From MAILER-DAEMON Fri Nov 30 07:18:29 2007 Date: 30 Nov 2007 07:18:29 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1196421509@mta.ca> X-IMAP: 1193925084 0000000060 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 Thu Nov 1 10:42:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Nov 2007 10:42:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1InaIg-0003ES-Kd for categories-list@mta.ca; Thu, 01 Nov 2007 10:41:34 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=MACINTOSH; format=flowed To: categories@mta.ca From: Paul Taylor Subject: categories: DVI, PDF and TAC Date: Thu, 1 Nov 2007 12:57:51 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: A X-Keywords: X-UID: 1 Mike Barr reported that Adobe Acrobat 8 tacitly suppresses all ligature glyphs of = the fi, fl, ff, ffi, and ffl sort and displays blanks in their place and then that I have to admit that I never tested it, just copied the = complaint from texhax but nevertheless this presumably gives us some idea why, at TAC, we still consider the dvi to be the official format. However, as I shall demonstrate, the rest of the world nowadays regards PDF as the standard format in which to publish technical documents. DVI (the normal output from LaTeX) was based on the 1950s Monotype typesetting system, and puts characters from various fonts at given positions on the page, but cannot rotate them, and has no graphics capability. The fonts also have to be supplied separately. On the other hand, it has the virtues of being a compact and simple format that future digital archeologists would have no difficulty in deciphering. Adobe's PDF and PostScript have general graphics capability. By insisting on DVI, "Theory and Applications of Categories" severely limits the ways in which authors can express their mathematical ideas. But its restrictions go further than this: the use of ANY macro package other than those by Mike Barr and Kris Rose is forbidden! Does anyone know of another journal that publishes primarily in DVI? One candidate might be the journal of the TeX Users' Group, tug.org/TUGboat but even that uses PDF. When I enquired about this, Karl Berry (who also wrote the Web2C Unix implementation of TeX) replied that DVI files are not self-contained, so they simply don't work for online archives. Turning to other respositories, arXiv.org generates papers in various formats on-the-fly. On its page for each paper, eg arxiv.org/abs/math/0512110 it offers PostScript and PDF, with other options (including DVI) only being available via another link. On my own web site, www.PaulTaylor.EU I also offer my papers in various formats, albeit statically, with DVI listed first. However, the downloads in September and October were DVI: 210 PDF: 1704 PS.GZ: 75 BKLT.GZ: 52 This includes 504 PDF and 69 other downloads of "Proofs and Types", but excludes the lecture slides and scanned manuscripts that I only offer in PDF. So, 85% of my readers choose PDF, even though I also offer=20 DVI. Looking at my colleagues' web pages, my impression is that most of them ONLY offer their papers in PDF nowadays. I can't give authoritative figures for this, as Hypatia has been dead for seven years. Papers that people send me as attachments to read or referee generally come in PDF=20= too. Even TAC has published a number of papers that it only offers in PS or PDF. Maybe Bob Rosebrugh could tell us how many downloads there have been in the various formats from the main TAC web site at MTA. Turning to software, although PDF legally belongs to Adobe, I really don't care about bugs in their programs, as I never use them. For me and anyone else who writes in LaTeX, PDF is de facto the language that is output by pdflatex and input by xpdf and ghostview (gv). It is a well documented open format, as Peter Selinger and Andrej Bauer have pointed out, citing www.adobe.com/devnet/pdf/pdf_reference.html and en.wikipedia.org/wiki/Portable_Document_Format See www.PaulTaylor.EU/technote.html for brief explanations and links to further information. When using my Mac OSX laptop, I use the program (Preview) that came with it to read PDF files. This also understands several other file=20 formats, but not DVI. When I run "latex" in the current tetex distribution of the TeX system, it actually invokes pdftex (the version of TeX that was modified by Jir=F5 Zlatuska to generate PDF instead of DVI) with \pdfoutput=3D0. Google indexes various other formats of web pages besides HTML, www.google.com/help/faq_filetypes.html including PDF and several versions of MicroS**t W*rd, but not DVI. Returning to TAC, I quote from its "Author Information" at tac.mta.ca/tac/authinfo.html An article must be submitted as a single source file. All macros must be included at the beginning of the file. Any macro that is not actually used should be deleted from the source file. The only exception is diagram macro packages. The currently acceptable diagram macro packages are those authored by Barr and Rose and Moore(Xy-pic). Recall however that authors are expected to provide source code which produces usable .dvi from these packages (see Note 4 above). Do not rely on .ps options. The author is responsible to ensure that the current version of a macro package has been used. Leaving the DVI/PS/PDF issue aside, why is it necessary for articles to be submitted as single source files? According to Mike Barr, in an email to me dated 10 July 2002, our rule is, no inclusions, with the following exceptions: diagram packages, including yours, and packages that are part of standard distributions. This is so that we can store each paper in a single file, without a growing (and essentially = unidentifiable) directory of inclusions. Now it has been customary for as long as I have had anything to do with computers that software (by which I mean both programs and papers in=20 this context) is developed in modular parts, divided into several files but collected in a directory or folder. In particular, programming with macros should be separate from writing text about mathematics. Indeed, the design of LaTeX2e presupposes this in its facility for passing options to macro packages via \usepackage[options]{package} Has Mike Barr not heard of sub-directories, or of tar-archives? Why does he create so much inconvenience for TAC authors for such a trivial benefit to himself? Notice, in particular, that my diagrams package was approved for use in TAC in 2002, but is now forbidden. Is this perhaps because Mike disgrees with me over DVI? The design decision to use PostScript inclusions to rotate diagonal arrows in my package was made in 1992, in consultation with users. Neither Mike nor Bob nor anyone else at the time argued against that decision, whilst several people said it was a good idea. The package web page, www.PaulTaylor.EU/diagrams explains the background to this decision, and also how to make half- decent diagrams using the UglyObsolete pre-1992 code if you REALLY need to use pure DVI. I would strongly request that anyone who uses my package, or who wishes to reply to my comments here, should READ this web page first.=09 TAC began two or three years AFTER this decision was made, but Mike and Bob did not discuss their pure DVI policy with me. If they had done so, we might have been able to lobby the maintainers of DVI programs (such as Paul Vojta of XDVI) to add support for rotation, or they might have persuaded me to improve the old DVI code in my package. The time for doing either of these things has now long past. In his email of 23 January 1990 to me and 24 other people that resulted in the establishment of the "categories" forum and subsequently the TAC journal, Bob Rosebrugh said it seems clear (maybe only to me) that a TeX-based journal is a starting point. My guess is that LaTeX together with some version of Mike's macros should be the starting standard. It would appear that they are still trying to impose this standard. There are plenty of people who regard MY package as the standard. For example, I recently heard from a blind mathematician who "draws"=20 diagrams by dictating the input language of my package to his wife, lalitalarking.blogspot.com/2007/09/great-dictator.html In conclusion, I call upon Mike, Bob and the Editorial and Advisory Committees of TAC to come out of the 1980s, and support the production=20= of papers using the modern typographical software that other journals use. Paul Taylor PS. Whichever /"Mr" Paul taylor/ it was who wrote a joint paper with Phil Scott on locally cartesian closed categories is no doubt greatly honored to find that "Monsieur" Jean benabou regards him as a special case. However, M. Benabou will perhaps be disappointed that he is neither the author of nor cited "alongside [the] lot of complete nonsense" to which I alluded. This may be found at computing.unn.ac.uk/staff/cgnr1/liege_quantum03.pdf it cites Baez, Barr, Bell, Birkhoff, Bishop, Bridges, Dirac, Dummett, Ehresmann, Einstein, Freyd, Heyting, Hilbert, Johnstone, Leibniz, Mac Lane, Peirce, Scedrov, Troelstra, Turing, Wells and me. From rrosebru@mta.ca Thu Nov 1 13:59:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Nov 2007 13:59:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IndJx-0002M4-Gb for categories-list@mta.ca; Thu, 01 Nov 2007 13:55:05 -0300 Date: Thu, 1 Nov 2007 12:35:08 -0300 (ADT) From: Bob Rosebrugh To: categories Subject: categories: Diplay Formats MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 2 This is to invoke a limit on the current categories discussion of document display formats to the next 48 hours. Paul's post redirects the discussion towards indirectly categorical relevance, but there are other forums where such matters are not peripheral. A brief comment on the Triassic tendencies of TAC and its editors will follow. regards to all, Bob From rrosebru@mta.ca Thu Nov 1 13:59:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Nov 2007 13:59:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IndIp-00029m-5V for categories-list@mta.ca; Thu, 01 Nov 2007 13:53:55 -0300 Date: Thu, 1 Nov 2007 09:47:45 -0700 From: Toby Bartels To: Categories Subject: categories: Surjective equivalences (Was: Historical terminology,.. and a few other things.) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 3 Jean Benabou wrote in part: >Surjective equivalences are much better than mere equivalences because : I would be grateful for a reference or list of references about "important properties and what they are good for" for surjective equivalences. --Toby From rrosebru@mta.ca Thu Nov 1 22:17:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Nov 2007 22:17:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Inkwt-0005vo-ML for categories-list@mta.ca; Thu, 01 Nov 2007 22:03:47 -0300 Date: Thu, 1 Nov 2007 21:56:01 -0300 (ADT) From: Bob Rosebrugh To: categories cc: Paul Taylor Subject: categories: Re: DVI, PDF and TAC MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 4 With regrets that this response is not briefer... On Thu, 1 Nov 2007, Paul Taylor wrote: > Does anyone know of another journal that publishes primarily in DVI? Many electronic journals in mathematics post dvi files. Like TAC, most of these post, and also archive, all of dvi, ps and pdf. TAC's policy on dvi has evolved since 1995. That policy will continue to change, no doubt at a slower rate than some would wish. None of us knows what the digital world will look like in 10 years, but careful choices made for TAC over a dozen years ago have been validated. > Maybe Bob Rosebrugh could tell us how many downloads there have > been in the various formats from the main TAC web site at MTA. Inevitably nowadays, most of the web traffic on sites like TAC's is for caching, so such figures for any TAC site mean nothing. If Paul's counts record human usage, then what is surprising is how many of the downloads were *not* pdf. ... > Why does he create so much inconvenience for TAC authors for such > a trivial benefit to himself? This request (for a single source file) is seen as, at worst, a trivial inconvenience by most authors, and it simplifies the lives of editors who volunteer their time and knowledge. Note that TAC's submission requirements for authors describe what we would like to see. Sometimes we don't. We are grateful on the many occasions when authors comply. In practice, TAC editors are flexible and are working with authors who gladly cooperate in publishing a visually pleasing article. We much prefer diagrams based on xypic, but a glance at recent numbers shows that leeway is available. ... > It would appear that they are still trying to impose this standard. Paul has made a heroic search for a conspiracy, but, alas, has not found one. I don't particularly remember writing the sentence he quotes from early 1990, and hadn't heard of his diagram package back then. My 1990 suggestion was not motivated by an intention to exclude him 15 years later. The misperception of malice is regretted. Bob Rosebrugh From rrosebru@mta.ca Fri Nov 2 16:23:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Nov 2007 16:23:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Io24c-0003Te-Po for categories-list@mta.ca; Fri, 02 Nov 2007 16:20:54 -0300 To: categories@mta.ca Subject: categories: TYPES small workshop on Effects and Type Theory Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Date: Fri, 02 Nov 2007 19:37:00 +0200 From: Tarmo Uustalu Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 5 In the frame of the extended EU FP6-funded TYPES project, we are organizing an ad hoc "small workshop" on integration of effects into type-theoretic programming/reasoning. This is an informal event and attendance is not confined to people involved in TYPES. On the contrary, attendance and contributions from outside the TYPES consortium are most welcome. The invited speakers are Paul Levy and Aleksandar Nanevski. --- Call for contibutions and participation Workshop on Effects and Type Theory, EffTT Tallinn, Estonia, 13-14 December 2007 http://cs.ioc.ee/efftt/ a "small workshop" of the TYPES project Background The syntax and semantics of impurities of computation known as effects have been an important challenge for functional programming. Today, we tend to employ categorically inspired tools such as monads, Lavwere theories and arrows, but also more pragmatic approaches such as uniqueness typing. Effects are an issue also for type-theoretic programming and reasoning, where a number of aspects make them specifically interesting. First, we do not yet know what the best dependently typed generalizations of our simply typed tools are, although we hope they would reinforce the dual utility of type-theoretic calculi as programming languages and logics. Second, this duality specifically forces that pure computations must terminate, so even nontermination is an impurity and potentially an effect. Third, is it not likely that the type-theoretic glasses can help us see more clearly the particularities of external-world effects such as true destructively updatable state and true interactive input-output? Thus, this workshop is exactly about effects and type theory. Topics of interest include * all kinds of dependent generalizations of monads and more * type-theoretic language design for effects * type-theoretic effectful programming methodology * time, nontermination and type theory * state and type theory, including combinations of Hoare-like logics and type theory * interactive input-output and type theory * theories of external-world effects * type theory and concurrency * type-theoretic descriptions of physical systems * and any further topics about effects and type theory Invited speakers Our invited speakers are Paul Levy (Birmingham) and Aleksandar Nanevski (Microsoft Research, Cambridge). Contributing a talk The rest of the programme will be based on contributed talks and discussions. If you would like to contribute a talk, send a title and abstract to efftt(at)cs.ioc.ee by 21 November 2007. Organizers The workshop organizers are Thorsten Altenkirch, Marino Miculan and Tarmo Uustalu. Venue The workshop will take place in the building of the Estonian Academy of Sciences on Tallinn's Dome Hill. The workshop dates are during the Tallinn Christmas market and the Christmas Jazz festival of Jazzkaar. Participating To register, please drop an email to efftt(at)cs.ioc.ee as soon as possible, but not later than 21 November 2007. Attendance is not confined to people involved in the TYPES project; the workshop is open to anyone interested. From rrosebru@mta.ca Fri Nov 2 16:23:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Nov 2007 16:23:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Io20Z-0002wx-Cz for categories-list@mta.ca; Fri, 02 Nov 2007 16:16:43 -0300 Date: Fri, 02 Nov 2007 16:26:41 +0100 From: Clemens Kupke MIME-Version: 1.0 To: cmcs08@cwi.nl Subject: categories: CMCS 2008: First call for papers Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 9th International Workshop on Coalgebraic Methods in Computer Science http://www.cwi.nl/projects/cmcs08/ Budapest, Hungary April 4-6, 2008 The workshop will be held in conjunction with the 11th European Joint Conferences on Theory and Practice of Software ETAPS 2008 March 29 - April 6, 2008 Aims and Scope During the last few years, it has become 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, and analysis. 