From MAILER-DAEMON Fri Nov 7 15:50:42 2003 Date: 07 Nov 2003 15:50:42 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1068234642@mta.ca> X-IMAP: 1068234632 0000000023 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From rrosebru@mta.ca Mon Sep 1 15:18:23 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Sep 2003 15:18:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ttE4-0006J5-00 for categories-list@mta.ca; Mon, 01 Sep 2003 15:16:28 -0300 Message-ID: <3F530B5B.9000100@uni-paderborn.de> Date: Mon, 01 Sep 2003 11:03:23 +0200 From: Reiko Heckel X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: grants on visual modelling techniques Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 1 OPEN CALL FOR APPLICATIONS SegraVis (http://www.segravis.org) is a European Research Training Network running from October 2002 until September 2006. It offers grants to post-doctoral researchers and advanced doctoral students with any of the following sites (researchers) * Universit=E4t Paderborn, Germany (Gregor Engels), Network Coordinator * University of Antwerp, Belgium (Dirk Janssens) * Universitat Polit=E9cnica de Catalunya, Barcelona, Spain (Fernando Orejas) * Technische Universitaet Berlin, Germany (Hartmut Ehrig) * University of Bremen, Germany (Hans-J=F6rg Kreowski) * University of Kent at Canterbury, United Kingdom ( Peter Rodgers) * University of Leiden , The Netherlands (Grzegorz Rozenberg) * University College London, United Kingdom (Wolfgang Emmerich) * Universit=E1 degli Studi di Milano, Bicocca, Italy (Mauro Pezze`) * Technical University of Darmstadt, Germany (Andy Sch=FCrr) * Universit=E0 di Pisa, Italy (Ugo Montanari) * Universit=E0 di Roma La Sapienza, Italy (Francesco Parisi-Presicce) The Call for Applications for the the period of October 2003 to September 2004 is now open for researchers from EU and associated countries, see see http://www.segravis.org/information.html for details. Deadline for applications is October 1st. (If you have trouble meeting the deadline, please contact me.) Yours Reiko Heckel --=20 Dr. Reiko Heckel URL: www.upb.de/cs/reiko.html Universit=E4t Paderborn, E4.130 Tel: ++49-05251-60-3356 33095 Paderborn, Germany Fax: ++49-05251-60-3431 Visit www.segravis.org, home of the SegraVis Research Training Network. Apply for a grant in one of 12 attractive locations throughout Europe. Join the graph transformation mailing list at www.gratra.org/list.html. From rrosebru@mta.ca Mon Sep 1 15:18:23 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Sep 2003 15:18:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ttCt-0006H4-00 for categories-list@mta.ca; Mon, 01 Sep 2003 15:15:15 -0300 Date: Mon, 1 Sep 2003 10:30:01 +0200 (CEST) From: Jonas Eliasson To: categories@mta.ca Subject: categories: Tree cover? 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 It seems that you can modify the construction of the localic Diaconescu cover to get an open surjective _filtered_ cover of a Grothendieck topos Sh(C). Instead of using the category String(C) of strings in C, you could construct the category Tree(C) of finite, rooted, binary trees in C. If given c and d in C you can find e such that e --> c and e --> d then Tree(C) is a poset with binary upper bounds, i.e. a filtered category. Could anyone provide a reference for such a construction? Grateful for any help, Jonas Eliasson ------------------------------------------ | Jonas Eliasson | | Department of Mathematics | | Uppsala University | | Sweden | | E-mail: jonase@math.uu.se | | Homepage: http://www.math.uu.se/~jonase/ | ------------------------------------------ From rrosebru@mta.ca Tue Sep 2 14:14:21 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Sep 2003 14:14:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19uEgN-0000Kh-00 for categories-list@mta.ca; Tue, 02 Sep 2003 14:11:07 -0300 Date: Tue, 2 Sep 2003 10:16:58 +0200 (CEST) From: Jonas Eliasson To: categories@mta.ca Subject: categories: Logic preserved in double negation subtopos? 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: 3 While writing a joint paper with Steve Awodey, we came to think about the following question: Given a Grothendieck topos Sh(C), what logic is preserved by the associated sheaf functor from Sh(C) to the double negation subtopos of Sh(C)? We know that a: Sh(C) --> DNSh(C) preserves geometric logic. Since it is double negation it also preserves 0 (falsehood), negation and implication. >From this you can draw the conclusion that a preserves the validity of formulas built up from double negation stable predicates without universal quantifiers. Presumably this has been studied in the literature, can something stronger be said about what validities are preserved, could anyone provide a reference for a general result of this kind? Grateful for any help, Jonas Eliasson ------------------------------------------ | Jonas Eliasson | | Department of Mathematics | | Uppsala University | | Sweden | | E-mail: jonase@math.uu.se | | Homepage: http://www.math.uu.se/~jonase/ | ------------------------------------------ From rrosebru@mta.ca Tue Sep 2 17:19:48 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Sep 2003 17:19:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19uHa6-0001mr-00 for categories-list@mta.ca; Tue, 02 Sep 2003 17:16:50 -0300 From: Oswald Wyler To: categories@mta.ca Subject: categories: Re: Uniform spaces In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-MIME-Autoconverted: from 8bit to quoted-printable by aphrodite.gwi.net id h82IZRWC039976 Sender: cat-dist@mta.ca Precedence: bulk Date: Tue, 02 Sep 2003 17:16:50 -0300 Status: O X-Status: X-Keywords: X-UID: 4 On Wed, 27 Aug 2003, Tom Leinster wrote: > Date: Wed, 27 Aug 2003 16:51:55 +0200 (CEST) > From: Tom Leinster > To: categories@mta.ca > Subject: categories: Uniform spaces >=20 > Hello, >=20 > Does anyone know of any account of the basic properties of the category= of > uniform spaces? I'm after things like (co)limits, cartesian closure, a= nd > (co)limit-preservation by the forgetful functor to Top. Bourbaki gets = me > some of the way, but his decision not to use categorical language and > the resulting circumlocutions make it a struggle. >=20 > Thanks, > Tom Hi Tom, The category UNIF of uniform spaces, without a separation axiom, is topological over sets, and hence complete and cocomplete, with concrete limits and colimits. UNIF=A0is not cartesian closed. Cook and