If surface mail has to be used, then 3 copies of the paper/abstract should be sent to: Martin Raussen, Dept. of Mathematical Sciences, Aalborg University, Fredrik Bajersvej 7G, DK-9220 Aalborg Øst, Denmark. Publication Accepted papers will be made available in the BRICS Notes series. Electronic Notes in Theoretical Computer Science has kindly offered to publish the proceedings of the workshop - full papers only - in a special volume. The programme committee will decide upon necessary revisions and acceptance of papers to this volume after the workshop. Registration The registration for GETCO'2001 is made through the registration for Concur'2001 with no extra-charge. Programme Committee: Patrick Cousot, Ecole Normale Superieure, Paris, France Lisbeth Fajstrup, Aalborg University, Denmark Eric Goubault, Commissariat a l'Energie Atomique, France Jeremy Gunawardena, Hewlett-Packard BRIMS, England Maurice Herlihy, Brown University, Providence, RI, USA Martin Raussen, Aalborg University, Denmark Vladimiro Sassone, Catania University, Italy Important Dates Call for Papers: March 13, 2001 Deadline for submission: May 20, 2001 Notification of acceptance: July 1, 2001 Final version due: July 29, 2001 CONCUR: August 21-August 24, 2001 GETCO: August 25, 2001 Local Organization Lisbeth Fajstrup, Aalborg University, Denmark Eric Goubault, Commissariat a l'Energie Atomique, France Martin Raussen, Aalborg University, Denmark Contact Person Martin Raussen Department of Mathematics Aalborg University Fredrik Bajersvej 7G DK-9220 Aalborg Øst Denmark Phone: (+45) 96 35 88 55 Fax: (+45) 98 15 81 29 Email: raussen@math.auc.dk Url : http://concur01.cs.auc.dk/workshop/workshops.html http://www-irma.u-strasbg.fr/~gaucher/getco/ From rrosebru@mta.ca Sun Apr 29 12:09:06 2001 -0300 Return-Path: Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f3TEcoH27584 for categories-list; Sun, 29 Apr 2001 11:38:50 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Received: from zent.mta.ca (zent.mta.ca [138.73.101.4]) by mailserv.mta.ca (8.11.1/8.11.1) with SMTP id f3R0W1b31409 for ; Thu, 26 Apr 2001 21:32:01 -0300 (ADT) X-Received: FROM guest.mathsoc.uottawa.ca BY zent.mta.ca ; Thu Apr 26 21:39:05 2001 -0300 X-Received: (from majordom@localhost) by guest.mathsoc.uottawa.ca (8.11.3/8.11.3) id f3R0VRL05070 for cmath-l-out; Thu, 26 Apr 2001 20:31:27 -0400 X-Received: from eo.cms.math.ca (eo.mathsoc.uottawa.ca [137.122.49.236]) by guest.mathsoc.uottawa.ca (8.11.3/8.11.3) with ESMTP id f3R0VP205066 for ; Thu, 26 Apr 2001 20:31:25 -0400 X-Received: from jardine.math.uwo.ca (root@jardine.math.uwo.ca [129.100.75.21]) by eo.cms.math.ca (8.9.0/8.9.0) with ESMTP id UAA32107 for ; Thu, 26 Apr 2001 20:31:24 -0400 (EDT) From: jardine@uwo.ca X-Received: from uwo.ca (jardine@cr405431-a.lndn1.on.wave.home.com [24.112.60.174]) by jardine.math.uwo.ca (8.10.2/8.10.0) with ESMTP id f3R0VMt10727 for ; Thu, 26 Apr 2001 20:31:22 -0400 Message-ID: <3AE8BDD8.5EA864FD@uwo.ca> Date: Thu, 26 Apr 2001 20:31:20 -0400 X-Mailer: Mozilla 4.75 [en] (X11; U; Linux 2.2.14 i686) X-Accept-Language: en MIME-Version: 1.0 To: Subject: categories: Stanford conference, July 30 - August 3 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 30 Third Announcement: Conference on Algebraic Topological Methods in Computer Science Stanford University July 30 - August 3, 2001 This meeting is supported by grants from the National Science Foundation, the Natural Sciences and Engineering Research Council of Canada, the Fields Institute, Hewlett-Packard, and the Stanford Department of Mathematics. The most up to date information on the conference appears on the conference web page http://math.stanford.edu/atmcs/index.htm. Housing is still available on campus at a cost of about 50.00 US per night. The deadline for registering for on campus housing at Stanford is *May 1, 2001*. There is a registration form available as a pdf file, to be printed, filled out and faxed to the Stanford Summer Conference Services office. There is also a short registration for the conference itself at that web page. If you are coming to the conference and have not yet registered for the conference, please do so. There will be no registration fee. Limited financial support may be available for travel and housing. Please make your request when registering for the conference. The following have agreed to speak at this meeting: John Baez (Math, UC Riverside) Marshall Bern (Xerox PARC) Anders Bjorner (Royal Institute of Technology, Stockholm) Tamal Dey (CS, Ohio State) Herbert Edelsbrunner (CS, Duke) David Eppstein (CS, UC Irvine) Michael Freedman (Microsoft) Philippe Gaucher (CNRS, Strasbourg) Eric Goubault (Commissariat a l'Energie Atomique, France) Jean Goubault-Larrecq (ENS Cachan) Marco Grandis (Dip. di Mat., Genova) Jeremy Gunawardena (HP BRIMS) John Harer (Math, Duke) Joel Hass (Math, UC Davis) Maurice Herlihy (CS, Brown) Reinhard Laubenbacher (Math, NMSU) Laszlo Lovasz (Microsoft) Vaughan Pratt (CS, Stanford) Christian Reidys (Los Alamos National Lab) Bernd Sturmfels (Math, UC Berkeley) Noson Yanofsky (CS, Brooklyn College) There will be some time for contributed talks. If you would like to give a short talk at the meeting, please send a title and abstract to one of the organizers. The organizers for this meeting are: Gunnar Carlsson: gunnar@math.stanford.edu Rick Jardine: jardine@uwo.ca