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); probabilistic systems as coalgebras; algebras versus coalgebras; coalgebraic specification and verification; coalgebras and (modal) logic; game theory in coalgebra; 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, Warsaw, Barcelona and Vienna. The proceedings appeared as Electronic Notes in Theoretical Computer Science (ENTCS) Volumes 11,19, 33, 41, 65.1, 82.1, 106 and 164.1. You can get an idea of the types of papers presented at the meeting by looking at the tables of contents of the ENTCS volumes from those workshops. Location CMCS 2008 will be held in Budapest on April 4-6, 2008. It will be a satellite workshop of ETAPS 2008, the European Joint Conferences on Theory and Practice of Software. Programme Committee Jiri Adamek (chair, Braunschweig), Corina Cirstea (Southampton), Neil Ghani (Nottingham), H. Peter Gumm (Marburg), Bart Jacobs (Nijmegen), Clemens Kupke (co-chair, Amsterdam), Alexander Kurz (Leicester), Ugo Montanari (Pisa), Larry Moss (Indiana), John Power (Edinburgh), Jan Rutten (Amsterdam), Lutz Schroder (Bremen), Tarmo Uustalu (Tallinn), Yde Venema (Amsterdam), Hiroshi Watanabe (Osaka). Submissions Two sorts of submissions will be possible this year: Papers to be evaluated by the programme committee for inclusion in the ENTCS proceedings: These papers must be written using ENTCS style files and be of length no greater than 20 pages. They must contain original contributions, be clearly written, and include appropriate reference to and comparison with related work. If a submission describes software, software tools, or their use, it should include all source code that is needed to reproduce the results but is not publicly available. If the additional material exceeds 5 MB, URL's of publicly available sites should be provided in the paper. Short contributions: These will not be published but will be compiled into a technical report of the Technical University of Braunschweig. They should be no more than two pages and may describe work in progress, summarise work submitted to a conference or workshop elsewhere, or in some other way appeal to the CMCS audience. Both sorts of submission should be submitted in postscript or pdf form as attachments to an email to cmcs08@cwi.nl. The email should include the title, corresponding author, and, for the first kind of submission, a text-only one-page abstract. After the workshop, we expect to produce a journal proceedings of extended versions of selected papers to appear in Theoretical Computer Science. Important Dates Deadline for submission of regular papers: January 13, 2008. Notification of acceptance of regular papers: February 11, 2008. Final version for the preliminary proceedings: February 18, 2008. Deadline for submission of short contributions: March 10, 2008. Notification of acceptance of short contributions: March 17, 2008. For more information, please write to cmcs08@cwi.nl From rrosebru@mta.ca Fri Nov 2 16:23:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Nov 2007 16:23:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Io23q-0003Kq-6v for categories-list@mta.ca; Fri, 02 Nov 2007 16:20:06 -0300 Subject: categories: Re: Comma categories Date: Fri, 02 Nov 2007 12:12:28 -0400 From: wlawvere@buffalo.edu To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 7 Dear Uwe You are right in thinking that there should be such=20 an exposition because the construction is explicitly=20 or implicitly involved in so many contexts that a=20 formal summary would be useful. Unfortunately,=20 I know of no such exposition though Hugo Volger=20 started one many years ago. As you can see from the TAC Reprint of my=20 thesis, the original motivation was to=20 be able to state the definition of adjointness in a=20 wholly elementary way for arbitrary categories=20 without involving enrichments in some fixed category of sets. If A is a reflective subcategory in=20 some X and if B is coreflective in the same X, then=20 composing the implicit functors yields an adjoint=20 pair between A and B. The point is that conversely=20 any adjoint pair can be so factored through a third=20 "adjunction" category X and the universally available=20 choice has this simple construction as a pullback. It proved to be the appropriate tool for calculating Kan extensions, adequacy comonads, fibrations,etc. Grothendieck defined slice categories and Artin the=20 gluing, both of which are special cases of this construction. Although inserters are interdefinable (like equalizers vs pullbacks), some consider inserters more basic:=20 given x:A->C and y:B->C, one can take the=20 inserter of the two composites AxB->C to obtain=20 the construction under discussion.=20 In the special case A=3DB=3D1 (when the inserter and the=20 "comma" category are the same) we obtain the homset=20 (x,y) of two objects of C. The latter was the reason=20 for my notation: it generalizes a frequent notation for=20 hom.[Recall that every object belongs to a unique=20 category; thus the standard notation C(x,y) is actually redundant (if C is not enriched), though easier to understand. Either notation is preferable to the=20 excessive HomsubC, a back formation not be confused with the informative HomsubR when C arises from=20 adjoining some additional structure R to a given base.] =20 Concerning the bizarre name: (1) I had neglected to give the construction any name,=20 so (2) one started giving it a name based on reading=20 aloud the notation: x comma y; (3) some continued the "name" but changed the notation to a vertical arrow. Since it is well justified to name a category for its=20 objects, and since the effect of insertion is to create=20 objects with one ingredient more of structure, recent=20 discussions here have proposed the name/notation Map(x,y) [or for emphasis Map(subC)(x,y)] for the category with its faithful functor to AxB. Although I often use the word "map" interchangeably with "morphism", note that the above suggests a more concrete content: philosophically, in order to confront=20 objects in two categories A and B, it is necessary to=20 first functorially transport them into a common=20 category C. For example to map a 2-truncated simplicial=20 set to a diffentiable manifold (such as a piece of paper) one first interprets each in appropriate ways as=20 topological spaces, and the resulting objects form a=20 category (having full subcategories of "cartographical"=20 interest). =20 I would be happy to offer a prize for the best exposition! Bill Quoting Uwe Egbert Wolter : > Dear all, >=20 > I'm looking for a comprehensive exposition of definitions and > results > around comma/slice categories. Especially, it would be nice to have > something also for non-specialists in category theory as young > postgraduates. Is there any book or text you would recommend? >=20 > Best regards >=20 > Uwe Wolter >=20 >=20 >=20 >=20 From rrosebru@mta.ca Fri Nov 2 16:23:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Nov 2007 16:23:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Io202-0002tG-Vf for categories-list@mta.ca; Fri, 02 Nov 2007 16:16:11 -0300 Date: Fri, 2 Nov 2007 12:44:10 +0100 (CET) From: Jiri Adamek To: categories net Subject: categories: Functor derivatives - a question and a result MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 8 Andre Joyal defined derivatives of analytic functors in his 1986 paper. Recently I heard the more general definition of a derivative F' of an endofunctor F defined via a universal sub-cartesian transformation from F'xId into F. Who is the author of this definition? The following result seems to indicate that outside of the realm of analytic functors derivatives may not be really useful: Theorem. Every non-faithful functor F:Set -> Set has the derivative F' = 0 (the constant functor to the empty set). xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx From rrosebru@mta.ca Fri Nov 2 20:40:00 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Nov 2007 20:40:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Io608-0002Lq-9I for categories-list@mta.ca; Fri, 02 Nov 2007 20:32:32 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories From: JeanBenabou Subject: categories: References Date: Fri, 2 Nov 2007 02:55:08 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 9 Dear colleagues, I hope someone, and in particular Prof. Peter Johnstone, will help me with the following information. I thought I had, with Jacques Roubaud, proved in our joint note at the "Comptes Rendus" which I mentioned in my previous mail proved a theorem on Monads and Descent. I must have been mistaken, and also the many persons who quoted this note, because in El Proposition 1.5.5 is the same theorem, but attributed to J. Beck. I immediately "rushed" to the monumental bibliography of El to find the reference, and there, big surprise, there was no J. Beck at all among the 1262 references. Thus i'd greatly appreciate to have the date and paper of the paper where Beck proved this theorem, and the precise statement he made, in particular, did he prove his theorem in the general context of fibered or indexed categories, or only in some very special case. Many thanks for your help From rrosebru@mta.ca Sat Nov 3 06:44:29 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 03 Nov 2007 06:44:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IoFOr-0004Zq-My for categories-list@mta.ca; Sat, 03 Nov 2007 06:34:41 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Ross Street Subject: categories: Apropos a couple of current topics Date: Sat, 3 Nov 2007 20:03:35 +1100 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 10 History can be harder than mathematics. Yet, it is a worthy goal to get it right. This can require discussion and feedback. Here are some of my memories which I am quite happy for people to correct if they have a fuller picture. Jean Benabou invented bicategories. In SLNM47 you will find the particular example of a bicategory Spn(E) whose morphisms are spans in a pullback-complete category E. You will also find the convention to refer to properties holding in the homs as local. I always thought it nice that the homs in Spn(E) were slice categories E / a x b, thereby unifying two uses of "local". You will also find in that SLNM47 paper, the notion of morphism of bicategories and of homomorphism of bicategory. These a very useful concepts. They do compose in their own way. I believe there was no attempt to deny that the "indexed categories" of Pare-Schumacher are category-valued homomorphisms. The 1969-70 academic year at Tulane University Math Dept was dedicated to Category Theory. Jack Duskin and I were there (doing some teaching as well as research) for the whole year. Saunders Mac Lane and Eduardo Dubuc were there for the first semester. Bernhard Banaschewski and Z. Hedrln were there for the second semester. However, we had a lot of visitors as well. In particular, Jean Benabou visited sometime in the first semester. In particular, I learnt from Benabou's lectures about the "Chevalley condition" for fibrations and how descent data were Eilenberg-Moore algebras. Jean gave me a copy of his Comptes Rendu article with Jacques Roubaud. Very soon after Jean Benabou left, Jon Beck arrived. He asked me what the various visitors had talked about. When I told him about Benabou's lecture on descent, he said that that was what he had planned to talk about ("triples" and descent). I encouraged him to do so but he decided to change his topic. His topic by the way was also very interesting: using monads -- sorry, triples -- in homotopy theory and categorical coherence. This was before operads! I wondered what would happen to Beck's work on descent. Category theorists were not prolific publishers. Then I found reference to the "Beck condition" in Bill Lawvere's papers of the time: it was what Benabou had called the "Chevalley condition". So, when I had need for a 2-categorical version of this involving comma objects instead of pullbacks, I called it the "Beck-Chevalley condition". This 2-categorical version expresses pointwiseness of Kan extensions and embodies Lawvere's formula for such extensions. Also by the way, Lawvere's comma categories are generalized slice constructions so I proposed (not really wishing to introduce new notation but somewhat worried about using (f, g) as more than just the pair) using f/g for functors f and g into the same category. Now, as much as I would love SIX bottles of GOOD champagne, I am not going to submit a suggestion for Jean's challenge. Composition of fibrations is a wonderful thing as is composition of homomorphisms of bicategories; but they do different jobs. It is hard enough to say fibrations are composable from the homomorphism viewpoint! There is a thing about this that requires a mixture of the two views. Regard one fibration p : E --> A as a homomorphism E_ : A --> Cat. Keep the other q : A -- > B as a fibration. Then the homomorphism corresponding to the composite q p is a generalized left Kan extension of E_ along q. Ross From rrosebru@mta.ca Sat Nov 3 06:44:29 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 03 Nov 2007 06:44:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IoFNY-0004YE-Av for categories-list@mta.ca; Sat, 03 Nov 2007 06:33:20 -0300 Date: Fri, 2 Nov 2007 19:30:31 -0500 (EST) From: Michael Barr To: Categories Subject: categories: Re: References MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 11 I certainly heard Jon lecture on this a number of times. PUblication? Lot's of luck. A quick glance at MathSciNet shows that there are an awful lot of J. Beck's, at least one J.M. Beck and at least one Jonathan Beck, but no paper by our Jon Beck on descent theory. As for precise statement, don't even think about it. But my recollection was only whether a triple could descend across a functor. There were cocyle conditions that were necessary and sufficient. I think the "Beck-Chevalley condition" was a simple example. At one point, Jon told my wife with some regret that, thanks to my insistence, he was finally published. He seemed constitutionally incapable of putting his thoughts in public. I think you can suppose that if PTJ couldn't find it, it isn't there except in the (increasingly feeble) memories of those who heard him. On Fri, 2 Nov 2007, JeanBenabou wrote: > Dear colleagues, > > I hope someone, and in particular Prof. Peter Johnstone, will help me > with the following information. I thought I had, with Jacques > Roubaud, proved in our joint note at the "Comptes Rendus" which I > mentioned in my previous mail proved a theorem on Monads and Descent. > I must have been mistaken, and also the many persons who quoted this > note, because in El Proposition 1.5.5 is the same theorem, but > attributed to J. Beck. > > I immediately "rushed" to the monumental bibliography of El to find > the reference, and there, big surprise, there was no J. Beck at all > among the 1262 references. > > Thus i'd greatly appreciate to have the date and paper of the paper > where Beck proved this theorem, and the precise statement he made, in > particular, did he prove his theorem in the general context of > fibered or indexed categories, or only in some very special case. > > Many thanks for your help > > From rrosebru@mta.ca Sun Nov 4 10:28:06 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 04 Nov 2007 10:28:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IogCc-00032P-U3 for categories-list@mta.ca; Sun, 04 Nov 2007 10:11:50 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: Reply to Jean Benabou Date: Sun, 04 Nov 2007 05:32:52 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 12 Dear Jean, There is an unpublished (untitled and undated) four-pages manuscript which John Beck gave to me (and I supposed also to many ohers) when he was at McGill. In it, he states and proves two theorems, the CTT (crude tripleableness theorem), and the PTT (precise tripleableness theorem). There is a connection between triples and descent implicit in the PTT. But this is not the same connection with descent as the Benabou-Roubaud theorem. The following remarks seem to be relevant to this issue. In M. Bunge and R. Pare, "Stacks and equivalece of indexed categories", Cahiers de Topologie et Geometrie Differentielle, vol XX-4 (1979) 373-399, we state and prove the following version of the Benabou-Roubaud theorem (which we quote): Proposition 2.3. (Benabou and Roubaud [10]). let A be an S-indexed category (S is a topos) for which Sigma (or Pi) exists and satisfies the Beck condition. Then A is a stack iff for every regular epi alpha:J--->>I in S, the functor alpha^*: A^I ---> A^J is tripleable (resp. cotripleable). What Bob Pare and I called "the Beck condition" above is the "Chevalley property" introduced in your paper. The proof of the Benabou-Roubaud theorem was not given in your Comptes Rendues note, and this is why we gave an explicit proof of it, since we needed to apply it to stacks. Indeed, from the Benabou-Roubaud theorem, using in addition the Beck's tripleableness theorem, one can obtain applications showing that a certain S-indexed category A is a stack. In Bunge-Pare, we obtain in this way, using Duskin's version of the tripleability theorem (J. Duskin, "Variations on Beck's tripleability criterion", Reports of the Midwest Categories Seminar III, LNM 106, Springer, 1969), the following (actually we give it a more generality): Corollary 2.5. Any topos S, indexed by itself in the usual way, is a stack. In turn, this is used in my sequel paper (M. Bunge,Stack completions and Morita equivalence for categories in a topos", Cahiers de Top. et Geo. Diff. XX-4 (1979) 401-436), using closure properties of stacks, to identify/construct stack completions of category objects in S. It seems then to be an error on the part of Peter Johnstone to have attributed Proposition 1.5.5 in E1 (page 297) to Beck and not to Benabou and Roubaud. At the end of this section on "Descent Conditions and Stacks" (page 303), the references given in El 1 are Bourn, Bunge and Pare, Giraud, Grothendieck, Reiterman and Tholen, but curiously enough, not Benabou-Roubaud. I am sure that Peter will repair this error should a second edition of the Elephant ever appear. I would like to add that people who write such monumental works are bound to make errors of this sort, particularly in this case, as the manuscript was not (to my knowledge) distributed around to the topos theorists and other mathematicians for comments and criticism prior to publication. With best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ >From: JeanBenabou >To: Categories >Subject: categories: References >Date: Fri, 2 Nov 2007 02:55:08 +0100 > >Dear colleagues, > >I hope someone, and in particular Prof. Peter Johnstone, will help me >with the following information. I thought I had, with Jacques >Roubaud, proved in our joint note at the "Comptes Rendus" which I >mentioned in my previous mail proved a theorem on Monads and Descent. >I must have been mistaken, and also the many persons who quoted this >note, because in El Proposition 1.5.5 is the same theorem, but >attributed to J. Beck. > >I immediately "rushed" to the monumental bibliography of El to find >the reference, and there, big surprise, there was no J. Beck at all >among the 1262 references. > >Thus i'd greatly appreciate to have the date and paper of the paper >where Beck proved this theorem, and the precise statement he made, in >particular, did he prove his theorem in the general context of >fibered or indexed categories, or only in some very special case. > >Many thanks for your help > > > _________________________________________________________________ Get Cultured With Arts & Culture Festivals On Live Maps http://local.live.com/?mkt=en-ca&v=2&cid=A6D6BDB4586E357F!2010&encType=1&style=h&FORM=SERNEP From rrosebru@mta.ca Sun Nov 4 10:28:06 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 04 Nov 2007 10:28:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IogFV-0003Dl-B7 for categories-list@mta.ca; Sun, 04 Nov 2007 10:14:49 -0400 Date: Sat, 3 Nov 2007 12:40:03 -0400 (EDT) From: Bill Lawvere To: Categories Subject: categories: Re: Apropos a couple of current topics MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 13 Concerning Ross Street's interesting remarks about history, I should clarify where the term "Beck condition" comes from. I would like to urge the readers of our categories list to study Jon's 1967 thesis which is now available via TAC Reprints. There one finds Jon's tripleability theorems which were used for example by Benabou and Roubaud in their 1970 paper on descent. Ross remembers that Jon arrived in Tulane in 1969 prepared to lecture on triples and descent. I heard Jon lecturing on that topic already in late 1967 at a meeting of the American Mathematical Society in Illinois. In particular, he explicitly stated the condition that I therefore called the "Beck condition" in my work on Hyperdoctrines (presented to an AMS meeting in NYC in early 1968). Later I saw this condition referred to as the Chevalley condition in a paper of J-L Verdier. I do not know whether Jon was familiar with that work of Chevalley. Some sort of "coequalizer in the base implies descent" property on a fibration F is of course true (in addition to F(X+Y) = F(X) x F(Y)) for those fibrations that exemplify a reasonable notion F of parameterized family. Tripleability provides a useful tool for analyzing