Fischer, Math. Ann. 173 (1967), 290-306, defined uniform converg= ence structures of a set X as sets \scrF of filters on XxX satisfying five axioms. With the obvious definition of uniform continuity, sets with a uniform convergence structure in this sense form a topological category over sets, but Gazik, Kent and Richardson in Bull.Austral.Math.soc 11 (19= 74), 413-424, showed that this category is not cartesian closed. In LNM 378, 591-637, I replaced the Cook-Fischer axiom that the principal filter generated by the diagonal of XxX is in \scrF by the less demanding axion that the principal filter generated by (x,x), for every x \in X, is in \scrF. This is now part of the accepted definition of uniform convergence spaces. In Bull.Austral.Math.Soc. 15 (1976), 461-465 my student R.S. Lee showed that the category of uniform convergence spaces with this definition is cartesian closed; this is not the cartesian close= d hull of UNIF. For quasitoposes, we must go to semiuniform spaces which have partial morphisms -- relations (m,g) with m an embedding -- represented by one-point extensions. Semiuniform convergence spaces and their uniformly continuous maps form a quasitopos, but not the quasitopos hull of UNIF. This has been determined by Ad=E1mek and Reiterman, The quasitopos hull o= f the category of uniform spaes -- a correction, in the journal Topology and its Applications. For more information and literature, see my book Lecture Notes on Topoi and Quasitopoi. Oswald Wyler From rrosebru@mta.ca Tue Sep 2 20:45:17 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Sep 2003 20:45:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19uKnF-00009P-00 for categories-list@mta.ca; Tue, 02 Sep 2003 20:42:37 -0300 Date: Tue, 2 Sep 2003 20:52:12 +0100 (BST) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: Logic preserved in double negation subtopos? Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *19uHCH-0006G6-53*ymcE9joHdyg* Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 5 The inclusion of double-negation sheaves is an example of what I called a sub-open map in my paper "Open maps of toposes" (Manuscripta Math. 31 (1980), 217-247). Sub-open maps have the property that their inverse image functors commute with implication -- indeed, one could take that as a definition, although it wasn't how I defined them in the paper. Peter Johnstone ---------- On Tue, 2 Sep 2003, Jonas Eliasson wrote: > While writing a joint paper with Steve Awodey, we came to think about the > following question: > > Given a Grothendieck topos Sh(C), what logic is preserved by the > associated sheaf functor from Sh(C) to the double negation subtopos of > Sh(C)? > > We know that a: Sh(C) --> DNSh(C) preserves geometric logic. Since it is > double negation it also preserves 0 (falsehood), negation and implication. > >From this you can draw the conclusion that a preserves the validity of > formulas built up from double negation stable predicates without universal > quantifiers. > > Presumably this has been studied in the literature, can something stronger > be said about what validities are preserved, could anyone provide a > reference for a general result of this kind? > > Grateful for any help, > Jonas Eliasson > > > > > ------------------------------------------ > | Jonas Eliasson | > | Department of Mathematics | > | Uppsala University | > | Sweden | > | E-mail: jonase@math.uu.se | > | Homepage: http://www.math.uu.se/~jonase/ | > ------------------------------------------ > > > > > > > > From rrosebru@mta.ca Fri Sep 5 14:11:12 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 05 Sep 2003 14:11:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19vK3e-0007b3-00 for categories-list@mta.ca; Fri, 05 Sep 2003 14:07:38 -0300 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 4 Sep 2003 10:41:26 +0100 To: AMAST04 mailing list:;;@cs.stir.ac.uk From: Carron Shankland Subject: categories: AMAST 2004: Call for Papers Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 Please distribute this announcement to your colleagues. --------------------------------------------- CALL FOR PAPERS AMAST 2004: 10th International Conference on Algebraic Methodology And Software Technology http://www.cs.stir.ac.uk/events/amast2004/ Paper Submissions: 19th January 2004 --------------------------------------------- AMAST 2004 July 12th - 16th, 2004 Stirling, Scotland, UK. SPEAKERS -------- * Roland Backhouse (Nottingham) * Don Batory (Texas) * Michel Bidoit (CNRS) * Muffy Calder (Glasgow) * Bart Jacobs (Nijmegen) * JJ Meyer (Utrecht) The major goal of the AMAST Conferences is to promote research that may lead to the setting of software technology on a firm, mathematical basis. This goal is achieved by a large international cooperation with contributions from both academia and industry. The virtues of a software technology developed on a mathematical basis have been envisioned as being capable of providing software that is (a) correct, and the correctness can be proved mathematically, (b) safe, so that it can be used in the implementation of critical systems, (c) portable, i.e., independent of computing platforms and language generations, and (d) evolutionary, i.e., it is self-adaptable and evolves with the problem domain. TOPICS ------ As in previous years, we will invite papers reporting original research on setting software technology on a firm mathematical basis. We expect two kinds of submissions for this conference: technical papers and system demonstrations. Of particular interest is research on using algebraic, logic, and other formalisms suitable as foundations for software technology, as well as software technologies developed by means of logic and algebraic methodologies. Topics of interest include, but are not limited to, the following: SOFTWARE TECHNOLOGY: * systems software technology * application software technology * concurrent and reactive systems * formal methods in industrial software development * formal techniques for software requirements, design * evolutionary software/adaptive systems PROGRAMMING METHODOLOGY: * logic programming, functional programming, object paradigms * constraint programming and concurrency * program verification and transformation * programming calculi * specification languages and tools * formal specification and development case studies ALGEBRAIC AND LOGICAL FOUNDATIONS: * logic, category theory, relation algebra, computational