these descents, but which are the tripleable (monadic) functors that could arise from descent in some fibration? Bill On Sat, 3 Nov 2007, Ross Street wrote: > History can be harder than mathematics. Yet, it is a worthy goal to get it > right. This can require discussion and feedback. Here are some of my > memories which I am quite happy for people to correct if they have a > fuller picture. > > Jean Benabou invented bicategories. In SLNM47 you will find > the particular example of a bicategory Spn(E) whose morphisms > are spans in a pullback-complete category E. You will also find the > convention to refer to properties holding in the homs as local. I always > thought it nice that the homs in Spn(E) were slice categories > E / a x b, thereby unifying two uses of "local". > > You will also find in that SLNM47 paper, the notion of morphism of > bicategories and of homomorphism of bicategory. These a very useful > concepts. They do compose in their own way. I believe there was no > attempt to deny that the "indexed categories" of Pare-Schumacher > are category-valued homomorphisms. > > The 1969-70 academic year at Tulane University Math Dept was dedicated to > Category Theory. Jack Duskin and I were there (doing some teaching as well > as research) for the whole year. Saunders Mac Lane and Eduardo Dubuc were > there for the first semester. Bernhard Banaschewski and Z. Hedrln were > there for the second semester. However, we had a lot of visitors as well. > In particular, Jean Benabou visited sometime in the first semester. In > particular, I learnt from Benabou's lectures about the "Chevalley > condition" for fibrations and how descent data were Eilenberg-Moore > algebras. Jean gave me a copy of his Comptes Rendu article with Jacques > Roubaud. > > Very soon after Jean Benabou left, Jon Beck arrived. He asked me what the > various visitors had talked about. When I told him about Benabou's lecture > on descent, he said that that was what he had planned to talk about > ("triples" and descent). I encouraged him to do so but he decided to > change his topic. His topic by the way was also very interesting: using > monads -- sorry, triples -- in homotopy theory and categorical coherence. > This was before operads! > > I wondered what would happen to Beck's work on descent. Category theorists > were not prolific publishers. Then I found reference to the "Beck > condition" in Bill Lawvere's papers of the time: it was what Benabou had > called the "Chevalley condition". So, when I had need for a 2-categorical > version of this involving comma objects instead of pullbacks, I called it > the "Beck-Chevalley condition". This 2-categorical version expresses > pointwiseness of Kan extensions and embodies Lawvere's formula for such > extensions. > > Also by the way, Lawvere's comma categories are generalized slice > constructions so I proposed (not really wishing to introduce new notation > but somewhat worried about using (f, g) as more than just the pair) using > f/g for functors f and g into the same category. > > Now, as much as I would love SIX bottles of GOOD champagne, I am not going > to submit a suggestion for Jean's challenge. Composition of fibrations is > a wonderful thing as is composition of homomorphisms of bicategories; but > they do different jobs. It is hard enough to say fibrations are composable > from the homomorphism viewpoint! > > There is a thing about this that requires a mixture of the two views. > Regard one fibration p : E --> A as a homomorphism E_ : A --> Cat. Keep > the other q : A -- > B as a fibration. Then the homomorphism corresponding > to the composite q p is a generalized left Kan extension of E_ along q. > > Ross > > > > From rrosebru@mta.ca Sun Nov 4 10:28:06 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 04 Nov 2007 10:28:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IogIa-0003Q9-As for categories-list@mta.ca; Sun, 04 Nov 2007 10:18:00 -0400 To: Categories Subject: categories: References From: JeanBenabou Date: Sun, 4 Nov 2007 00:37:46 +0100 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 14 Dear Michael, Thank you for your "trying" to answer. I have waited for a few days for a "real" answer from Peter Johnstone to whom my question was, for obvious reasons, primarily addressed. When my mail to you was completed I received a very complete and nice answer from Ross Street, and then, two more by Marta Bunge. I want to thank them, and tell them that I shall try to give, in detail, complete answers to their mails.That is, if the "higher authorities" who control this list consider that this "mathematico- historical" disicussion could be as important as, say, the more than 20 mails devoted to the over all important discussion about role versus r\ole. I shall wait a week before answering Ross and Marta, in case I get more answers, and, who knows, one can always dream, one by "Peter Johnstone himself". I shall make do with your answer, but before I make a few comments about it, in order for you to understand them, it would be better to read carefully my comments on the "non-answer" by Johnstone to a question which concerned his book; After all "he" was responsible for not mentioning a 1970 "Comptes Rendus" note, very frequently referred to, and attributing the result to John Beck, without any reference to a published, or unpublished, paper of his. Not even to a paper of another author, dating of before 1970, and crediting John Beck with precisely, I insist on it, the same theorem that was given in my joint note with Roubaud, and which is attributed to Beck in "The Elephant"! He is no "baby in the woods", and if he writes something in an important book, published by Oxford University Press, he must be able to explain his decision. I hope to hear from him soon, and; since I am on this unpleasant matter, I hope he shall also answer the following questions: (i) In the long Appendix of his Topos Theory (TT), there is only one theorem.It is due to a student of mine, Jean Celeyrette, whose thesis is mentioned in the bibliography of (TT). Why has the name of J.Geleyrette totally disappeared from the much much longer bibliography of (El)? (ii) Same question about my Louvain paper on "Distributors" or "Profunctors", which he uses in an essential manner in (El) without ever mentioning my name. It was also in the bibligraphy of (TT) and again absent from the bibliography of (El) With the note on descent, and many other examples, this is getting to be "a habit" with Peter Johnstone. I advise him to lose very quickly such habits, they might become dangerous for one's health. ------------------------------------------------------------------------ - Considering what I said about Johnstone, you won't be surprised if I tell you that your answer does not fully satisfy me (and that is an understatement) Since oblique, bold, etc fonts are not accepted in this list, I shall write between quotation marks any parts of your mail I want to comment upon, and without quotation marks my comments. (i) "I certainly heard Jon lecture on this a number of times" What does your "this" precisely refer to? (ii) " PUblication? Lot's of luck. A quick glance at MathSciNet shows that there are an awful lot of J. Beck's, at least one J.M. Beck and at least one Jonathan Beck, but no paper by our Jon Beck on descent theory. As for precise statement, don't even think about it". I cannot, and will not, take this for an answer. My joint theorem with Roubaud is a precise statement, and so is its reformulation by Johnstone. He was obviously too young in 1970 to have heard it, if was the same, directly from Beck. How can he be sure it was the same, if you are not? Moreover I am vain enough to consider it was an important result, because it established a connection between two important theories, namely: descent and triples. There were enough good mathematicians in North America in the late 60's, and certainly, at least one of them, would have grasped its importance, and given one, or many, applications, as we did, Roubaud and myself, in the same note where we stated the theorem. Where are these applications? (iii) "But my recollection was only whether a triple could descend across a functor. There were cocyle conditions that were necessary and sufficient. I think the "Beck-Chevalley condition" was a simple example" I do not merely "think", I am sure, that I learnt Chevalley's condition, from Chevalley, in 1964. At that time fibered categories, "invented" by Grothendieck, were almost "unheard of" in the "North American" category community. The first reference I know of is Gray's paper in the 1965 conference of La Jolla, where he refers to Chevalley's lost notes for 1962 lectures at Berkeley. Even now, they are often presented in terms of "indexed categories", under the influence of W.V. Lawvere's 1971 "Perugia Notes". I thus doubt very much that, whatever Beck's talent,in 1964, when his PhD thesis was not yet completed, he might have had anything like Chevalley's condition for arbitrary fibrations. I "think" I was wrong to "compromise" and to accept that what I called the Chevalley Condifion should have Beck's name assocoated to it, and I'm sure that, from now on, I shall call Chevalley condition what was up to now called Beck-Chevalley condition, only because I insisted that it was historically a nonsense to call it, as the North American school did, "Beck" condition ! (iii) "At one point, Jon told my wife with some regret that, thanks to my insistence, he was finally published. He seemed constitutionally incapable of putting his thoughts in public." When and where was he "finally published"? (iv) "I think you can suppose that if PTJ couldn't find it, it isn't there except in the (increasingly feeble) memories of those who heard him." I do not merely "think", I know that you are a mathematician, (and that of course for me means a good one). Thus, if Beck's formulation had been so blatantly simple and precise as mine and Roubaud's you wouldn't need an effort of memory to remember it, with precision. And this is of course also true for many of the mathematicians "who heard him". Although I was not among the happy few who heard him, I don't need a great effort of memory to remember Beck' Triplability Theorem. I "think" also that Johnstone had better find a more credible justification than mere "hear so" and "think that", Jean Benabou, out of solidarity with the so-called "category-community" might not, even if he is angry, rise such a fuss. But Jacques Roubaud has no such solidarity, is very angry, and he is known, and respected, in "circles" much wider that the few handful of persons that some of us tend to "think of" as the center of the world. I do not "think", we are the center of the world, I am even sure, we are not ! From rrosebru@mta.ca Sun Nov 4 13:56:31 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 04 Nov 2007 13:56:31 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IojdS-0003tZ-0N for categories-list@mta.ca; Sun, 04 Nov 2007 13:51:46 -0400 Date: Sun, 4 Nov 2007 16:16:16 +0000 (GMT) From: "Prof. Peter Johnstone" To: Categories mailing list Subject: categories: Partial respponse to Jean Benabou MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 15 I intend in due course to reply to the long message posted by Jean Benabou on 30 October. Unfortunately I am very busy at present (I am lecturing six days a week this term) and I shall not have time to do so for a while. However, Jean's evident desire for a quick response from me on the subject of Proposition B1.5.5 in the Elephant cannot be ignored. Since I wasn't around at the time this result was first proved, I had to rely on the recollections of those older than myself as to its provenance: I am grateful to Ross Street and Bill Lawvere for their recent postings confirming the impression I received many years ago that Jon Beck had indeed had this idea in the late 1960s. It seems clear from their accounts that Beck had the idea independently of Benabou and Roubaud; which of them had it first seems impossible to establish at this stage. Since their work was independent, I should of course have credited Benabou--Roubaud as well as Beck at this point in the text of the Elephant, and I apologize for not having done so. The reason, I must confess, was that I had simply not come across the Benabou--Roubaud paper; clearly, this was a failure of due diligence on my part. Incidentally, in reply to a comment in Marta Bunge's posting, I did distribute draft copies of the Elephant to a number of colleagues, and invite their comments, before it was published. None of them picked up this particular point -- though I am not blaming them for that; the fault was of course mine. Regarding the fact that Celeyrette's thesis, and Jean's Louvain notes on "Les Distributeurs", are not in the bibliography of the Elephant, there is a simple reason for this: I decided at an early stage that, in order to keep the size of the bibliography within bounds, it was necessary to limit it to published books and papers, and to exclude unpublished theses and other articles circulated only in preprint form. (Hence also the absence of Beck from the bibliography.) I did relent in one or two cases (actually as a result of comments from one of the people who saw the drafts before publication) where a result appearing in someone's thesis, and not subsequently published, was specifically referred to in the text. Peter Johnstone From rrosebru@mta.ca Mon Nov 5 07:15:55 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Nov 2007 07:15:55 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ioznp-0003zM-44 for categories-list@mta.ca; Mon, 05 Nov 2007 07:07:33 -0400 From: "Marta Bunge" To: P.T.Johnstone@dpmms.cam.ac.uk, categories@mta.ca Subject: categories: RE: Partial response to Jean Benabou Date: Sun, 04 Nov 2007 17:12:15 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 16 Dear Peter, >Incidentally, in reply to a comment in Marta Bunge's posting, I did >distribute draft copies of the Elephant to a number of colleagues, and >invite their comments, before it was published. None of them picked up >this particular point -- though I am not blaming them for that; the >fault was of course mine. > I have now read in the Preface to the Elephant that you did indeed circulate a draft to "a number of colleagues (including Ieke Moerdijk and Andrew Pitts, as well as Martin Hyland, Anders Kock and Gavin Wraith) for their comments and suggestions". Apologies for my swift comment. I had gotten a different impression from the reaction of several participants to the Fields Institute Workshop on Galois Theory etc, where the first two volumes of the Elephant were exhibited, including comments by Bill Lawvere and several other people whose work was prominently represented in the book. Aurelio Carboni and I perused quickly section B4.5 on the symmetric monad, and were satisfied on the spot with your account of it, but it did not occur to me to look up section B1.5, even though my paper with Bob Pare on stacks was cited at the end of the section. Had I done so, I would have pointed out that Proposition 1.5.5 is due to Benabou and Roubaud as we ourselves had pointed out in our paper. I take this opportunity to apologize to Jean Benabou for stating his theorem with Jacques Roubaud in the context of indexed categories and not on that of (bi)fibrations without warning the reader, but I also want to point out that, since the proof itself only refers to two fibers and a transition map between them, it is equally meaningful in both contexts. I believe that I only became aware of the gross difference afterwards, when lecturing on stack completions at the Benabou Seminar in Paris (whenever that was -- 1979?). I still think now that it is easier to work with the indexed presentations of fibrations than with the fibrations themselves, without attempting to turn this into a philosophical statement of any kind. Beck's Tripleability theorem is indeed useful in the applications of the Benabou-Roubaud theorem, and among them are those we give in the Cahiers paper on stacks, and in its sequel by myself on stack completions. Perhaps the relabelling of the Chevalley condition as "the Beck condition" led to some confusion as to whether the Benabou-Roubaud theorem had been proved (also) by Jon Beck? I myself never heard him speaking on this, and never saw any draft written by him of a proof of this theorem. Sadly, we cannot consult Jon himself on this issue, so we might as well drop it. Perhaps you would consider, prior to publication of Volume III of the Elephant, enlarging your list of commentators to include at least those whose work you include in some form or other. With best regards, Marta From rrosebru@mta.ca Mon Nov 5 14:45:21 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Nov 2007 14:45:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ip6ki-0006Cl-C8 for categories-list@mta.ca; Mon, 05 Nov 2007 14:32:48 -0400 Date: Mon, 5 Nov 2007 13:21:13 +0100 (CET) From: claudio pisani Subject: categories: Re: Comma categories To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 17 The following facts about slice categories may be worth noticing: 1 In the equivalence between df/X (discrete fibrations over a category X) and presheaves on X, the slices X/x -> X correspond to the representable presheaves. 2. (Yoneda Lemma) The reflection of x:1->X (as an object of Cat/X) in df/X is (isomorphic to) X/x (with its terminal object as reflection map). In particular, the full subcategory sl/X of df/X generated by the slices over X is isomorphic to X. 3. For any functor p:P->X, a morphism p->X/x in Cat/X is a cone of base p and vertex x. 4. So, a reflection of p->X/x of p in sl/X is a colimiting cone. 5. A functor f:X->Y has a right adjoint iff the pullback f*Y/y of any slice of Y is (isomorphic to) a slice of X. 6. If ex_f -| f* : df/Y -> df/X is the "left Kan extension" along f, then the counit=20 e: ex_f f* Y/y -> Y/y=20 is an iso for any y iff f is "dense" (aka "connected") while it is a colimiting cone for any y iff f is "adequate" (aka "dense"). Using instead the adjunction=20 f_! -| f* : Cat/Y -> Cat/X the counit is a colimiting cone for any y iff f is adequate (as before), while it is an absolute colimit iff f is dense. Best regards. Claudio --- Uwe Egbert Wolter ha scritto: > Dear all, >=20 > I'm looking for a comprehensive exposition of > definitions and results > around comma/slice categories. Especially, it would > be nice to have > something also for non-specialists in category > theory as young > postgraduates. Is there any book or text you would > recommend? >=20 > Best regards >=20 > Uwe Wolter >=20 >=20 >=20 From rrosebru@mta.ca Mon Nov 5 18:58:05 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Nov 2007 18:58:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpAot-0003Qp-Mk for categories-list@mta.ca; Mon, 05 Nov 2007 18:53:23 -0400 Date: Mon, 5 Nov 2007 16:04:57 -0500 From: "Zinovy Diskin" To: Categories Subject: categories: Re: on the history of the Beck-Chevalley condition MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 18 Dear Categories, You may find it interesting that a particular case of the condition (in the particular context of hyperdoctrines-as-logical-theories) was formulated by Paul Halmos in his Algebraic Logic (1962). Halmos considered trivial algebraic theories with variables as the only terms and their substitutions as morphisms. A direct comparison is not straightforward because Halmos' formulation was for FOL without equality. Yet with the help of Robert Seely's paper "Hyperdoctrines, natural deduction and the Beck condition" (1983), one can find what part/form of the condition was stated by Halmos. ZD On Nov 3, 2007 11:40 AM, Bill Lawvere wrote: > > Concerning Ross Street's interesting remarks about history, > I should clarify where the term "Beck condition" comes from. > > I would like to urge the readers of our categories > list to study Jon's 1967 thesis which is now available > via TAC Reprints. There one finds Jon's > tripleability theorems which were used for example by > Benabou and Roubaud in their 1970 paper on descent. > > Ross remembers that Jon arrived in Tulane in 1969 prepared to > lecture on triples and descent. I heard Jon lecturing on that topic > already in late 1967 at a meeting of the American Mathematical > Society in Illinois. In particular, he explicitly stated the > condition that I therefore called the "Beck condition" in my > work on Hyperdoctrines (presented to an AMS meeting in NYC > in early 1968). > > Later I saw this condition referred to as the Chevalley condition in a > paper of J-L Verdier. I do not know whether Jon was familiar > with that work of Chevalley. > > Some sort of "coequalizer in the base implies descent" property on a > fibration F is of course true (in addition to F(X+Y) = F(X) x F(Y)) > for those fibrations that exemplify a reasonable notion F of > parameterized family. Tripleability provides a useful > tool for analyzing these descents, but which are the tripleable (monadic) > functors that could arise from descent in some fibration? > > Bill > > On Sat, 3 Nov 2007, Ross Street wrote: > > > History can be harder than mathematics. Yet, it is a worthy goal to get it > > right. This can require discussion and feedback. Here are some of my > > memories which I am quite happy for people to correct if they have a > > fuller picture. > > > > Jean Benabou invented bicategories. In SLNM47 you will find > > the particular example of a bicategory Spn(E) whose morphisms > > are spans in a pullback-complete category E. You will also find the > > convention to refer to properties holding in the homs as local. I always > > thought it nice that the homs in Spn(E) were slice categories > > E / a x b, thereby unifying two uses of "local". > > > > You will also find in that SLNM47 paper, the notion of morphism of > > bicategories and of homomorphism of bicategory. These a very useful > > concepts. They do compose in their own way. I believe there was no > > attempt to deny that the "indexed categories" of Pare-Schumacher > > are category-valued homomorphisms. > > > > The 1969-70 academic year at Tulane University Math Dept was dedicated to > > Category Theory. Jack Duskin and I were there (doing some teaching as well > > as research) for the whole year. Saunders Mac Lane and Eduardo Dubuc were > > there for the first semester. Bernhard Banaschewski and Z. Hedrln were > > there for the second semester. However, we had a lot of visitors as well. > > In particular, Jean Benabou visited sometime in the first semester. In > > particular, I learnt from Benabou's lectures about the "Chevalley > > condition" for fibrations and how descent data were Eilenberg-Moore > > algebras. Jean gave me a copy of his Comptes Rendu article with Jacques > > Roubaud. > > > > Very soon after Jean Benabou left, Jon Beck arrived. He asked me what the > > various visitors had talked about. When I told him about Benabou's lecture > > on descent, he said that that was what he had planned to talk about > > ("triples" and descent). I encouraged him to do so but he decided to > > change his topic. His topic by the way was also very interesting: using > > monads -- sorry, triples -- in homotopy theory and categorical coherence. > > This was before operads! > > > > I wondered what would happen to Beck's work on descent. Category theorists > > were not prolific publishers. Then I found reference to the "Beck > > condition" in Bill Lawvere's papers of the time: it was what Benabou had > > called the "Chevalley condition". So, when I had need for a 2-categorical > > version of this involving comma objects instead of pullbacks, I called it > > the "Beck-Chevalley condition". This 2-categorical version expresses > > pointwiseness of Kan extensions and embodies Lawvere's formula for such > > extensions. > > > > Also by the way, Lawvere's comma categories are generalized slice > > constructions so I proposed (not really wishing to introduce new notation > > but somewhat worried about using (f, g) as more than just the pair) using > > f/g for functors f and g into the same category. > > > > Now, as much as I would love SIX bottles of GOOD champagne, I am not going > > to submit a suggestion for Jean's challenge. Composition of fibrations is > > a wonderful thing as is composition of homomorphisms of bicategories; but > > they do different jobs. It is hard enough to say fibrations are composable > > from the homomorphism viewpoint! > > > > There is a thing about this that requires a mixture of the two views. > > Regard one fibration p : E --> A as a homomorphism E_ : A --> Cat. Keep > > the other q : A -- > B as a fibration. Then the homomorphism corresponding > > to the composite q p is a generalized left Kan extension of E_ along q. > > > > Ross > > > > > > > > > > > From rrosebru@mta.ca Mon Nov 5 18:58:05 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Nov 2007 18:58:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpAnW-0003GO-Sv for categories-list@mta.ca; Mon, 05 Nov 2007 18:51:58 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: Beck, Benabou-Roubaud, etc. Date: Mon, 05 Nov 2007 15:49:17 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 19 Dear colleagues, You can find in the following site http://www.math.mcgill.ca/bunge/abstracts.ps the abstracts of the Oberwolfach Meeting on Descent from 1995. Look in particular at the abstract for Jack Duskin's lecture on "Triples and Descent". In it, Jack gives as a corollary to the main theorem what he calls "the Benabou-Roubaud-Beck theorem". Jack mentions that the theorems exposed in that lecture were part of a thesis project of his student M. Alsani. There seems to be a precedent in print then in adding Beck's name to this theorem, unlike what I previously thought. Other authors (like Claudio Hermida) refer to it as "the Beck-Benabou-Roubaud theorem". This, added to the recollections by Ross Street and Bill Lawvere, seem to contradict my previous contention (from lack of evidence) that the theorem is exclusively due to Benabou and Roubaud. It seems now clear that Jon Beck must have had it (independently) too, which of course is not at all surprising. With best wishes, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ I found the abstract of Jack Duskin's lecture at the Oberwolfach meeting on Descent, 1995. ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ _________________________________________________________________ Gear up with the exclusive HALO 3 theme back for Windows Live Messenger. http://entertainment.sympatico.msn.ca/WindowsLiveMessenger/halo3.aspx From rrosebru@mta.ca Tue Nov 6 08:56:29 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Nov 2007 08:56:29 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpNqz-0000MX-50 for categories-list@mta.ca; Tue, 06 Nov 2007 08:48:25 -0400 Date: Mon, 5 Nov 2007 21:56:34 -0500 (EST) From: Bill Lawvere To: categories@mta.ca Subject: categories: Beck's 1967 Descent Talk Message-ID: MIME-Version: 1.0 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 In the Bulletin of the AMS Volume 74 (1968) page 91 one sees that a meeting of the Society was held on Saturday November 25th, 1967 at the University of Illinois in Urbana, with over 200 people attending. There was a session of selected 20-Minute papers on Categorical Algebra, arranged by Professor Saunders Mac Lane. Papers by Barr, Beck, Gray, Lawvere, Linton were included. The abstracts for the talks at that meeting were published in the Notices of the AMS Volume 14 (1967). On page 938 one finds Jon Beck's abstract: 652-8. Jon Beck, Cornell University, Ithaca, New York. Descent and standard constructions (triples). There is a close relationship between descent theory in algebraic geometry and the theory of categories which are definable by means of standard constructions (tripleable categories). The "tripleableness theorem" sheds some light on descent criteria. The form of Cech cohomology used in descent theory is an appropriate triple cohomology theory. Its interpretation is discussed from the triple point of view. (Received October 2, 1967.) It is possible that someone still has notes of that lecture 40 years later. Bill Lawvere *********************************************************** From rrosebru@mta.ca Tue Nov 6 15:45:05 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Nov 2007 15:45:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpU8f-0005aG-T2 for categories-list@mta.ca; Tue, 06 Nov 2007 15:31:05 -0400 Date: Tue, 06 Nov 2007 10:49:54 +0000 From: David Pym MIME-Version: 1.0 To: categories@mta.ca, fom@cs.nyu.edu Subject: categories: Paper announcement: Categorical Models of Classical Logic and GoI Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 21 Readers of these lists may be interested to hear that the following paper has appeared: C. F=FChrmann and D. Pym. On categorical models of classical logic and th= e=20 geometry of interaction. Mathematical Structures in Computer Science 17(5), 2007, 957-1027. Abstract. It is well known that weakening and contraction cause naive=20 categorical models of the classical sequent calculus to collapse to=20 Boolean lattices. In previous work, summarised briefly herein, we have=20 provided a class of models called /classical categories/ that is sound=20 and complete and avoids this collapse by interpreting cut reduction by a=20 poset enrichment. Examples of classical categories include boolean=20 lattices and the category of sets and relations, where both conjunction=20 and disjunction are modelled by the set-theoretic product. In this=20 article, which is self-contained, we present an improved axiomatisation=20 of classical categories, together with a deep exploration of their=20 structural theory. Observing that the collapse already happens in the=20 absence of negation, we start with negation-free models called /Dummett=20 categories/. Examples of these include, besides the classical categories=20 mentioned above, the category of sets and relations, where both=20 conjunction and disjunction are modelled by the disjoint union. We prove=20 that Dummett categories are MIX, and that the partial order can be=20 derived from hom-semilattices, which have a straightforward=20 proof-theoretic definition. Moreover, we show that the=20 Geometry-of-Interaction construction can be extended from multiplicative=20 linear logic to classical logic by applying it to obtain a classical=20 category from a Dummett category. Along the way, we gain detailed insights into the changes that proofs=20 undergo during cut elimination in the presence of weakening and=20 contraction. -- Prof. David J. Pym t: +44 (0) 117 312 8012 Principal Scientist f: +44 (0) 117 312 9250 HP Labs e: david.pym@hp.com Bristol, UK w: http://www.hpl.hp.com/personal/davpym/ Professor of Logic & Computation, University of Bath, UK From rrosebru@mta.ca Tue Nov 6 15:45:05 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Nov 2007 15:45:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpU9R-0005jA-Mm for categories-list@mta.ca; Tue, 06 Nov 2007 15:31:53 -0400 Date: Tue, 6 Nov 2007 11:38:26 +0000 (GMT) From: Marcelo Fiore To: categories@mta.ca Subject: categories: Re: Functor derivatives - a question and a result References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 22 On a related matter to the message below by Jiri, let me point out the following paper: M. Fiore. Differential structure in models of multiplicative biadditive intuitionistic linear logic. In Typed Lambda Calculi and Applications (TLCA 2007), LNCS 4583, pp. 163-177, 2007. [Available from ] presenting a categorical framework for differentiation, directly synthetised from the differential calculus of generalised species of structures. Though, as it transpired in conversation with Anders Kock, the setting is also applicable to convenient vector spaces and some models of SDG. On Fri, 2 Nov 2007, Jiri Adamek wrote: > > Andre Joyal defined derivatives of analytic functors > in his 1986 paper. Recently I heard the more general definition > of a derivative F' of an endofunctor F defined via a universal > sub-cartesian transformation from F'xId into F. Who is the author > of this definition? The following result seems to indicate that > outside of the realm of analytic functors derivatives may not > be really useful: > > Theorem. Every non-faithful functor F:Set -> Set has the derivative > F' = 0 (the constant functor to the empty set). > > xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx > alternative e-mail address (in case reply key does not work): > J.Adamek@tu-bs.de > xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx > From rrosebru@mta.ca Tue Nov 6 15:45:05 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Nov 2007 15:45:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpUA0-0005pt-ML for categories-list@mta.ca; Tue, 06 Nov 2007 15:32:29 -0400 Mime-Version: 1.0 (Apple Message framework v752.3) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable From: Peter Hines Subject: categories: Postdoctoral Position in Quantum Computing / Theoretical Computer Science Date: Tue, 6 Nov 2007 13:13:00 +0000 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 23 POSTDOCTORAL POSITION ANNOUNCEMENT Dear categorists : Hopefully the following is of interest. The post is not exclusively =20 category-theoretic, but should be suitable for people with a =20 background in category theory from either a theoretical computer =20 science or quantum mechanical perspective. ---------------------------- Applications are invited for a 3 year appointment on a project funded =20= by the EPSRC into the fundamentals of low level quantum computation =20 and applications. The project is led by Professor Sam Braunstein, and you will work =20 with partners on the project in Mathematics at Heriot-Watt, =20 Manchester and Newcastle and in Computer Science at Glasgow and =20 Newcastle. You must have a relevant PhD. Ideally; you should have a background =20 in relevant areas of either Mathematics or theoretical Computer =20 Science as well as research experience in quantum computing. Previous =20= research experience is highly desirable, and you will be expected to =20 work independently and contribute to the overall direction of the =20 research project, as well as assisting with coordination between the =20 various participating institutions. The project aims to establish a coherent and natural framework for =20 low level quantum processing, analogous to the framework for =20 classical computation provided by low-level computational models such =20= as state machines, transition systems, petri nets, etc. The framework =20= developed will be applied to algorithms in group theory, cryptography =20= and secure distributed computation. For more information about the =20 proposed work see http://www.cs.york.ac.uk/~schmuel/proposal.pdf The salary will be =A326,666 per annum. The post is available from =20 January 2008 for a period of up to three years. From rrosebru@mta.ca Tue Nov 6 15:45:05 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Nov 2007 15:45:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IpUAf-0005wg-Gv for categories-list@mta.ca; Tue, 06 Nov 2007 15:33:09 -0400 Date: Mon, 05 Nov 2007 08:33:47 -0500 From: jim stasheff Subject: categories: Re: References To: Categories MIME-version: 1.0 (Apple Message framework v752.3) Content-type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-transfer-encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 24 Perhaps some one took notes on paper and hasn't thrown them out? It would be great to ad them to Jon's small nachlass. jim On Nov 2, 2007, at 8:30 PM, Michael Barr wrote: > I certainly heard Jon lecture on this a number of times. PUblication? > Lot's of luck. A quick glance at MathSciNet shows that there are > an awful > lot of J. Beck's, at least one J.M. Beck and at least one Jonathan > Beck, > but no paper by our Jon Beck on descent theory. As for precise > statement, > don't even think about it. But my recollection was only whether a > triple > could descend across a functor. There were cocyle conditions that > were > necessary and sufficient. I think the "Beck-Chevalley condition" > was a > simple example. > > At one point, Jon told my wife with some regret that, thanks to my > insistence, he was finally published. He seemed constitutionally > incapable of putting his thoughts in public. > > I think you can suppose that if PTJ couldn't find it, it isn't there > except in the (increasingly feeble) memories of those who heard him. > > On Fri, 2 Nov 2007, JeanBenabou wrote: > >> Dear colleagues, >> >> I hope someone, and in particular Prof. Peter Johnstone, will help me >> with the following information. I thought I had, with Jacques >> Roubaud, proved in our joint note at the "Comptes Rendus" which I >> mentioned in my previous mail proved a theorem on Monads and Descent. >> I must have been mistaken, and also the many persons who quoted this >> note, because in El Proposition 1.5.5 is the same theorem, but >> attributed to J. Beck. >> >> I immediately "rushed" to the monumental bibliography of El to find >> the reference, and there, big surprise, there was no J. Beck at all >> among the 1262 references. >> >> Thus i'd greatly appreciate to have the date and paper of the paper >> where Beck proved this theorem, and the precise statement he made, in >> particular, did he prove his theorem in the general context of >> fibered or indexed categories, or only in some very special case. >> >> Many thanks for your help >> >> > > From rrosebru@mta.ca Wed Nov 7 11:51:50 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Nov 2007 11:51:50 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ipn4E-0000M6-HM for categories-list@mta.ca; Wed, 07 Nov 2007 11:43:46 -0400 From: Gaucher Philippe To: categories@mta.ca Subject: categories: Preprint: Homotopical equivalence of combinatorial and categorical semantics of process algebra Date: Wed, 7 Nov 2007 11:51:56 +0100 MIME-Version: 1.0 Content-Disposition: inline Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 26 Dear All, Here is a new preprint. Sincerely yours. pg. Author: P. Gaucher Title: Homotopical equivalence of combinatorial and categorical semantics of process algebra Abstract: It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomorphic if and only if the associated flows are weakly equivalent. The interest of the categorical semantics is that combinatorics totally disappears. Last but not least, a part of the categorical semantics of CCS goes down to a pure homotopical semantics of CCS using A. Heller's privileged weak limits and colimits. These results can be easily adapted to any other process algebra for any synchronization algebra. URL: http://www.pps.jussieu.fr/~gaucher/cubeflow.pdf