algebra * algebraic foundations for languages and systems, coinduction * theorem proving and logical frameworks for reasoning * logics of programs SYSTEMS AND TOOLS (for system demonstrations or ordinary papers): * software development environments * support for correct software development * system support for reuse * tools for prototyping * component based software development tools * validation and verification * computer algebra systems * theorem proving systems PUBLICATION ----------- As in the past, the proceedings of AMAST 2004 will be published by Springer in the Lecture Notes in Computer Science series. We invite prospective authors to submit electronically previously unpublished papers of high quality. Submissions should not have been published and should not be under consideration for publication elsewhere. Papers must be no longer than 15 pages (6 pages for system demonstrations) and should be prepared using LaTeX and the LNCS style that can be downloaded from http://www.springer.de/comp/lncs/authors.html. Please send a fully self-contained PostScript file to amast@cs.stir.ac.uk. If for any reason it is impossible to submit a paper electronically, authors should send six copies of their submission to the program chair at the address below. All papers will be refereed by the programme committee, and will be judged based on their significance, technical merit, and relevance to the conference. Papers should be received by January 19, 2004. Address for non-electronic submissions: Charles Rattray AMAST'2004 Program Chair Department of Computing Science and Mathematics University of Stirling Stirling FK9 4LA UK PRIZES ------ There will be a prize for the best paper overall, and for the best student paper. These prizes are sponsored by BCS-FACS (the British Computing Society special interest group Formal Aspects of Computing Science). Each prize winner will receive a year's membership of BCS-FACS and a year's subscription to the Formal Aspects of Computing journal. IMPORTANT DATES --------------- * Paper submissions: January 19, 2004. * Notification of paper acceptance: March 1, 2004 * Camera ready papers due: April 5, 2004 * AMAST'2004 Conference: July 12-16, 2004 LOCATION -------- The conference will be held at the University of Stirling http://www.stir.ac.uk/ CONTACT ------- For further information, send email to amast@cs.stir.ac.uk --------------------------------------------------------------------- Dr Carron Shankland Email: ces@cs.stir.ac.uk Computing Science & Mathematics Tel: 01786-467444 University of Stirling http://www.cs.stir.ac.uk/~ces Stirling FK9 4LA --------------------------------------------------------------------- -- The University of Stirling is a university established in Scotland by charter at Stirling, FK9 4LA. Privileged/Confidential Information may be contained in this message. If you are not the addressee indicated in this message (or responsible for delivery of the message to such person), you may not disclose, copy or deliver this message to anyone and any action taken or omitted to be taken in reliance on it, is prohibited and may be unlawful. In such case, you should destroy this message and kindly notify the sender by reply email. Please advise immediately if you or your employer do not consent to Internet email for messages of this kind. From rrosebru@mta.ca Tue Sep 9 21:04:18 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Sep 2003 21:04:18 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19wsOT-0001Qa-00 for categories-list@mta.ca; Tue, 09 Sep 2003 20:59:33 -0300 From: "M.M. Bonsangue" Date: Tue, 9 Sep 2003 17:24:25 +0200 Message-Id: <200309091524.h89FOPP02339@tin.liacs.nl> To: categories@mta.ca Subject: categories: Formal Methods for Components and Objects 2003 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 7 (We apologize for the reception of multiple copies) *********************** CALL FOR PARTICIPATION ********************** Second International Symposium on Formal Methods for Components and Objects (FMCO 2003) DATES 4 - 7 November 2003 PLACE Lorentz Center, Leiden University, Leiden, The Netherlands URL http://fmco.liacs.nl/fmco03.html >>>>>> EARLY REGISTRATION DEADLINE (15/09/2004) IS APPROACHING <<<<< OBJECTIVES The objective of this symposium is to bring together researchers and practioners in the areas of software engineering and formal methods to discuss the concepts of reusability and modifiability in component-based and object-oriented software systems. FORMAT The symposium is a four days event in the style of the former REX workshops, organised to provide an atmosphere that fosters collaborative work, discussions and interaction. The program consists of keynote and technical presentations, and contains an exquisite social event. Speakers' contributions will be published after the symposium in Lecture Notes in Computer Science by Springer-Verlag. PRELIMINARY PROGRAM TUESDAY 4th, November 2003 8:45 - 9:00 Welcome 9:00 - 10:00 Keynote: David Parnas (University of Limerick, IE) Mathematical Documentation of Software 10:00 - 10:30 Break 10:30 - 11:15 Razvan Diaconescu (IMAR, RO) Behavioural specification for hierarchical object composition 11:15 - 12:00 Heike Wehrheim (University of Oldenburg, DE) Preserving Properties under Change 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Andrew D. Gordon (Microsoft Research Cambridge, UK) Formal Tools for Securing Web Services 14:30 - 15:00 Break 15:00 - 15:45 Jeannette Wing (Carnegie Mellon University, USA) Vulnerability Analysis Using Attack Graphs 15:45 - 16:00 Break 16:00 - 16:45 Albert Benveniste (IRISA/INRIA - Rennes, FR) Heterogeneous reactive systems formal modeling 16:45 - 17:30 Yassine Lakhnech (University of Grenoble, FR) t.b.a. WEDNSDAY 5th, November 2003 9:00 - 10:00 Keynote: Tony Hoare (Microsoft Research Cambridge, UK) The Verifying Compiler: a Grand Challenge for Computing Research 10:00 - 10:30 Break 10:30 - 11:15 Willem-Paul de Roever (University of Kiel, DE) Data Refinement: model-oriented proof methods and their comparison 11:15 - 12:00 Frank de Boer (CWI, Amsterdam, NL) Hoare Logics for Object-Oriented Programming: State of the Art 12:00 - 13:30 Lunch break 13:30 - 14:15 Jean-Marc Jezequel (IRISA, Rennes, FR) Model-Driven Engineering: Basic Principles and Open Problems 14:15 - 15:00 Jan Friso Groote (Eindhoven University of Technology, NL) t.b.a. 17:00 - 19:15 Social Event 19:30 - Dinner THURSDAY 6th, November 2003 9:00 - 10:00 Keynote: Yuri Gurevich (Microsoft Research Redmond, USA) The Semantics of AsmL 10:00 - 10:30 Break 10:30 - 11:15 Egon Boerger (Pisa University, IT) Exploiting the "A" in Abstract State Machines for Specification Reuse. A Java/C# Case Study. 