http://www.pps.jussieu.fr/~gaucher/cubeflow.ps Comments: 23 pages From rrosebru@mta.ca Wed Nov 7 20:18:28 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Nov 2007 20:18:28 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IputM-0005Vu-Lq for categories-list@mta.ca; Wed, 07 Nov 2007 20:05:04 -0400 From: Bill Lawvere To: categories@mta.ca Subject: categories: Re: Comma categories MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Date: Wed, 07 Nov 2007 20:05:04 -0400 Status: O X-Status: X-Keywords: X-UID: 27 I recently noticed that in Abstract no. 652-4 in the Notices of the AMS volume 14 (1967) page 937, John Gray advocates a systematic treatment of the calculus of comma categories and lists five operations which should be explicitly accounted for in such a calculus. He also mentions that Jon Beck contributed to that discussion. Probably John Gray's notes, if they still exist, would be a helpful guide to someone planning to write a systematic treatment as suggested recently Uwe Wolters. Bill On Mon, 5 Nov 2007, claudio pisani wrote: > > The following facts about slice categories may be > worth noticing: > > 1 In the equivalence between df/X (discrete fibrations > over a category X) and presheaves on X, the slices X/x > -> X correspond to the representable presheaves. > > 2. (Yoneda Lemma) The reflection of x:1->X (as an > object of Cat/X) in df/X is (isomorphic to) X/x (with > its terminal object as reflection map). > In particular, the full subcategory sl/X of df/X > generated by the slices over X is isomorphic to X. > > 3. For any functor p:P->X, a morphism p->X/x in Cat/X > is a cone of base p and vertex x. > > 4. So, a reflection of p->X/x of p in sl/X is a > colimiting cone. > > 5. A functor f:X->Y has a right adjoint iff the > pullback f*Y/y of any slice of Y is (isomorphic to) a > slice of X. > > 6. If ex_f -| f* : df/Y -> df/X is the "left Kan > extension" along f, then the counit > e: ex_f f* Y/y -> Y/y > is an iso for any y iff f is "dense" (aka "connected") > while it is a colimiting cone for any y iff f is > "adequate" (aka "dense"). > Using instead the adjunction > f_! -| f* : Cat/Y -> Cat/X > the counit is a colimiting cone for any y iff f is > adequate (as before), while it is an absolute colimit > iff f is dense. > > Best regards. > > Claudio > > > > --- Uwe Egbert Wolter ha > scritto: > >> Dear all, >> >> I'm looking for a comprehensive exposition of >> definitions and results >> around comma/slice categories. Especially, it would >> be nice to have >> something also for non-specialists in category >> theory as young >> postgraduates. Is there any book or text you would >> recommend? >> >> Best regards >> >> Uwe Wolter >> >> >> > > > > From rrosebru@mta.ca Thu Nov 8 22:21:40 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 08 Nov 2007 22:21:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IqJJn-0002fN-2w for categories-list@mta.ca; Thu, 08 Nov 2007 22:09:59 -0400 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Wed, 7 Nov 2007 17:32:28 -0800 To: categories@mta.ca Subject: categories: Re: Comma categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 28 Bill Lawvere writes: > > > I recently noticed that in Abstract no. 652-4 in the Notices > of the AMS volume 14 (1967) page 937, John Gray advocates a > systematic treatment of the calculus of comma categories > and lists five operations which should be explicitly accounted > for in such a calculus. > He also mentions that Jon Beck contributed to that discussion. > > Probably John Gray's notes, if they still exist, would be > a helpful guide to someone planning to write a systematic > treatment as suggested recently Uwe Wolters. > > Bill As a followup to Bill's note, here is a slightly more recent positing by John Gray to another mailing list on this very topic. * To: types@theory.LCS.MIT.EDU * Subject: Re: Cobig, Coproduct, and Comma * From: gray@symcom.math.uiuc.edu (John Gray) * Date: Mon, 20 Mar 89 17:13:53 EST * Sender: meyer@theory.LCS.MIT.EDU Date: Mon, 20 Mar 89 15:32:11 CST >Cobig, Coproduct, and Comma Vaughan Pratt 3/19/89 >Formally a comma category is most slickly described as a lax pullback. >I've attempted an understandable account of this 2-category concept in >an appendix below. I'd appreciate pointers to other accounts. Comma categories are an ancient tool in category theory. They were introduced in F. W. Lawvere, Functorial Semantics of Algebraic Theories Thesis, Columbia University, 1963. He used them in --, The category of categories as a foundation for mathematics, Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag, New York. I discussed them in several places: J. W. Gray, Fibred and cofibred categories, same proceedings as above, 21-83. I gave a brief calculus of comma categories in: --, The categorical comprehension scheme, Category theory, Homology theory and their Applications III, Lecture Notes in Mathematics 99, Springer-Verlag, New York 1969, 242-312. They are described as "Cartesian quasi-limits" in the book: --, Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer-Verlag, New York 1974. which is the first place where the lax description of them can be found. I don't credit it to anybody there, since I assumed it was general knowledge. The name was changed to "lax limits" in: G. M. Kelly and R. Street, Review of the elements of 2-categories, Category Seminar, Lecture Notes in Mathematics 420, Springer- Verlag, New York 1974. The general theory of the properties of lax limits in 2-categories was discussed independently by Street and me in various publications. E. g., J. W. Gray, The existence and construction of lax limits, Cahiers Top. et Geom. Diff. 21 (1980), 277-304. --, Closed categories, Lax limits and homotopy limits, J. Pure Appl. Algebra 19 (1980), 127-158. --, The representation of limits, lax limits, and homotopy limits as sections, in Mathematical Applications of Category Theory, Contemporary Mathematics 30 (1984), AMS, 63-83. R. Street, Two constructions on lax functors, Cahiers Top. et Geom. Diff. 13, (1972), 217-264. --, Limits indexed by category-valued 2-functors, J. Pure and Applied Alg. 8 (1976), 149-181. It is of course very gratifying to see these ideas coming around again as useful tools in the semantics of programming languages. John Gray -- Bob From rrosebru@mta.ca Thu Nov 8 22:21:40 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 08 Nov 2007 22:21:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IqJLG-0002l2-BC for categories-list@mta.ca; Thu, 08 Nov 2007 22:11:30 -0400 From: Vladimiro Sassone To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v912) Subject: categories: The Bulletin of the EATCS goes Open Access Date: Thu, 8 Nov 2007 02:16:27 +0000 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 29 Dear colleagues, Since 2003 all issues of the Bulletin of the EATCS, the European Association for Theoretical Computer Science, have been produced entirely electronically and made available on the web for members only. The EATCS is now piloting the idea of Open Access for the Bulletin, in the spirit of best serving its research community. So, until further notice the volumes from no 79 onwards of the Bulletin of the EATCS will be available from http://www.eatcs.org/publications/bulletin.html With best regards (and apologies for cross-posting) \vs Prof. Vladimiro Sassone Bulletin of the EATCS, Editor-in-chief ECS, University of Southampton SO17 1BJ United Kingdom Tel: +44 23 8059 9009 Fax: 3045 www.ecs.soton.ac.uk/people/vs From rrosebru@mta.ca Thu Nov 8 22:21:40 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 08 Nov 2007 22:21:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IqJMb-0002qW-6i for categories-list@mta.ca; Thu, 08 Nov 2007 22:12:53 -0400 Date: Thu, 8 Nov 2007 16:14:44 +0100 (CET) Subject: categories: International Category Theory Conference 2008 From: CategoryTheory-2008@lmpa.univ-littoral.fr To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 30 =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D INTERNATIONAL CATEGORY THEORY CONFERENCE 2008 (CT08): First Announcement Universit=E9 du Littoral C=F4te d'Opale, Calais, France, 22-28 June 2008 In the tradition of previous meetings held in Carvoeiro (2007), White Point(2006), Vancouver (2004), Como (2000) and Coimbra (1999), an International Conference on Category Theory will be held at the Laboratoire de Math=E9matiques Pures et Appliqu=E9es J. Liouville of the Universit=E9 du Littoral C=F4te d'Opale from June 22 until June 28, 2008. The scientific programme will begin on Monday morning, June 23, and will finish before lunch on Saturday June 28. The programme will consist of conferences delivered by invited speakers and contributed talks. The main scientific topics of the conference will include: General Category Theory Topos Theory Higher Categorical Structures and Homotopy Theory Categories in Algebra and Logic Descent and Galois Theory. Categorical Topology Categories in Computer Science The members of the Scientific Committee are: Clemens Berger (Universit=E9 de Nice, France) Dominique Bourn (Universit=E9 du Littoral C=F4te d'Opale, France) George Janelidze (University of Cape Town, South Africa) Peter Johnstone (University of Cambridge, United Kingdom) Manuela Sobral (Universidade de Coimbra, Portugal) Robert Walters (Universit=E0 dell'Insubria, Italy) Updated information (deadlines, fees, registration, abstract submissions, accommodation, social program, etc.) will be provided in the conference web page http://saxo.univ-littoral.fr/CT08/ The Organizing Committee, Dominique Bourn, Marino Gran and Shalom Eliahou From rrosebru@mta.ca Fri Nov 9 12:03:54 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Nov 2007 12:03:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IqWES-0004tO-5Q for categories-list@mta.ca; Fri, 09 Nov 2007 11:57:20 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories From: JeanBenabou Subject: categories: References Date: Fri, 9 Nov 2007 00:28:50 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 31 Dear colleagues, I have had, so far, no less than 22 answers to my mail about references for the "Beck" theorem mentioned in the Elephant, all of them supporting Peter Johnstone. An I am not counting "references" which "referred" to other "reliable references" such as Dusko Pavlovic an Claudio Hermida in one of the THIRTEEN mails sent by Marta Bunge! I intend to answer in detail to all these mails. It might take a few days, because I don't have such a powerful team helping me in my research: bibliography, recollections, etc. It takes ALL of my time, 12 hours a day, but I enjoy EVERY MINUTE OF IT. I congratulate Peter Johnstone to have such a numerous and faithful army of supporters. But as you say in English: "The more the merrier". So Johnstone might use the delay before my answer to find a few dozen more supporters. It will make me even more happy, and I'm afraid he will need ALL the support he can get! From rrosebru@mta.ca Fri Nov 9 19:33:32 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Nov 2007 19:33:32 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IqdGD-000372-9e for categories-list@mta.ca; Fri, 09 Nov 2007 19:27:37 -0400 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: References Date: Fri, 09 Nov 2007 13:07:10 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 32 Dear Jean, Here is what seems to be my 14th letter to you in this connection. It is a friendly reminder that I did not "support" Peter Johnstone. Here is an extract at the end of my November 4 official intervention ("Response to Benabou") in categories. >It seems then to be an error on the part of Peter Johnstone to have >attributed Proposition 1.5.5 in E1 (page 297) to Beck and not to Benabou >and Roubaud. At the end of this section on "Descent Conditions and Stacks" >(page 303), the references given in El 1 are Bourn, Bunge and Pare, Giraud, >Grothendieck, Reiterman and Tholen, but curiously enough, not >Benabou-Roubaud. I am sure that Peter will repair this error should a >second edition of the Elephant ever appear. In fact, I do not think that Peter Johnstone "supports" himself in this matter -- he has admitted the error. What more do you want him to say? However, in view of the evidence, provided by Street and Lawvere, of the possibility that Jon Beck may have discovered this theorem independently and even talked about it, as the abstract in the Notices of the AMS (1967) seems to indicate, it seems fair after all to add his name to yours in connection with it. It is common pratice in mathematics to give credit for ideas disseminated at lectures, even more so if these ideas are mentioned in an abstract. This does not take away your own credit. This is my very last letter on the subject, private or official. With best wishes, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ >From: JeanBenabou >To: Categories >Subject: categories: References >Date: Fri, 9 Nov 2007 00:28:50 +0100 > >Dear colleagues, > >I have had, so far, no less than 22 answers to my mail about >references for the "Beck" theorem mentioned in the Elephant, all of >them supporting Peter Johnstone. An I am not counting "references" >which "referred" to other "reliable references" such as Dusko >Pavlovic an Claudio Hermida in one of the THIRTEEN mails sent by >Marta Bunge! I intend to answer in detail to all these mails. >It might take a few days, because I don't have such a powerful team >helping me in my research: bibliography, recollections, etc. >It takes ALL of my time, 12 hours a day, but I enjoy EVERY MINUTE OF >IT. I congratulate Peter Johnstone to have such a numerous and >faithful army of supporters. But as you say in English: "The more the >merrier". So Johnstone might use the delay before my answer to find a >few dozen more supporters. It will make me even more happy, and I'm >afraid he will need ALL the support he can get! > > > _________________________________________________________________ Express yourself with free Messenger emoticons. Check out freemessengeremoticons.ca From rrosebru@mta.ca Fri Nov 9 19:39:17 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Nov 2007 19:39:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IqdQk-0003xv-DF for categories-list@mta.ca; Fri, 09 Nov 2007 19:38:30 -0400 Date: Fri, 9 Nov 2007 18:39:41 +0000 (GMT) From: Dusko Pavlovic To: categories@mta.ca Subject: categories: Re: References References: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 33 [Note from moderator: I agree with Dusko's remarks below. After 48 hours no further items will be posted in this thread. Moderate language will be used in any contributions posted during that time.] am i the only one who is not enjoying these arguments on CATEGORIES any more? i started worrying: ** is there something special in category theory that attracts all these ** argumentative people? is it that being argumentative somehow helps you ** with math, and then arguing with the symbols on a piece of paper and on ** the blackboards then somehow overflows into your social life?... and then i remembered that sometime in the early 90s, when the mailing lists started spreading, there were such flame wars on every mailing list. and they went on deep into the mid 90es. so i thought, maybe the category theorists are not more argumentative than other people. maybe it's just that some of them started using email a bit later, so they are discovering the medium of flame war just now. so i waved my hand. but now great jean benabou brought my little name into his argument, together with claudio hermida. i am not sure what he means, but it sounds like claudio and i are the examples of unreliable sources. well, honestly, jean, i really don't have the remotest idea what i have done to deserve the unhonorable mention. not having worked in anything related to this area for more than 10 years, and not having done much worth mentioning before that, AND not really depending on this community either for my job or for my reputation, i am simply just surprised... even after 10 min of scouring my memory, the only *remote* possibility that occurs to me is that i gave a talk in oberwolfach, cca 1994, describing the preservation conditions necessary and sufficient for descent, and effective descent ((monadicity is sufficient, but not necessary)). after my talk, or maybe in the middle, jean benabou stood up and said: "Mathematics Should Be Beautiful." i started mumbling that i was sorry if mine was so ugly ((was it my hand drawn diagrams on the slides?)), but that some people would say that mathematics should first of all be true... by which point we were both talking --- and max kelly hushed us both down. (we miss you, max!) i really really respect jean benabou's work. i also respect paul taylor's work. i have learned a great deal from both of them. but i really don't like how they argue their cases on this mailing list. (and if i am alone in that, then maybe i dont even belong here.) i agree that mathematics should be beautiful. short of that, since the truth doesn't always obey everyone's taste, math should at least be elegant, or decent. but if we are able to produce beautiful, or elegant, or at least decent mathematical arguments --- why is it that we generate such unpleasant nonmathematical arguments? can we please please stop? all the best, -- dusko On Fri, 9 Nov 2007, JeanBenabou wrote: > Dear colleagues, > > I have had, so far, no less than 22 answers to my mail about > references for the "Beck" theorem mentioned in the Elephant, all of > them supporting Peter Johnstone. An I am not counting "references" > which "referred" to other "reliable references" such as Dusko > Pavlovic an Claudio Hermida in one of the THIRTEEN mails sent by > Marta Bunge! I intend to answer in detail to all these mails. > It might take a few days, because I don't have such a powerful team > helping me in my research: bibliography, recollections, etc. > It takes ALL of my time, 12 hours a day, but I enjoy EVERY MINUTE OF > IT. I congratulate Peter Johnstone to have such a numerous and > faithful army of supporters. But as you say in English: "The more the > merrier". So Johnstone might use the delay before my answer to find a > few dozen more supporters. It will make me even more happy, and I'm > afraid he will need ALL the support he can get! > > > From rrosebru@mta.ca Sun Nov 11 19:48:43 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 11 Nov 2007 19:48:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IrML4-0000Wn-BF for categories-list@mta.ca; Sun, 11 Nov 2007 19:35:38 -0400 Date: Sat, 10 Nov 2007 10:33:41 -0500 From: "Keith Harbaugh" To: "Dusko Pavlovic" , categories@mta.ca Subject: categories: Re: References MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 35 The recent postings of Jean Benabou have, it seems to me, had at least two effects: 1. promoting a better understanding of the historical truth of how, in what order, and by whom various categorical ideas were discovered and disseminated, and 2. in confronting possible inaccuracies in some other mathematicians understanding of the same, making some of those other mathematicians, and no doubt some bystanders, uncomfortable (to say the least) at the inflammatory tone of some of M. Benabou's remarks. I think the first effect is salutary and beneficial to the understanding of the past, and thus also, to the future. It would be unfortunate and, I