11:15 - 12:00 Werner Damm (University of Oldenburg, DE) t.b.a. 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Desmond D'Souza (Kinetium, Austin, USA) t.b.a. 14:30 - 15:00 Break 15:00 - 15:45 Rob van Ommering (Philips Research Laboratories, Eindhoven, NL) Component Based Architectures and Formalization 15:45 - 16:30 Jose Luiz Fiadeiro (University of Leicester, UK) CommUnity on the move: architectures for distribution and mobility 16:30 - 16:45 Break 16:45 - 17:30 Gregor Engels (University of Paderborn, DE) Consistent interaction of components FRIDAY 7th, November 2003 9:00 - 10:00 Keynote: E. Allen Emerson (The University of Texas at Austin, USA) Model Checking Many Components 10:00 - 10:30 Break 10:30 - 11:15 Amir Pnueli (The Weizmann Institute of Science, ISR) t.b.a. 11:15 - 12:00 Natalia Sidorova (Eindhoven University of Technology, NL) Practical approaches for the verification of asynchronous components: model checking, abstraction and static analysis 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Joseph Sifakis (Verimag, FR) t.b.a. 14:30 - 15:00 Break 15:00 - 15:45 Philippe Schnoebelen (CNRS, Cachan, FR) The Verification of Lossy Channel Systems 15:45 - 16:30 Bengt Jonsson (Uppsala University, SE) t.b.a. 16:30 - 17:15 Jan Rutten (CWI, Amsterdam, NL) A case study in coinductive stream calculus: signal flow graphs for dummies MOBI-J WORKSHOP On Monday 3rd, November 2003, there will be a one-day Mobi-J workshop on "Assertional Methods for Java and its Extension with Mobile Asynchronous Channels". REGISTRATION Participation is limited to about 80 people, using a first-in first-served policy. To register, please fill in the registration form at http://fmco.liacs.nl/fmco03.html. The EARLY registration fee (BEFORE September 15, 2003) is 375 euro for regular participants and 250 euro for students It includes the participation to the symposium, a copy of the proceedings, all lunches and refreshments, and a social event (with dinner). ORGANIZING COMMITTEE F.S. de Boer (CWI and Utrecht University) M.M. Bonsangue (LIACS-Leiden University) S. Graf (Verimag) W.P. de Roever (CAU) For more information about participation and registration see the FMCO site above or consult either F.S. de Boer (frb@cwi.nl) or M.M. Bonsangue (marcello@liacs.nl). From rrosebru@mta.ca Tue Sep 16 17:05:48 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2003 17:05:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19zM0G-0007fE-00 for categories-list@mta.ca; Tue, 16 Sep 2003 17:00:48 -0300 Message-ID: <3F674462.6090401@bluewin.ch> Date: Tue, 16 Sep 2003 19:12:02 +0200 From: Krzysztof Worytkiewicz User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.0.1) Gecko/20020823 Netscape/7.0 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: associated sheaf functor Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 8 Dear All, Is anybody aware of a variant of the "plus" construction giving the "associated separated presheaf" wrt to a Grothendieck topology which works on a basis as only piece of data (ie without generating the whole topology and then applying the classical plus functor)? Any hint welcome... Cheers Krzysztof From rrosebru@mta.ca Wed Sep 17 12:18:07 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Sep 2003 12:18:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ze0j-0006wl-00 for categories-list@mta.ca; Wed, 17 Sep 2003 12:14:29 -0300 To: Subject: categories: CMCS '04, FIRST ANNOUNCEMENT, CALL FOR PAPERS Date: Wed, 17 Sep 2003 13:26:00 +0200 Organization: Techn. Uni, Inst. f. Theoretische Informatik Message-ID: <004401c37d0e$7c1ceb30$6f27a986@trek> X-Priority: 3 (Normal) X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook, Build 10.0.4510 Sender: cat-dist@mta.ca Precedence: bulk From: cat-dist@mta.ca Status: RO X-Status: X-Keywords: X-UID: 9 + + + CMCS '04 + + + FIRST ANNOUNCEMENT + + + CALL FOR PAPERS + + + Apologies if you receive multiple copies of this message. +----------------------------------------------------------+ | | | | | 7th International Workshop on | | Coalgebraic Methods in Computer Science | | | | C M C S 2004 | | | | | | Barcelona, March 27-29, 2004 | | http://www.iti.cs.tu-bs.de/~cmcs/ | | | +----------------------------------------------------------+ The workshop is held in conjunction with ETAPS 2004 (7th European Joint Conferences on Theory Theory and Practice of Software, March 27- April 4,2004) http://www.lsi.upc.es/etaps04/ AIMS AND SCOPE During the last few years, it is becoming increasingly clear that a great variety of state-based dynamical systems, like transition systems, automata, process calculi and class-based systems can be captured uniformly as coalgebras. Coalgebra is developing into a field of its own interest presenting a deep mathematical foundation, a growing field of applications and interactions with various other fields such as reactive and interactive system theory, object oriented and concurrent programming, formal system specification, modal logic, dynamical systems, control systems, category theory, algebra, analysis, etc. The aim of the workshop is to bring together researchers with a common interest in the theory of coalgebras and its applications. The topics of the workshop include, but are not limited to: - the theory of coalgebras (including set theoretic and categorical approaches); - coalgebras as computational and semantical models (for programming languages, dynamical systems, etc.); - coalgebras in (functional, object-oriented, concurrent) programming; - coalgebras and data types; - (coinductive) definition and proof principles for coalgebras (with bisimulations or invariants); - coalgebras and algebras; - coalgebraic specification and verification; - coalgebras and (modal) logic; - coalgebra and control theory (notably of discrete event and hybrid systems). The workshop will provide an opportunity to present recent and ongoing work, to meet colleagues, and to discuss new ideas and future trends. Previous workshops of the same series have been organized in Lisbon, Amsterdam, Berlin, Genova, Grenoble, and Warsaw. The proceedings appeared as "Electronic Notes in Theoretical Computer Science (ENTCS)", Volumes 11, 19, 33, 41, 65.1 and 82.1. Selected papers have been/are being published in Theoretical Computer Science, Theoretical Informatics and Applications, and Mathematical Structures in Computer Science. You can get an idea of the types of papers presented at previous meetings by looking at the tables of content of the above ENTCS volumes from these meetings. They are available via the ENTCS page http://www.elsevier.nl/gej-ng/31/29/23/show/Products/notes/contents.htt PROGRAM COMMITTEE Jiri Adamek, chair (Braunschweig), Corina Cirstea (Oxford), H. Peter Gumm (Marburg), Alexander Kurz (Amsterdam), Ugo Montanari (Pisa), Larry Moss (Bloomington, IN), Ataru T. Nakagawa (Tokyo), Dirk Pattinson (Muenchen) Grigore Rosu (Urbana, ILL), Jan Rutten (Amsterdam), James Worrell (New Orleans). LOCATION CMCS 2004 will be held in Barcelona on March 27-29, 2004. It is a satellite workshop of ETAPS 20034, the European Joint Conferences on Theory and Practice of Software. For venue, registration and suggested accommodation see the ETAPS 2004 Web page: http://www.lsi.upc.es/etaps04/ SUBMISSIONS Submissions will be evaluated by the Program Committee for inclusion in the proceedings, which will be published in the ENTCS series. Papers must contain original contribution, be clearly written, and include appropriate reference to and comparison with related work. Papers (of at most 15 pages) should be submitted electronically as PostScript files at the address J.Adamek@tu-bs.de. A separate message should also be sent, with a text-only one-page abstract and with mailing addresses (both postal and electronic), telephone number and fax number of the corresponding author. IMPORTANT DATES Deadline for submission: January 1, 2004 Notification of acceptance: February 1, 2004 Final version due: February 16, 2004 Workshop dates: March 27-29, 2004 For more information, please contact: Jiri Adamek, Technical University of Braunschweig phone: (0049) 5319521 fax: (0049) 5319529 e-mail: J.Adamek@tu-bs.de + + + CMCS '04 + + + FIRST ANNOUNCEMENT + + + CALL FOR PAPERS + + + From rrosebru@mta.ca Wed Sep 17 12:18:07 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Sep 2003 12:18:07 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ze1n-00073p-00 for categories-list@mta.ca; Wed, 17 Sep 2003 12:15:35 -0300 Message-ID: <3F684865.4040808@bluewin.ch> Date: Wed, 17 Sep 2003 13:41:25 +0200 From: Krzysztof Worytkiewicz X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: associated sheaf functor References: Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 10 > I don't quite understand this question. > I was interested since I am looking at the plus construction as part > of my work at the moment. Let P be a presheaf on the site (C,J) and consider the "classical" plus construction P^+(c) = colim_{R \in J(c)}Match(R,P) where Match(R,P) is the set of matching families for the cover R \in J(c) and the colimit is taken over J(c) ordered by reverse inclusion (cf. McLane & Moerdijk) . This is a nice filtered colimit so x \in P^+(c) can be expressed as an equivalence class of matching families. Suppose now that J is given by a basis K. It is not immediately clear (at least not for me) what happens in a variant of the above where Match(R,P) is taken as the set of matching families for the K-cover R. Indeed, the notion of "common refinement" for K-covers is not as handy as the one for J-covers for the task at hand since op-ordering K-covers will not necessarily give a filtered category. The other obvious candidate for a "category K(c)" where a factorisation witnessing a refinement (of K-covers) is a morphism (in the opposite category) will probably fail to be filtered as well, so I wondered if anybody allready looked at such things. I agree that a one-sentence prose might have been a bit messy... Cheers Krzysztof From rrosebru@mta.ca Fri Sep 19 14:48:19 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2003 14:48:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A0PJL-0004NC-00 for categories-list@mta.ca; Fri, 19 Sep 2003 14:44:51 -0300 Message-ID: <3F698F71.9070601@mcs.le.ac.uk> Date: Thu, 18 Sep 2003 11:56:49 +0100 From: "V. Schmitt" User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.0.0) Gecko/20020623 Debian/1.0.0-0.woody.1 X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Preprint: Flatness, preorders and general metric spaces Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 11 Dear all, I put a recent paper (just submitted) in the math Arxiv at http://front.math.ucdavis.edu/math.CT/0309209 Your comments are most welcome. Thanks. Vincent. *Title:* Flatness, preorders and general metric spaces *Author:* Vincent Schmitt *Categories:* CT Category Theory *Abstract:* This paper studies a general notion of flatness in the enriched context: P-flatness where the parameter P stands for a class of presheaves. One obtains a completion of a category A by considering the category Flat_P(A) of P-flat presheaves over A. This completion is related to the free cocompletion of A under a class of colimits defined by Kelly. For a category A, for P = P0 the class of all presheaves, Flat_P0(A) is the Cauchy-completion of A. Two classes P1 and P2 of interest for general metric spaces are considered. The P1- and P2-flatness are investigated and the associated completions are characterized for general metric spaces (enrichemnts over R+) and preorders (enrichments over Bool). We get this way two non-symmetric completions for metric spaces and retrieve the ideal completion for preorders. From rrosebru@mta.ca Fri Sep 19 14:48:19 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2003 14:48:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A0PKm-0004Vr-00 for categories-list@mta.ca; Fri, 19 Sep 2003 14:46:20 -0300 Message-ID: <3F6AB761.7010707@uni-paderborn.de> Date: Fri, 19 Sep 2003 09:59:29 +0200 From: Reiko Heckel To: categories@mta.ca Subject: categories: CfP: GT-VMT @ ETAPS 2004 Content-Type: text/plain; charset=3DISO-8859-1; format=3Dflowed Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 18 [apologies for multiply copies] Call for Papers: International Workshop on Graph Transformation and Visual Modeling Techniques http://www.upb.de/cs/ag-engels/GT-VMT04/ A satellite of ETAPS, March 27 - 28 2004, Barcelona, Spain supported by the SegraVis resarch training network http://www.segravis.org Scope and Objectives Effective applications of visual modelling