believe, harmful, to prevent M. Benabou from assisting the mathematical community in obtaining better understanding of the categorical past. On the other hand, at least one thing should not have been said on this mailing list, or elsewhere: "I advise [Peter Johnstone] to lose very quickly such habits, they might become dangerous for one's health." (Benabou's message of November 3). Threats, explicit or implict, to the reputation of individuals are permissible (one's reputation is a function of the collective body), while threats to one's "health" are not. As a proposal, and request to the moderator to modify his "48 hour" limit imposed in his addition to Dusko Pavlovic's message of November 9, I suggest the following rule: The discussion and argument may continue to the point of diminishing returns (repetition and triviality). Threats of illegal action (such as the one quoted above) will not be allowed. A positive and respectful tone, accepting the good faith and intentions of all parties, is required. Again, the intent is to keep the good (increased understanding), while avoiding the bad (unnecessary insults and threats). By the way, I am, shall we say, an interested bystander to the discussions on this list. The fact that my contributions to category theory are nil may, on the one hand, mean that I do not have the right to intrude in this matter, or on the other hand, may mean that I can be more objective, without axes to grind or anything to defend or assert. You choose which to believe. In any case, thanks to all parties, Rosebrugh, Benabou, Johnstone and all the others for your vast and much appreciated contributions both to mathematical research and to this list. Sincerely, Keith Harbaugh From rrosebru@mta.ca Sun Nov 11 19:48:43 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 11 Nov 2007 19:48:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IrMJt-0000S0-Q9 for categories-list@mta.ca; Sun, 11 Nov 2007 19:34:25 -0400 Date: Sat, 10 Nov 2007 08:22:06 +0100 From: "George Janelidze" To: Categories Subject: categories: Monadicity and Descent MIME-Version: 1.0 Content-Type: text/plain; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 36 Dear Colleagues, I agree with Dusko that it is much better to avoid any offensive remarks and I agree with every word Marta says in what she calls her "seems to be 14th" message. On the other hand, it really feels as everyone is trying to argue against Jean, gently or less gently=85 and then it is understandable if Jean's remarks become a bit (or more than a bit) angry. Removing "offensive", "against", and "angry", I think this discussion was actually very useful since it gave new information on the history of an important mathematical discovery (I essentially knew what Ross said because I heard that from Ross before, but I did not know about Jon's talk in 1967 mentioned by Bill). Let me also add that [G. Janelidze and W. Tholen, Facets of descent I, Applied Categorical Structures, Vol. 2, No 2, 1994, 245-281] refers (for the "monadicity=3Ddescent") to both B=E9nabou and Roubaud and to Beck. However: while the precise reference on the B=E9nabou-Roubaud paper is given there, Beck's name is mentioned without any reference to a paper (since as far as we knew he did not publish such a paper). I also had feeling confirmed by this discussion that nobody really remembers what exactly did Jon do, while the B=E9nabou-Roubaud paper is available to everyone=85 George Janelidze From rrosebru@mta.ca Sun Nov 11 19:48:43 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 11 Nov 2007 19:48:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IrMLr-0000Za-VJ for categories-list@mta.ca; Sun, 11 Nov 2007 19:36:27 -0400 Date: Sat, 10 Nov 2007 22:46:45 +0200 (EET) Subject: categories: Category Theory papers From: "Georgios Nassopoulos" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-7 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 37 Some Category Theory papers are referenced at http://users.uoa.gr/~gnassop Pr. George F. Nassopoulos From rrosebru@mta.ca Sun Nov 11 19:48:43 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 11 Nov 2007 19:48:43 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IrMML-0000bc-R0 for categories-list@mta.ca; Sun, 11 Nov 2007 19:36:57 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories From: JeanBenabou Subject: categories: Droit de reponse Date: Sun, 11 Nov 2007 05:43:41 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 38 NOTE TO THE MODERATOR You have tried, and succeeded, to stop me in the beginning of 2006, when I wanted to oppose to the preposterous change of "cartesian" and "cocartesian" into "prone" and "supine". You shall not stop me now! I have received more than 30 mails in ten days. I HAVE A RIGHT OF REPLY, which is admitted by every civilized community, and I explicitly refer to it ! Thus, I ASK for TEN DAYS ACCESS TO THIS LIST STARTING AT THE MOMENT I RECEIVE AN ANSWER FROM YOU TO THE PRESENT MAIL Even this is unfair, because I am alone to do answer an ever growing group of correspondents. But I will do with 10 days. Unless of course I receive more mails during these 10 days, in which case the delay for my answers will be extended. If you try to stop me I shall send my answers, together with this mail, directly to the members of the category list whose mail address I have. But not only to them. There are a lot of mathematicians, in category theory and other fields, French and of all citizenships, who would be very interested to receive it. Some of them would just "adore" to receive it. Especially if I join the pdf. file I made in 2006 which is on my computer under the name "Kafka in category-land". It contains, among many other things, a "lesson of typography" by Wood, a very elegant mail by Paul Taylor suggesting that I was now too old and should concentrate my activities to harmless "playing with my rattles and fluffs", and the mails we exchanged during that period I shall also send a copy to the French "Academie des Sciences" where my joint note was published, to the "Societe Mathematique de France", where some persons literally worship Grothendieck, and others tend to like some of my mathematical "rattles and fluffs". And also to any other institution I can think of. So far, in all the mails I have received, there is not the slightest proof that Beck had proved the theorem attributed to him by Johnstone, let alone any justification of the fact that he "forgot" to mention in his bibliography the Benabou-Roubaud (B.R) note. The problem is not only a matter of "priority" of the (B.R) note. In some of the answers, in particular in Johnstone's, other subjects have been evoked e.g. Celeyrette's thesis, my Louvain paper, and many other "omissions". I shall mention them. I intend to give proofs to support each of my statements So I "beg" you Mr. Moderator, don't interfere! The persons who sent more than 30 mails on this question, and in particular Johnstone, are adults. Let them SPEAK FOR THEMSELVES. And let the mathematical community be the sole judge! As I am invoking my RIGHT TO ANSWER, I shall tolerate NOT A SINGLE CHANGE in the present mail. I am addressing it to the "Category list", not only to you Mr. Moderator, thus of course I want it to be sent to all members of this list. From rrosebru@mta.ca Mon Nov 12 11:45:56 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 12 Nov 2007 11:45:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IrbN9-00064z-IF for categories-list@mta.ca; Mon, 12 Nov 2007 11:38:47 -0400 Date: Mon, 12 Nov 2007 15:34:52 +0100 (CET) From: Jiri Adamek To: categories Subject: categories: A Position in Theoretical Computer Science in Braunschweig MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 39 6-year PhD or Post Doc Position in Theoretical Computer Science At the Institute of Theoretical Computer Science of the Gauss Faculty of the Technical University of Braunschweig a position is available from December 2007. Candidates must have an outstanding undergraduate degree (German Diplom, Master or equivalent) in computer science with a strong theoretical background or a degree in mathematics. Post-doctoral candidates can also be considered. The successful candidate is expected to combine research in the field of algebraic and coalgebraic methods in computer science with teaching duties of 4 hours a week which are expected to be held in German. The position will initially be filled for two years and is extendable for up to six years. The salary ranges between EUR 2800 and 3100 per month depending on the background of the candidate. Candidates are requested to send their applications until November 15 to me, preferably by e-mail. According to current German legal rules preference must be given to equally qualified female or disabled candidates. Prof. Dr. Jiri Adamek Chair of the Institute for Theoretical Computer DScience Technical University of Bruanschweig Postfach 3386 38 106 Braunschweig, Germany xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx From rrosebru@mta.ca Mon Nov 12 18:44:38 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 12 Nov 2007 18:44:38 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IrhwR-0001uW-E2 for categories-list@mta.ca; Mon, 12 Nov 2007 18:39:39 -0400 Subject: categories: Postdoc positions at Dalhousie To: categories@mta.ca (Categories List) Date: Mon, 12 Nov 2007 17:10:16 -0400 (AST) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: <20071112211016.817E15C27F@chase.mathstat.dal.ca> Sender: cat-dist@mta.ca Precedence: bulk From: cat-dist@mta.ca Status: RO X-Status: X-Keywords: X-UID: 40 Killam and AARMS Postdoctoral Fellowships in Category Theory and the Foundations of Computation, Dalhousie University, Halifax, Canada The Category Theory group at Dalhousie University invites applications for postdoctoral positions. The successful candidates will participate in the activities of our group, and will be part of the Atlantic Category Theory Seminar, which includes faculty, postdocs, and students from Dalhousie University, Saint Mary's University, and Mount Allison University. See http://www.mathstat.dal.ca/~selinger/atcat/. Postdoc positions are for 2 years, and normally start on or near September 1, 2008. There are two types of positions that can be applied for: Killam Postdoctoral Fellowships (http://www.dalgrad.dal.ca/killam/kpdf/) Note: Ph.D. must have been obtained January 2006 or later. AARMS Postdoctoral Fellowships (http://www.aarms.math.ca/pdf/) Note: Ph.D. must have been obtained December 2003 or later. See the respective websites for the full application requirements and procedures. To meet internal deadlines, applications for either (or both) types of position, including three letters of reference, should be sent by December 17, 2007, to: Dr. Karl Dilcher, Chair Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia Canada, B3H 4H6 We suggest that any candidates applying in category theory or foundations of computation also contact one of us so that we are aware of the application. Bob Pare Richard Wood Dorette Pronk Peter Selinger From rrosebru@mta.ca Tue Nov 13 21:24:56 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 13 Nov 2007 21:24:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Is6m9-0003rW-6v for categories-list@mta.ca; Tue, 13 Nov 2007 21:10:41 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed To: Categories , From: JeanBenabou Subject: categories: First answers Date: Tue, 13 Nov 2007 08:49:45 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 41 1- TYPOGRAPHY AND MODERATION In this mailing list, such fonts as "bold face", "italics", etc are =20 not accepted, only lower and upper case letters can be used. I =20 mention these trivialities because in february 2006 I received a =20 lesson of typography from Wood. As I receive more and more "lessons" =20 through this mailing list, I have made a file in my computer, with =20 these lessons, an a few other things, under the name of : " Kafka in =20 Category-land". I have consulted my "Kafka-file" thus I can quote =20 precisely Wood's lesson: "Jean Benabou...should also be told that ordinary words written in =20 upper case are understood to be SHOUTED ". I have not the slightest intention to shout in this mail, nor in any =20 other I send through this mailing list. But, in spite of Wood's =20 lesson, because of the drastic typographical limitations of this =20 list, I shall use upper case letters for titles, subtitles, or =20 whenever I want to emphasize a word or a sentence. And if Professor Wood disapproves my decision, I refer him to Freyd =20= and Scedrov who used quite liberally upper case letters in their =20 book, although they had huge typographical possibilities, including =20 very fancy "punctured diagrams". I'm getting a bit tired by the ever increasing number of lessons I =20 receive in this mail. About every subject. Except mathematics, which =20 I'd very much like to receive. Unfortunately in this domain I'm =20 usually on the "giving" side. As for example in my mail on locally =20 cartesian closed categories. Most of you know that English is not my mother language. I try to =20 write it as well as I can. If you find something wrong in my typing, =20 my spelling, my grammar, or even my "style", forgive me. But no more =20 "lessons' please! I shall try to answer to each of the persons who have participated to =20= this discussion. There are many, and there might even be more since I =20= received this morning mails from Janelidze and Hardbaugh. You'll =20 understand that, even if I work full time at it, and I do, it will =20 not be possible to "sort out" the order of my answers, so i'll give =20 each of them as soon as I have enough documentation to complete it. I =20= apologize if some of you will have to wait longer than others. I have been advised many times to use "moderation" in my answers. I =20 am usually a moderate person, but I shall try to be even more so. =20 Please try to make the the same "effort" when addressing to me. I =20 shall use in each individual answer the same "degree of moderation" =20 as the one I found in the mail (or mails) I'm answering to. for =20 example in my : 2- ANSWER TO DUSKO PAVLOVIC =46rom his mail I quote: "but now great jean benabou brought my little name into his argument, together with claudio hermida. i am not sure what he means, but it =20 sounds like claudio and i are the examples of unreliable sources." And I answer : Great Dusko Pavlovic, with all due respect of course, =20 little Jean Benabou would like to point out that it was never him, =20 but Marta Bunge who brought your great name, together with Claudio =20 Hermida's, into this argument. As for your reliability, and the relevance of your testimony in this =20 subject, I do not have the slightest doubt about them. It seems that Peter =20 Johnstone has some, because from one of his mails to Marta I quote: "Without wishing to be rude, I'm not sure that I would take Dusko as an authority on the history of category theory" I do not know what he meant by that, thus I'm afraid that 4 persons: =20 Claudio, Marta Peter, and great Dusko, will have to "sort out" their =20= big problems. They are no concern of little Jean Benabou. Respectfully yours,Jean. 3- ANSWER TO JIM STASHEFF I quote: "Perhaps some one took notes on paper and hasn't thrown them out? It would be great to ad them to Jon's small nachlass." A very good question Jim, even if your purpose is different from mine =20= I totally agree with it. I'm surprised nobody did come out with an =20 answer. And I ask you a straightforward question: Aren't YOU surprised that of this "mythical" result, there is not a =20 single written trace, no publication by Beck or by ANYONE who would =20 have used his result in his work. And no written notes? I won't be as choosy as Peter Johnstone, and will accept any =20 reference in a thesis, a prepublication in the most obscure =20 university, anterior to my joint paper with Roubaud. Fair enough? 4- ANSWER TO MICHAEL BARR =46rom the second answer of Barr I quote "Jon had a precise formulation of a rather simple special case that =20 did not mention either fibrations or indexed categories (the latter =20 didn't exist at the time" This, for me, settles the problem on two counts: (i) I learnt the Chevalley conditions from Chevalley in 1964, in a =20 public course, he is dead, but many people who attended this =20 course,in particular Jacques Roubaud, but many others I know, will be =20= quite ready to testify if you "force me to ask them". The Chevalley condition is about FIBRATIONS, it can of course be =20 reformulated in terms of "indexed categories" which, according to =20 Barr, didn't exist at the time, neither did fibrations, the first =20 mention of them in "Category-land" is in John Gray's paper in La =20 Jolla (1965). Thus Beck couldn't have discovered this condition in =20 1964. And I promise to rise a public debate, outside of this list, if =20= the name of Jon Beck is mentioned about them in the future. Some people seem to believe that there were no categories outside of =20 "Category-land". Typical of this attitude, sorry Mike, is your =20 statement that indexed categories DIDN'T EXIST at the time. They were invented by Grothendieck in 1961, NOT by Lawvere in 1971. The main mistake of this "fertile brain" (El; preface) was to have =20 called them "pseudo-functors" which isn't such a bad name after all. =20 At the same period he did a much much more INEXCUSABLE mistake, =20 namely: not realizing that: The Times They Are A'changin' (Bob Dylan, =20= 1964). And that "prone" and "supine" were obviously more "chic" than =20 the "overworked" "cartesian" and "cocartesian". (I shall examine this question of "terminology" in more detail in my =20 answer to Peter Johnstone) Thus the Benabou-Roubaud result cannot have been proved by Beck. It's =20= a result about Chevalley bi-fibrations, which can be,but why should =20 they be?, reformulated in terms of indexed-categories. But none of =20 the two "existed" according to Barr, and not in "category-land" =20 according to Benabou. (For more details about this "non-existence", I also refer to my =20 answer to Bill Lawvere, which will be come next) 5- ANSWER TO WILLIAM LAWVERE I quote you: "I would like to urge the readers of our categories list to study =20 Jon's 1967 thesis which is now available via TAC Reprints. There one =20 finds Jon's tripleability theorems which were used for example by =20 Benabou and Roubaud in their 1970 paper on descent" Sorry Bill, don't CONFUSE THE ISSUES: We are talking about =20 Chevalley's condition, and my joint result with Roubaud about the =20 relation between "descent data" and algebras for a monad. ABSOLUTELY =20 NOT about Beck's tripleability theorems, which we used, with due =20 reference, in our note. Is there ANY mention in this "now available" =20 thesis of fibrations, of ANY KIND of condition on fibrations, and of =20 "descent data". If there is none, mentioning this thesis is totally =20 IRRELEVANT to the present discussion. One of the major interests of our result, was that it gave the =20 possibility TO USE Beck's theorems, in a totally "new" domain, namely =20= "descent", and that IN ORDER TO DO IT, Chevalley's condition was =20 sufficient. It is also necessary, which means our result CANNOT be =20 improved! Can anyone after 37 years, give a better result, of course =20 in the general case of