techniques require tool support at a semantic level, e.g., for model analysis, transformation, and consistency management. Due to the variety of languages and methods used in different domains, an engineering approach is required which allows for the generation of such tools from high-level specifications. Graph transformations provide means to specify, at a conceptual level, complex operations on diagrams. Complementing this by techniques like * meta modelling (including OCL) * compiler construction * logic and algebraic semantics the workshop aims to bring together researchers from different communities to discuss their respective contributions to different aspects of modelling and modelling languages, like * syntax and well-formedness, * static and dynamic semantics, * analysis and verification, * refinement and transformations, and * integration and consistency of models. History This workshop is the third in the series of GT-VMT workshops: * GT-VMT 2000 in Geneva (Switzerland) at ICALP'00. * GT-VMT 2001 on Crete (Greece) at ICALP'01. * GT-VMT 2002 in Barcelona (Spain) at ICGT 2002. Program Committee The PC consist of members of the graph transformation community and external experts for complementary techniques and application areas. * Jan Aagedal (Norway) * Luciano Baresi (Italy) * Andrea Corradini (Italy) * Jose Fiadeiro (UK) * Martin Gogolla (Germany) * Martin Gro=DFe-Rhode (Germany) * Reiko Heckel (Germany) [chair] * Uwe Kastens (Germany) * Joost Kok (The Netherlands) * Mark Minas (Germany) * Andy Sch=FCrr (Germany) Submission Authors are invited to submit extended abstracts of 5 to 10 pages in ENTCS format until December 19, 2003 electronically via our submission web form (to be published here in due time). The contributions should report about ongoing research in the areas of graph transformation and visual modeling techniques according to the scope and objectives of the workshop. Position papers and contributions making methodological statements are strongly encouraged. Accepted contributions will appear in an issue of Elsevier's Electronic Notes in Theoretical Computer Science. A preliminary version of the issue will be available at the workshop. Important Dates December 19, 2003 Submission Deadline January 23, 2004 Notification of Acceptance Februar 20, 2004 Camera Ready Version March 27 - 28 Time of the Workshop --=20 Dr. Reiko Heckel URL: www.upb.de/cs/reiko.html Universit=E4t Paderborn, E4.130 Tel: ++49-05251-60-3356 33095 Paderborn, Germany Fax: ++49-05251-60-3431 Visit www.segravis.org, home of the SegraVis Research Training Network. Apply for a grant in one of 12 attractive locations throughout Europe. Join the graph transformation mailing list at www.gratra.org/list.html. From rrosebru@mta.ca Fri Sep 19 14:48:19 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2003 14:48:19 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A0PKL-0004Si-00 for categories-list@mta.ca; Fri, 19 Sep 2003 14:45:53 -0300 Date: Fri, 19 Sep 2003 07:43:00 +0400 From: Natalie Reply-To: Natalie X-Priority: 3 (Normal) Message-ID: <1631960559.20030919074300@myrealbox.com> To: categories@mta.ca Subject: categories: duality theory 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: 19 I'm interested in duality theory for an arbitrary category, especially for "classical" algebraic categories( e.g., SEMI, the category of semigroups and their homomorphisms). I want to obtain such result: to build general "dualization algorithm" (for varieties), and "to hang" fundamental operations and identities at each step of this "algorithm", where they arise. But I haven't possibility to get books (I'm think these books would help me) such as Borceux, "Categorical Algebra"; Clark/Davey, "Natural Dualities for the Working Algebraist"; Johnstone, "Stone Spaces"; Manes, "Algebraic theories" and more others. IS THERE DUALITY THEORY FOR THE CATEGORIES described above? ------------------------------------------------------------- My ideas in this direction are restricted only by the next: 1. Using factorization systems(in particular, via congruences lattice) for the category of algebras(but HOW in general situation, without special methods?) 2. Using inclusion of the category TH^op (considering as theory in the sense of (Barr/Wells)'s "Toposes, triples and theories") in the category MOD(TH) of models for this theory. 3. Via iso of categories (SET^(W))^op = CABA_(W^op) (for given endofunctor( or, narrow concept, functor part of triple) W on SET). 4.(main!!) Via generalization of the standart duality example (ComRing1)^op ~=~ AffSchemes What is the role of Birkhoff's subdirect representation theorem for algebras in the construction of the topological space SPEC, how we can construct (in general situation) the sheaf of algebras on this space? And the main: what the grounds of this construction( if it is possible)? How to prove directly the duality between algebraic and geometric theories ( if it is available)? -------------- The next questions/exersices parallels this "algorithm": A. The best test for this general theory --- to apply it for the well-known duality (ComRing1)^op ~=~ AffSchemes, mentioned above. B. If (4.) is available, how we can in general terms to obtain the equivalence between the category CABA_(W^op) in (3.) and the correspondent category given by construction in (4.)? C.(deeper) How the duality theory connect algebra, logics and topology? D. What is this "algorithm in terms 2-categories?" ------------------------------------------------------------- Natalie natalie_reznik@myrealbox.com From rrosebru@mta.ca Tue Sep 23 16:09:24 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2003 16:09:24 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A1sSl-0002lc-00 for categories-list@mta.ca; Tue, 23 Sep 2003 16:04:39 -0300 X-Originating-IP: [194.66.147.4] X-Originating-Email: [cft71@hotmail.com] From: "Christopher Townsend" To: categories@mta.ca Subject: categories: Higher Order Yoneda? Date: Mon, 22 Sep 2003 15:03:26 +0000 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Message-ID: X-OriginalArrivalTime: 22 Sep 2003 15:03:26.0861 (UTC) FILETIME=[AD07D3D0:01C3811A] Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 20 I was looking for a reference (or correction!) to the following observation in indexed category theory. Let E be a cartesian category and H an E-indexed category (that is H is a functor from E^op to CAT, where CAT is some background category of possibly large categories). Then, if C is an internal category in E we have a categorical equivalence Nat[Cat(_,C),H]=H(C_0) where C_0 is the object of objects of C. The objects of Nat[Cat(_,C),H] are the natural transformations and the morphisms are the modifications (see, e.g. definition B1.2.1(c) in Johnstone's Elephant). On objects, this equivalence is just Yoneda's lemma, so surely it has been observed already that it extends to this 2-categorical statement? Best wishes, Christopher Townsend From rrosebru@mta.ca Tue Sep 30 13:32:51 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Sep 2003 13:32:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A4NK4-0006xm-00 for categories-list@mta.ca; Tue, 30 Sep 2003 13:26:00 -0300 Mime-Version: 1.0 X-Sender: street@icsmail.ics.mq.edu.au Message-Id: In-Reply-To: References: Date: Thu, 25 Sep 2003 12:34:14 +1000 To: categories@mta.ca From: Ross Street Subject: categories: Re: Higher Order Yoneda? Content-Type: text/plain; charset="us-ascii" ; format="flowed" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 [Note from Moderator: Apologies to Ross for the inadvertent delay in posting this.] >I was looking for a reference (or correction!) to the following observation >in indexed category theory. Try, for example, Theorem (5.15) of 13. Cosmoi of internal categories, Transactions American Math. Soc. 258 (1980) 271-318; MR82a:18007. Regards, Ross PS Allow me to correct an annoyingly wrong gratuitous word on the same page as that Theorem; the word "full" should be deleted on the second line of (5.13). From rrosebru@mta.ca Tue Sep 30 13:34:21 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Sep 2003 13:34:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A4NPT-0007Pa-00 for categories-list@mta.ca; Tue, 30 Sep 2003 13:31:35 -0300 Date: Fri, 26 Sep 2003 16:01:50 +0100 (BST) From: Tom Leinster To: categories@mta.ca Subject: categories: New address: T. Leinster 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 Hello, I'm moving to Scotland to start a new life. My address there is t.leinster@maths.gla.ac.uk and my web page, once I arrive and set it up, will be linked to from http://www.maths.gla.ac.uk/people/?id=295 My IHES email account will probably expire in a month or so (and my Cambridge account as soon as anyone notices I haven't been there for a year and takes exception). Best wishes, Tom From rrosebru@mta.ca Tue Sep 30 13:34:30 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Sep 2003 13:34:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1A4NPi-0007QQ-00 for categories-list@mta.ca; Tue, 30 Sep 2003 13:31:50 -0300 From: Jpdonaly@aol.com Message-ID: <191.1f8940ae.2ca4976e@aol.com> Date: Thu, 25 Sep 2003 15:09:34 EDT Subject: categories: Categories of elements (Pat Donaly) To: categories@mta.ca 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: 23 To all category theorists: In various textbooks, I see reference to the common comma category Elts(G), which is called the "category of elements of functor G". This category seems to be drastically misnamed. Does anyone agree? Here is my side of the story, beginning with a review of the nature of Elts(G) and some of its significance. G is a functor from a small category C into the category F of small functions. Denoting the singleton {0} of the void set 0 by 1 (as usual), Elts(G) consists of all triples (g,a,f) with a in the domain category C and g:1--->G(codomain a), f:1--->G(domain a) such that g = G(a) o f, where "o" denotes function composition. (Warning: By my conventions, a.domain a = a; codomain a.a = a.) The composition of Elts(G) is defined by (h,b,g)(g,a,f) = (h,ba,f). The objects are the Elts(G)-morphisms of the form (f,u,f), where u is a C-object, and the map (f,u,f)-->f(0) identifies each of these with an element of the set G(u), so that the convention of naming categories after their objects (to the extent possible) is what presumably leads to calling Elts(G) "the category of elements of the values of G at objects" or, for short, "the category of elements of G". Several basic features of Elts(G) are exposed by treating it as a subcategory of a product BxC of a transition category B with the domain category C of G. To define B, let X be the set of functions f:1--->G(u) as u varies over the objects of C; B is then the full transition category or groupoid of X, that is, the self-product XxX with the transition composition (h,g)(g,f) = (h,f). But rather than taking the ordinary cartesian product for BxC, one uses the attachment product consisting of those triples (g,a,f) with (g,f) in B and a in C, so that C-morphism a is viewed as being attached on its left to g and on its right to f. Then Elts(G) inherits by restriction the projection functor (g,a,f)-->a, which is reasonably called the detaching functor from Elts(G) into C---this functor will be generically denoted by "det". There is also the transition projection (g,a,f)-->(g,f) which maps Elts(G) functorially onto a transitive relation on X, and the rule (g,a,f)-->g defines the entwining function of the canonical natural transformation which entwines the constant functor (g,a,f)-->1 on Elts(G) with the function composite functor G o det: Elts(G)--->F. (A constant functor with value 1 will be denoted generically by "delta(1)".) There are many important examples: If C is a group with object e so that G is group action with action set Y=G(e), then, to within the identification (g,a,f)-->(g(0),a,f(0)), Elts(G) is the traditional idea of a G-action as a function from CxY into Y after correction to remove the categorically problematic product CxY. If C is actually the group RxR, R being the additive group of real numbers and G the action of C on the real affine plane by translation, then Elts(G) is essentially the category of attached planar vectors as used in Engineering Statics 101, which is why it seems appropriate to continue to use the word "attach" in the context of a more general function-valued functor G:C--->F. If C is a certain type of monoid, then Elts(G) is a semiautomaton. Among its theoretical services is the fact that Elts(G) plays a role in the construction of Kan extensions along inclusion functors, thus in particular in the theory of induced group actions. It plays an analogous part in sheafification relative to a Grothendieck site, and it is used to show that representable functors are dense in the set of