fibrations (or indexed categories, if you =20 still prefer them!) Of course, in special cases you can get "better" results. Going to =20 the extreme, I can "prove" that, for the identity fibration, every =20 map is a descent map! Our result can be GENERALIZED, if one generalizes the notion of =20 descent. I have had such a huge generalization for many years, in a =20 paper called "what is descent in year 2000?" I am NOT going to talk about it, or about many new results of mine =20 concerning not only fibered categories, but the much more general =20 FOLIATED ones, I have defined and studied, because in these sad, sad =20 times, I'm SURE I would find these results in some other "Elephant", =20 re-named, duly "re-indexated", and with NO reference to me. For =20 "space-saving" reasons, of course! I quote you again: "Ross remembers that Jon arrived in Tulane in 1969 prepared to lecture on triples and descent" First big mistake Bill. Ross mentioned that Jon arrived in Tulane =20 after I left. And I was there in early 1970. So the earliest Jon =20 could have been there was spring 1970, not 1969! Although I have many =20= answers to give, in each case I check all the "details"; and in this =20 case it is NOT a detail. Ross also "remembers" that Jon DID NOT speak. Neither in a formal =20 lecture, nor in INFORMAL DISCUSSIONS, with ANYBODY. I was in Tulane =20 at about the same period, there was quite a concentration of category =20= theorists, and "I remember", that apart from the formal lectures, =20 there were many informal discussions, as there are in any category =20 theory meeting, or in ANY mathematical meeting. Quite surprising, isn't it, that Jon spoke to NOBODY who can =20 "remember" ANYTHING Quotation again "I heard Jon lecturing on that topic already in late 1967 at a =20 meeting of the American Mathematical Society in Illinois. In =20 particular, he explicitly stated the condition that I therefore =20 called the "Beck condition" in my work on Hyperdoctrines (presented =20 to an AMS meeting in NYC in early 1968) On what "topic"? If it is "the" Chevalley condition, Chevalley =20 lectured about it in 1964, i.e. four years BEFORE the 1968 meeting =20 you mentioned (He probably had it before 1964, but 1964 is good =20 enough for me!) You probably remember that I spent the academic year 1966-67 in =20 Chicago. We even shared the SAME office. There were many meetings of =20 the Midwest Category Seminar, And "I remember", with precision, that =20 I was AMAZED by the fact that fibered categories and descent were not =20= mentioned A SINGLE TIME during that whole year. My surprise was all the greater since in Paris, the Grothendieck. =20 school was devoting a lot of time and energy on this "topic"; I might =20= remind to some of those who have small problems with their memory, =20 that Giraud's thesis was published in 1964 and the title was "Methode =20= de la Descente". And of course, of the fundamental paper of =20 Grothendieck, "Categories fibrees et Descente" (1961). As a side remark, you are of course aware that "Hyperdoctrines" are =20 an important, but very special case of fibrations . And I am still =20 waiting for an answer to the question I asked in my previous mail: =20 Has ANYBODY defined a notion of morphism of Hyperdoctrines, and, if =20 nobody has, WHY NOT? Thus sorry Bill, with all due respect, I don't think Jon Beck, =20 "invented" the Chevalley condition, and if he proved ANYTHING about =20 Monads and descent, it must have been in a very, very special case, =20 since as Michael Barr stated it a bit curiously, "indexed =20 categories", let alone fibrations, "did not exist" when we published =20 our note. One more "detail" , I greatly admire your deep mathematical insight, =20 therefore I am convinced that if you had heard Beck speaking about =20 our joint theorem, you would immediately have understood its meaning, =20= and its possible consequences, and either you, or some of your =20 students, or both, would IMMEDIATELY have put the theorem "at work". =20= And it would have found its way very quickly in many papers and =20 books. Just as Beck's tripleability theorems did, and were frequently =20= quoted, and "refined". e.g. in a long paper by Duskin in the Reports =20 of the Midwest Category Seminar (RMCS) , Springer Lecture Notes (LN) =20 106 (1969). No mention of fibrations or descent in Duskin's paper, although Jack =20 quoted in his bibliography: BECK , J., untitled manuscript, Cornell, =20 1966. No mention in this whole (LN). No mention in the next RMCS, LN137,1970. WHERE ARE THE PAPERS BILL? Was all "Category-land" ASLEEP between =20 1967 and 1971? Again a quotation "Later I saw this condition referred to as the Chevalley condition in =20= a paper of J-L Verdier. I do not know whether Jon was familiar with =20 that work of Chevalley" Later than what? Again I insist on the fact that Chevalley's =20 condition is about fibrations. You "do not know", but ,concerning the =20= question of priority of Chevalley, with all due respect, what YOU =20 know is not the issue. Neither is what I know, but at least, I was in =20= Paris in the years 1960-1965, where all these things happened, I =20 participated in some of them, thus what I know, I know "first hand". "Final" quotation: "The abstracts for the talks at that meeting were published in the Notices of the AMS Volume 14 (1967). On page 938 one finds Jon Beck's abstract: 652-8. Jon Beck, Cornell University, Ithaca, New York. Descent and standard constructions (triples). There is a close relationship between descent theory in algebraic geometry and the theory of categories which are =20 definable by means of standard constructions (tripleable categories). The "tripleableness theorem" sheds some light on descent criteria. The form of Cech cohomology used in descent theory is an appropriate triple cohomology theory. Its interpretation is discussed from the triple point of view. (Received October 2, =20 1967.) It is possible that someone still has notes of that lecture 40 years =20 later" Again sorry Bill. There is A HUGE difference between noting that in =20 one, or perhaps a few, SPECIAL CASES, such as Cech cohomology, there =20 was a "close relationship", which "sheds some light" on descent =20 criteria, and a precise GENERAL theorem, which, in order to be =20 stated, needs fibrations, AND the Chevalley condition! As I already mentioned,Beck's tripleability theorems found their =20 place IMMEDIATELY in countless books and papers,WHY NOT "his" other =20 theorem? WHEN is the first precise mention of it before, say, 1975. There is A WHOLE CHAPTER in Mac Lane's CWM book (1971) devoted to =20 monads, no mention of Beck's "contribution" to descent. Not in the =20 exercises also, not even in the informal historical note, where he =20 could have mentioned it without giving details. So WHERE ARE the =20 publications, by Beck or ANYBODY, mentioning "his" theorem, or USING it? Sorry Bill you have totally failed to convince me. If you have =20 convinced other persons, when I answer them, I shall carefully study =20= their arguments and shall try to answer them. But I won't even bother =20= to answer to "religious" arguments of the form "I believe that" Beck =20 had the theorem because Bill Lawvere "said so". Absolutely no offense =20= meant to you Bill, "I believe" in your deep insight in mathematics, =20 "I believe" in the importance of your contributions to category =20 theory. But, please Bill, don't ask me to believe in more. Maybe it is high time that some persons realize and/or admit that, =20 NOT ALL of category theory was born in "category-land", and some of =20 it still lives "abroad". 6- ANSWER TO ROSS STREET I quote : "the "indexed categories " of Pare-Schumacher" !? Sorry Ross, the "indexed categories" are not Pare-Schumacher, and not =20= Lawvere. They are, as I mentioned to Barr, also due to the "fertile =20 brain" of "you know who". I have repeated this for more than 35 =20 years. Vainly it seems. And since you are kind enough to mention my =20 1967 paper on bicategories that many a'member of "Category land", and =20= quite a few "outlandish" mathematicians have read, they are in this =20 paper, in detail, page 47, under the name of pseudo-functors and and =20= in the bibliography, clearly attributed to you know who. Can't most =20 category-landers read? That paper was written in English if my memory =20= doesn't " fail me" . By the way, "descent data" were also in this paper, "profunctors" =20 were announced page 49, and the fundamental theorem extending =20 Grothrndieck's construction to morphisms: E=B0-->Prof was clearly stated, together with the applications I had in mind. Thank you for having "forced" me to re-read that 40 years old paper =20 which i had not looked at for more than 30 years. It hasn't aged at =20 all, and I can still recommend it for the variety of important =20 examples it contained. Some of which I didn't remember I had =20 mentioned at that "prehistorical" age. I quote you again "I learnt from Benabou's lectures about the "Chevalley condition" for =20= fibrations and how descent data were Eilenberg-Moore algebras. Jean =20 gave me a copy of his Comptes Rendus article with Jacques Roubaud" "Very soon after Jean Benabou left, Jon Beck arrived ...When I told =20 him about Benabou's lecture on descent, he said that that was what he =20= had planned to talk about ("triples" and descent). I encouraged him =20 to do so but he decided to change his topic." Thus, you didn't hear Beck speak, not even in a private conversation, =20= on a "mythic" theorem he MIGHT have had about fibered or indexed =20 categories, which "DIDN'T EXIST". And that, of course, fully justifies Jonhstone not only for giving =20 Beck full credit, but "forgetting" to mention in his bibliography our =20= joint note. Which I gave you personally, and lectured publicly about. =20= And, by a strange "coincidence", the theorem stated by Johnstone, was =20= PRECISELY, the one in our note! I said I wouldn't shout, so I don't. =20 I'm not sure that Roubaud won't when he learns about the very =20 "special" sense of history and honesty which seems to prevail in =20 "Category-land"! And "operads", comma objects, and other 2-categorical "whatnots" =20 won't hide the real issue of Johnstone, KNOWING WHAT HE WAS DOING, =20 falsified "history" as you call it. But he did that SO MANY TIMES, =20 with MY work in particular, and with the approbation of many members =20 of the establishment of "Category-land", that he thinks he can get =20 away with "anything". On a totally different question, I quote you again: "Now, as much as I would love SIX bottles of GOOD champagne, I am not =20= going to submit a suggestion for Jean's challenge. Composition of =20 fibrations is a wonderful thing as is composition of homomorphisms of bicategories; =20= but they do different jobs. It is hard enough to say fibrations are =20 composable from the homomorphism viewpoint! There is a thing about this that requires a mixture of the two views. Regard one fibration p : E --> A as a homomorphism E_ : A --> Cat. Keep the other q : A -- > B as a fibration. Then the homomorphism =20 corresponding to the composite q p is a generalized left Kan extension of E_ along q." In my paper on bicategories, I gave a lot of examples. One of which =20 was pseudo-functors, now called indexed categories, and attributed to =20= Lawvere. But I NEVER mentioned fibrations in that paper, although I =20 knew about them, and about Chevalley's condition, which I remind you, =20= I learned in 1964. Because I KNEW, already at that time, that in =20 spite of what "The Elephant" says: FIBRATIONS AND "INDEXED CATEGORIES", ARE NOT THE SAME THING. Because: (i) The theory of fibrations is a FIRST ORDER theory, the "theory" of =20= indexed categories IS NOT. It is not even a "higher order theory" =20 however "high" you are ready to go. (ii) Fibrations can be INTERNALIZED in a topos (a logical category =20 suffices), indexed categories CANNOT. (iii) To go from an "indexed category" to a fibration, by the =20 Grothendieck construction, DOES NOT require the axiom of choice (AC) =20 whereas in the other direction YOU NEED AC. For sets, if you restrict =20= to small categories, and for classes (whatever that means) if you =20 deal with big ones, even if they are locally small. In most definitions, constructions and proofs with fibered =20 categories, all we need is FINITE DIAGRAMS involving vertical and =20 cartesian maps (sorry, "prone" maps if you understand those better). =20 We almost never need a cleavage of the fibration, which requires AC. =20 And in the very, very few cases where we do need it, a special =20 emphasis should be put on this necessity. (iv ) Last, but not least, fibrations DO compose, and "indexed =20 categories" DON'T And because of this last fact, I knew I was taking no big risk by =20 offering champagne! But of course, MY OFFER IS STILL STANDING. Sorry Ross, you don't qualify for even A GLASS of champagne. =20 Nevertheless I'll be very happy to offer you more than a glass if you =20= visit me in Paris, for old friendship's sake, but not for having come =20= any close to answering my question. And your so called "mixture of two views", requires, if you start =20 with two "indexed categories", FIRST TO REPLACE one of them by a =20 fibration, and THEN, to use generalized left Kan extensions. You =20 CANNOT avoid the first step, can you? All this complication compared =20 with the well known, 5 lines proof or no proof at all, result : =20 fibrations are stable under composition. Quite (un) surprisingly, I could not find any "trace" of this result =20 in the "monumental" Elephant. Am I wrong professor Johnstone? Are you =20= going to explain, AGAIN, as for my Louvain Paper, Celeyrette's =20 thesis, and the Comptes Rendus note that it was because of a "space-=20 saving decision"? Well, I've tried to explain, as I have done now for more than 30 =20 years, why I thought, and think more and more now, especially after =20 reading some "surprising" pages of the Elephant, that "indexed =20 categories" were to put it mildly, a WRONG manner to view fibered =20 categories.(See again (i) (ii) and (iv) even if you want to assume =20 any form of AC). But as we say in France: "Il n'est pire sourd que qui ne veut entendre" Even Peter Johnstone, who has been coeditor with Bob Pare of LN 611 =20= (1978) "Indexed Categories and their applictions" has been" forced" =20 to introduce fibered categories in his Elephant. Very badly I must =20 say, And, of course, with the same "space-saving" attitude concerning =20= good references.Especially to my work! As for me, I have never been "forced" to change my position. "Fibered =20= man" I was, from the beginning, and "fibered man" I remain. I don't =20 even have to compromise, and become "half-fibered and half-indexed". I"ll interrupt this mail now because: (i) I don't want it to be rejected because it is too long (ii) I really need some sleep! (iii) Although many other answers are almost finished, I need a =20 little more documentation to be absolutely sure about a few facts. I shall continue my answers on tuesday, meanwhile any further =20 "testimonies" or "comments" from any of you, even from the very =20 "prolific" Marta, will be welcome! Before I stop, I'd like to add a very special 7- ANSWER TO RONNIE BROWN Dear Ronnie, Please forgive me for not having answered earlier to your very kind, =20 and nice message of october 31. I appreciated it all the more because =20= it was the only mathematical "reaction" I received after my long mail =20= concerning locally cartesian categories, and "a few other things". I want to thank you publicly. And I want also to tell you that in the =20= "help!" discussion about what to tell to absolute beginners about =20 Category Theory, your contribution was, by far, the one I appreciated =20= most. You'll understand why, if I tell you that in 1999, in a colloquium on =20= history and philosophy of mathematics I gave a talk with the title =20 "Une analogie en theorie des categories". Moreover in the =20 introduction of this talk I explained that I had intended initially =20 to speak about : "Analogies et theorie des categories". And a first =20 draft of more than 120 pages, covering only a part of what I wanted =20 to say, convinced me that "une analogie" would suffice. I hope when this unpleasant phase is finished, if it is ever =20 finished, that we'll have more time to compare our ideas. They will =20 probably not coincide, but I'm sure we will very quickly agree on =20 many many points. Meanwhile, many thanks again, Jean= From rrosebru@mta.ca Wed Nov 14 12:19:41 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 14 Nov 2007 12:19:41 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IsKrG-00079j-D4 for categories-list@mta.ca; Wed, 14 Nov 2007 12:12:54 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) To: categories@mta.ca From: Kathryn Hess Subject: categories: Postdoc positions in Lausanne Date: Wed, 14 Nov 2007 17:02:33 +0100 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 42 The Institute of Geometry, Algebra and Topology (IGAT) of the Ecole =20 Polytechnique F=E9d=E9rale de Lausanne (EPFL) invites applications for =20= full-time postdoctoral positions from 1 September 2008 through 31 =20 August 2009, with possibility of extension for a second year. =20 Priority will be given to applicants in geometry or topology, though =20 excellent applications in algebra or category theory will be =20 considered as well. In addition to research, duties include teaching within the framework =20= of the Mathematics Section of the EPFL. Candidates must have completed their PhD no earlier than 2004 and =20 have shown promise of excellence in research in geometry, topology, =20 algebra or category theory. Applications, including curriculum vitae, publication list, research =20 plan, statement of teaching experience, and three references, must be =20= submitted electronically by 15 January 2008 to Prof. Kathryn Hess =20 (kathryn.hess@epfl.ch). Applications may be submitted in French or =20 English. For further information concerning the IGAT, see http://igat.epfl.ch/=20 igat/.