function-valued functors on C, providing, according to Mac Lane and Moerdijk, "a plethora of tensor products". Even more basically, if the domain C of G is discrete, then Elts(G) is a coproduct a.k.a. a disjoint union of (the object values) of G. As will be noticed in a moment, the set of functors A:C--->Elts(G) which are right inverse to the detaching functor det on Elts(G)---that is, the "attaching functors" into Elts(G)---constitute a (small) limit object of G, thus, in the case of discrete C, a product of the sets G(u). From these examples it appears that Elts(G) is sufficiently important to require a unique and unambiguous nomenclature, but, unfortunately, the things in Elts(G) are really not the elements of G. The (global) elements of an object u in a category are generally agreed to be the morphisms from a given terminal object t to u. This convention terminologically extends the observation that the functions f from the terminal object 1 into a (small) set X can be identified with the elements of X by the mapping f-->f(0). The general definition has the virtue that each terminal object has only one element, as should surely be the case, and the representable functor of t provides a plausible (but not necessarily effective) attempt to convert a given category into a category of functions between sets of elements. In fact, this language seems to have found broad acceptance. But then the functor G is an object in the morphismwise (i.e. "vertical") composition category F^C of natural transformations whose (fully extended) entwining functions map from C into F, and, because 1 is terminal in F, the constant functor delta(1) on C is terminal in F^C. So G already has a set of elements, namely, those natural transformations which entwine delta(1) with G. Such elements of G are not in Elts(G) in any sense. At first sight this terminological conflict might seem to be innocuous, since Elts(G) and global elements of G occur in somewhat disparate contexts, but the apparent separation does not hold up well when one considers how close the set of global elements of G is to being a limit object of G. The only problem with it is that it is not small; that is, it is not in the codomain category F of G, and the only reason for this defect is that F, the common codomain of the entwining functions of the things in F^C, is not small. Mac Lane in CWM gives an ad hoc workaround which replaces, for a given G, the category F with a small, G-dependent category of small functions, but this approach effectively isolates G by artificially depriving it of morphisms into functors which do not happen to map into Mac Lane's ad hoc replacement category; so one needs a more perspicuous method of eliminating F and its untoward largeness. F. W. Lawvere was apparently motivated by such considerations to introduce comma categories in his thesis, an approach which works very well in addressing the present awkwardnesses. One defines a category Law(C) whose objects are the categories Elts(G) as G ranges through the functors in F^C and whose morphisms are the cocompatible functors S:Elts(G)--->Elts(H) between such objects. The composition is function composition of functors, and "cocompatible" means that S does not disturb middle components of attached C-morphisms or, alternatively put, det o S = det, where "det" continues to be the generic symbol for a detaching functor. Then, if s in F^C entwines functor G with functor H, there is a cocompatible functor S:Elts(G)--->Elts(H) which is evaluated at an attached C-morphism (y*,a,x*) by S(y*,a,x*)=(s(codomain a)(y)*, a, s(domain a)(x)*), where I use y*, for example, to denote that function f:1--->G(codomain a) whose value is y. Then the assignments s-->S define a functorial isomorphism---which I call the Lawvere isomorphism (but should this be attributed to someone else?) from F^C onto Law(C). Moreover, the objects Elts(G) of Law(C) are small. This implies, of course, that the homset of cocompatible functors from Elts(G) into Elts(H) is also small. Elts(delta(1)) evidently consists of triples of the form (0*,a,0*) and can thus be identified with C by the detaching functor (0*,a,0*)-->a on Elts(delta(1)). With this identification, a cocompatible functor from Elts(delta(1)) into Elts(G) becomes an attaching functor into Elts(G), so that, in the Lawvere picture, the global elements of G are the attaching functors into the category of...uh...elements of G. The set of such global elements is plainly small and therefore must be what God intends to be the standard limit object of G, except that it is difficult to believe that God would use such a verbal collision to say what a global element is. These are my grounds for believing that Elts(G) has to be renamed and redenoted. As a related suggestion, I might recommend dropping the habit of referring to categories by the names of their objects. This illogicality immediately inhibits use of the subcategory concept (I still don't know what categorists use to refer to the subcategory formed by the monomorphisms in an abstractly given category), and then it just goes looking for the sort of trouble which has turned up as "the category of elements of G". Besides this, the terminology "comma category" is disrespectful of category theory, itself, due to the inappropriateness of naming a fundamental, overarching categorical concept after a punctuation mark. ("Slice category" doesn't seem to be any better.) Given the precedent of attached vectors, which are used in a rough sense even by sophisticated diagrammaticists, the category Elts(G) is obviously some kind of attachment category, and since the transition components of any of its morphisms all have domain 1, it is a based attachment category with base 1 or just a basement category---or even just a basement denoted by something like G/1, if you're used to placing domains on the right. Anyway, this is approximately what I use in my study notes, and so far it works fine. At the same time, I would be interested in seeing sharp, well reasoned criticisms of this note provided that they are written at about the same technical level so that I can understand them. I would like to emphasize that, aside from what may be terminologically or notationally novel here, I am not making a substantial research proposal or claiming priority for any discoveries. I have no reason at all to doubt that all of the mathematics here is well known. Pat Donaly