= From rrosebru@mta.ca Thu Nov 15 21:16:40 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 15 Nov 2007 21:16:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ispdq-0005OJ-IK for categories-list@mta.ca; Thu, 15 Nov 2007 21:05:06 -0400 Date: Wed, 14 Nov 2007 23:25:18 -0200 (BRST) Subject: categories: WoLLIC 2008 - Call for Papers From: ruy@cin.ufpe.br To: wollic@cin.ufpe.br MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 43 [** sincere apologies for duplicates **] Call for Papers 15th Workshop on Logic, Language, Information and Computation (WoLLIC 2008) Edinburgh, Scotland July 1-4, 2008 WoLLIC is an annual international forum on inter-disciplinary resear= ch involving formal logic, computing and programming theory, and natura= l language and reasoning. Each meeting includes invited talks and tutorials as well as contributed papers. The Fifteenth WoLLIC will be held in Edinburgh, Scotland, from July 1 to July 4, 2008. It is sponsored by the Association for Symbolic Logic (ASL), the Interest Group in Pure and Applied Logics (IGPL), the European Association for Logic, Language and Information (FoLLI), the European Association for Theoretical Computer Science (EATCS), the Sociedade Brasileira de Computacao (SBC), and the Sociedade Brasileira de Logica (SBL). PAPER SUBMISSION Contributions are invited on all pertinent subjects, with particular interest in cross-disciplinary topics. Typical but not exclusive areas of interest are: foundations of computing and programming; novel computation models and paradigms; broad notions of proof and belief; formal methods in software and hardware development; logical approac= h to natural language and reasoning; logics of programs, actions and resources; foundational aspects of information organization, search, flow, shar= ing, and protection. Proposed contributions should be in English, and consist of a schola= rly exposition accessible to the non-specialist, including motivation, background, and comparison with related works. They must not exceed 10 pages (in font 10 or higher), with up to 5 additional pages for references and technical appendices. The paper's main results must not be published or submitted for publication in refereed venues, including journals and other scientific meetings. It is expected that each accepted paper be presented at the meeting = by one of its authors. Papers must be submitted electronically at www.cin.ufpe.br/~wollic/wollic2008/instructions.html A title and single-paragraph abstract should be submitted by February 24, and the full paper by March 2 (firm date). Notifications are expected by April 13, and final papers for the proceedings will be due by April 27 (firm date). PROCEEDINGS The proceedings of WoLLIC 2008, including both invited and contribut= ed papers, will be published in advance of the meeting as a volume in Springer's Lecture Notes in Computer Science (tbc). In addition, abstracts will be published in the Conference Report section of the Logic Journal of the IGPL, and selected contributions will be published as a special post-conference WoLLIC 2008 issue of the Journal of Logic and Computation (tbc). INVITED SPEAKERS Olivier Danvy (BRICS) Anuj Dawar (Cambridge, UK) Makoto Kanazawa (Nat Inst of Informatics, Japan) Mark Steedman (Edinburgh U) Henry Towsner (CMU) (more to come...) STUDENT GRANTS ASL sponsorship of WoLLIC 2008 will permit ASL student members to apply for a modest travel grant (deadline: April 1, 2008). See www.aslonline.org/studenttravelawards.html for details. IMPORTANT DATES February 24, 2008: Paper title and abstract deadline March 2, 2008: Full paper deadline (firm) April 13, 2008: Author notification April 27, 2008: Final version deadline (firm) PROGRAM COMMITTEE Lev Beklemishev (Utrecht) Eli Ben-Sasson (Technion) Xavier Caicedo (U Los Andes, Colombia) Mary Dalrymple (Oxford) Martin Escardo (Birmingham) Wilfrid Hodges (Queen Mary, U London) (Chair) Achim Jung (Birmingham) Louis Kauffman (Maths, U Ill at Chicago) Ulrich Kohlenbach (Darmstadt) Leonid Libkin (Edinburgh U) Giuseppe Longo (Ecole Normal Superieure, Paris) Michael Moortgat (Utrecht) Valeria de Paiva (PARC, USA) Andre Scedrov (Maths, U Penn) Valentin Shehtman (Inst for Information Transmission Problems, Mosco= w) Joe Wells (Heriot-Watt U, Scotland) ORGANISING COMMITTEE Mauricio Ayala-Rincon (U Brasilia, Brazil) Fairouz Kamareddine (Heriot-Watt U, Scotland, co-chair) Anjolina de Oliveira (U Fed Pernambuco, Brazil) Ruy de Queiroz (U Fed Pernambuco, Brazil, co-chair) WEB PAGE www.cin.ufpe.br/~wollic/wollic2008/ --- From rrosebru@mta.ca Fri Nov 16 08:35:19 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 16 Nov 2007 08:35:19 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1It0Jt-0002uS-Er for categories-list@mta.ca; Fri, 16 Nov 2007 08:29:13 -0400 Mime-Version: 1.0 (Apple Message framework v752.3) To: Categories Subject: categories: CT2007 Proceedings From: Ross Street Date: Fri, 16 Nov 2007 16:49:35 +1100 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 44 ====================================================== Proceedings Volume for CT2007: approaching deadline for submission International Conference on Category Theory CT2007 June 17-23 Hotel Tivoli Almansor Carvoeiro, Portugal http://www.mat.uc.pt/~categ/ct2007 Recall that the proceedings of CT2007 will be published in a Special Issue of the journal Applied Categorical Structures (ISSN 0927-2852). We strongly encourage speakers at the conference to submit a paper. As guest-editors we are responsible for the handling and refereeing of all papers for the special issue. Authors should submit their papers through one of us. The deadline for submission is Wednesday 28 November 2007. Instructions for authors and other information about the journal can be found at: . Authors will use the Editorial Manager system . In particular, when submitting their paper, they should choose Special Issue CT2007 as Article Type and then designate one of the three guest-editors to handle their paper. We look forward to your cooperation in producing a fine scientific record. Yours truly, Guest Editors for the Special Issue: Samson Abramsky Maria Manuel Clementino Ross Street From rrosebru@mta.ca Fri Nov 16 13:25:38 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 16 Nov 2007 13:25:38 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1It4r0-0003RN-Ng for categories-list@mta.ca; Fri, 16 Nov 2007 13:19:42 -0400 Date: Fri, 16 Nov 2007 11:17:56 -0500 To: categories@mta.ca From: "Ellis D. Cooper" Subject: categories: Categories in thermodynamics Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 45 I would like to know whether a category theoretic "rationalization" of the mathematical theory of thermodynamics published by Elliot H. Lieb and Jakob Yngvason has been published or at least undertaken. Their axioms add structure to a preorder: \begin{description} \item [(A1) Reflexivity.] $X \stackrel {A}{\sim} X$. \item [(A2) Transitivity.] $X \prec Y$ and $Y \prec Z$ imply $X \prec Z$. \item [(A3) Consistency.] $X \prec X'$ and $Y \prec Y'$ imply $(X,Y) \prec (X',Y')$. \item [(A4) Scaling Invariance.] If $X \prec Y$, then $tX \prec tY$ for all $t>0$. \item [(A5) Splitting and recombination.] For $0 < t < 1$, \begin{center}$X\stackrel {A}{\sim} (tX,(1-t)X)$.\end{center} \item [(A6) Stability.] If, for some pair of states, $X$ and $Y$, \begin{center} $(X,\epsilon Z_0) \prec (Y, \epsilon Z_1)$\end{center} \noindent holds for a sequence of $\epsilon$'s tending to zero and some some states $Z_0, Z_1$, then $X \prec Y$. \item [(CH) Comparison hypothesis.] For any two states $X$ and $Y$ in the same state space, either $X \prec Y$ or $Y \prec X$. \end{description} REFERENCES Elliot H. Lieb, Jakob Yngvason, "A guide to entropy and the second law of thermodynamics," Notices of the AMS, May, 1998, pp. 571-581. Elliot H. Lieb, Jakob Yngvason, "The physics and mathematics of the second law of thermodynamics," Physics Reports, Volume 310, Issue 1, March 1999, pp. 1-96. (This has the proofs.) Elliot H. Lieb, Jakob Yngvason, "A Fresh look at entropy and the second law of thermodynamics," Physics Today, April 2000, pp. 32-37. (See also text only preprint at http://www.esi.ac.at/preprints/ESI-Preprints.html .) Respectfully yours, Ellis D. Cooper From rrosebru@mta.ca Sat Nov 17 12:38:35 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 17 Nov 2007 12:38:35 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ItQS2-0002f5-KY for categories-list@mta.ca; Sat, 17 Nov 2007 12:23:22 -0400 From: Thomas Streicher Subject: categories: morphisms between (particular) hyperdoctrines To: categories@mta.ca Date: Sat, 17 Nov 2007 14:35:28 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 46 Jean Benabou has asked whether morphisms between hyperdoctrines have been considered in categorical logic. In work on "parametric polymorphism" (for polymorphic lambda calculus), see Birkedal, Lars; Mogelberg, Rasmus E. Categorical models for Abadi and Plotkin's logic for parametricity. Math. Structures Comput. Sci. 15 (2005), no. 4, 709--772. based on previous work by Robinson and Rosolini (1994) on "Reflexive graphs and parametric polymorphism" (Proc. LICS 1994) reflexive graphs in the category of PL-hyperdoctrines (models for polymorphic lambda calculus) find an essential use. Moreover, the involved PL-hyperdoctrines live over different bases. In B.Jacobs book "Categorical Logic and Type Theory" (Elsevier, 1999) the work of Robinson and Rosolini is briefly addressed and a few hundred pages earlier he discusses morphisms between hyperdoctrines for equational logic (however, without further exploiting this notion). In work on "tripos theory" (late 70ies) Hyland, Johnstone and Pitts have considered geometric morphisms between triposes and shown that via the tripos-to-topos construction give rise to (localic) geometric morphisms between the associated toposes. This has been used over and over again in subsequent work on realizability and related structures (e.g. modified realizability). Currently a student of mine (Jonas Frey) is writing up in his diploma thesis a universal characterisation of the "tripos-to-topos" construction as an (appropriately lax) left adjoint to the functor sending a topos to its associated subobject fibration (which is a tripos). Even in order to formulate this result one has to consider a category of triposes over different bases. (I am confident that his work will be available from my homepage end of this year). As far as I know there is no systematic study of morphism between hyperdoctrines (of the kind as Jean probably has in mind). But now and then particular instances have been considered and put to use for particular purposes. Thomas Streicher From rrosebru@mta.ca Mon Nov 19 11:51:01 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 19 Nov 2007 11:51:01 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iu8iY-0000fD-6J for categories-list@mta.ca; Mon, 19 Nov 2007 11:39:22 -0400 Date: Mon, 19 Nov 2007 08:04:24 +0000 To: categories@mta.ca Subject: categories: CiE08 - 2nd Call for Papers MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: A.Beckmann@swansea.ac.uk (Arnold Beckmann) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 47 [Apologies for multiple copies] ****************************************************************** SECOND CALL FOR PAPERS CiE 2008 http://www.cs.swan.ac.uk/cie08/ Computability in Europe 2008: Logic and Theory of Algorithms University of Athens Athens, June 15-20 2008 PAPER SUBMISSION is now OPEN: http://www.cs.swan.ac.uk/cie08/submission.php This is the fourth in a series of conferences organised by CiE (Computability in Europe), a European network of mathematicians, logicians, computer scientists, philosophers, physicists and others interested in new developments in computability and their underlying significance for the real world. Previous meetings took place in Amsterdam (2005), Swansea (2006) and Siena (2007). CiE 2008 aims at bridging the gap from the logical methods of mathematical and meta-mathematical flavour to the applied and industrial questions that are involved in devising and choosing the right algorithms and analysing their effectiveness and efficiency. IMPORTANT DATES: Submission of papers: January 4, 2008 Notification of authors: February 15, 2008 Final revisions: March 7, 2008 TUTORIALS will be given by: John V. Tucker (Swansea) Moshe Y. Vardi (Houston, TX) PLENARY SPEAKERS will include: Keith Devlin (Stanford, CA) Rosalie Iemhoff (Utrecht) Antonina Kolokolova (Vancouver, BC) Janos Makowsky (Haifa) Dag Normann (Oslo) Prakash Panangaden (Montreal, QC) Christos Papadimitriou (Berkeley, CA) Jan van Leeuwen (Utrecht) & Jiri Wiedermann (Prague) SPECIAL SESSIONS Algorithms in the history of mathematics (organized by J. Hoyrup, Roskilde, and K. Chemla, Paris) Formalising mathematics and extracting algorithms from proofs (organized by H. Barendregt, Nijmegen, and M. Seisenberger, Swansea) Higher type recursion theory and applications (organized by U. Berger, Swansea, and D. Normann, Oslo) Algorithmic game theory (organized by E. Koutsoupias, Athens, and B. von Stengel, London) Quantum algorithms and complexity (organized by V. Kendon, Leeds, and B. Coecke, Oxford) Biology and computation (organized by N. Jonoska, Tampa FL, and G. Mauri, Milano) CiE 2008 conference topics include, but not exclusively * Admissible sets * Analog computation * Artificial intelligence * Automata theory * Classical computability and degree structures * Complexity classes * Computability theoretic aspects of programs * Computable analysis and real computation * Computable structures and models * Computational and proof complexity * Computational learning and complexity * Concurrency and distributed computation * Constructive mathematics * Cryptographic complexity * Decidability of theories * Derandomization * DNA computing * Domain theory and computability * Dynamical systems and computational models * Effective descriptive set theory * Finite model theory * Formal aspects of program analysis * Formal methods * Foundations of computer science * Games * Generalized recursion theory * History of computation * Hybrid systems * Higher type computability * Hypercomputational models * Infinite time Turing machines * Kolmogorov complexity * Lambda and combinatory calculi * L-systems and membrane computation * Mathematical models of emergence * Molecular computation * Neural nets and connectionist models * Philosophy of science and computation * Physics and computability * Probabilistic systems * Process algebra * Programming language semantics * Proof mining * Proof theory and computability * Quantum computing and complexity * Randomness * Reducibilities and relative computation * Relativistic computation * Reverse mathematics * Swarm intelligence * Type systems and type theory * Uncertain reasoning * Weak systems of arithmetic and applications Contributed papers will be selected from submissions received by the PROGRAMME COMMITTEE consisting of: L. Aiello (Roma) T. Altenkirch (Nottingham) K. Ambos-Spies (Heidelberg) G. Ausiello (Roma) A. Beckmann (Swansea, co-chair) L. Beklemishev (Moscow) P. Bonizzoni (Milano) S. A. Cook (Toronto ON) B. Cooper (Leeds) C. Dimitracopoulos (Athens, co-chair) R. Downey (Wellington) E. Koutsoupias (Athens) O. Kupferman (Jerusalem) S. Laplante (Orsay) H. Leitgeb (Bristol) B. Loewe (Amsterdam) E. Mayordomo Camara (Zaragoza) F. Montagna (Siena) M. Mytilinaios (Athens) (+) M. Nielsen (Aarhus) I. Oitavem (Lisboa) C. Palamidessi (Palaiseau) T. Pheidas (Heraklion) Ramanujam (Chennai) A. Schalk (Manchester) U. Schoening (Ulm) H. Schwichtenberg (Muenchen) A. Selman (Buffalo NY) A. Sorbi (Siena) I. Soskov (Sofia) C. Timpson (Leeds) S. Zachos (New York NY) We cordially invite all researchers (European and non-European) in computability related areas to submit their papers (in PDF- format, max 10 pages) for presentation at CiE 2008. We particularly invite papers that build bridges between different parts of the research community. The CONFERENCE PROCEEDINGS will be published by LNCS, Springer-Verlag. There will also be journal special issues, collecting invited contributions related to the conference. So far we have secured special issues in the journals "Theory of Computing Systems" and "Archive for Mathematical Logic". From rrosebru@mta.ca Tue Nov 20 08:59:01 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 20 Nov 2007 08:59:01 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IuSaj-0004o8-29 for categories-list@mta.ca; Tue, 20 Nov 2007 08:52:37 -0400 Date: Tue, 20 Nov 2007 13:10:54 +0100 From: Lutz Schroeder MIME-Version: 1.0 To: categories Subject: categories: Complete Heyting algebras Content-Type: text/plain; charset=ISO-8859-15 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 48 Is the Heyting algebra of global elements of the classifier in an elementary topos always complete? (Of course, the classifier is complete internally; here, I mean externally complete.) I suspect not, but I can't presently think of a natural counterexample. Thanks, Lutz Schr=F6der --=20 ------------------------------------------------------------------ PD Dr. Lutz Schr=F6der office @ Universit=E4t Bremen: Senior Researcher Cartesium 2.051 Safe and Secure Cognitive Systems Enrique-Schmidt-Str. 5 DFKI-Lab Bremen FB3 Mathematik - Informatik Robert-Hooke-Str. 5 Universit=E4t Bremen D-28359 Bremen P.O. Box 330 440 D-28334 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki,de www.dfki.de/sks/staff/lschrode ------------------------------------------------------------------ ------------------------------------------------------------- Deutsches Forschungszentrum f=FCr K=FCnstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Gesch=E4ftsf=FChrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- From rrosebru@mta.ca Tue Nov 20 15:27:23 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 20 Nov 2007 15:27:23 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IuYYV-00004I-TB for categories-list@mta.ca; Tue, 20 Nov 2007 15:14:44 -0400 Date: Tue, 20 Nov 2007 14:57:28 +0000 (GMT) From: "Prof. Peter Johnstone" To: categories Subject: categories: Re: Complete Heyting algebras MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 49 On Tue, 20 Nov 2007, Lutz Schroeder wrote: > Is the Heyting algebra of global elements of the classifier in an > elementary topos always complete? The answer is no: any Boolean algebra, complete or not, can occur as Sub(1) in a topos (see Exercise 9.11 in my old Topos Theory book), and any quotient of a complete Heyting algebra (by a finitary Heyting congruence -- such quotients needn't be complete) can occur, by use of the filterpower construction (cf. my paper with Murray Adelman "Serre classes for toposes", Bull.Austral.Math.Soc. 25 (1982), 103-115). There are examples due to Peter Freyd of Heyting algebras which can't occur as Sub(1) in a topos generated by subobjects of 1. For a long time it was an open problem whether any Heyting algebra can occur as Sub(1) in a topos (without the restriction on generators): Dito Pataraia has recently announced a positive solution to this problem. I have heard a seminar talk about his solution, and seen half of a preprint, but haven't yet managed to understand the other half of his construction. Peter Johnstone From rrosebru@mta.ca Tue Nov 20 15:27:23 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 20 Nov 2007 15:27:23 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IuYcU-0000gE-1K for categories-list@mta.ca; Tue, 20 Nov 2007 15:18:50 -0400 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed To: Categories Content-Transfer-Encoding: quoted-printable From: JeanBenabou Subject: categories: Re: Complete Heyting algebras Date: Tue, 20 Nov 2007 18:08:49 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 50 Of course not ! Elementary toposes are models of a first order theory (this is also =20 true for elementary toposes with an NNO) . Thus there are countable =20 models. In such a model the answer is no. Le 20 nov. 07 =E0 13:10, Lutz Schroeder a =E9crit : > Is the Heyting algebra of global elements of the classifier in an > elementary topos always complete? (Of course, the classifier is =20 > complete > internally; here, I mean externally complete.) I suspect not, but I > can't presently think of a natural counterexample. > > Thanks, > > Lutz Schr=F6der > > > --=20 > ------------------------------------------------------------------ > PD Dr. Lutz Schr=F6der office @ Universit=E4t Bremen: > Senior Researcher Cartesium 2.051 > Safe and Secure Cognitive Systems Enrique-Schmidt-Str. 5 > DFKI-Lab Bremen FB3 Mathematik - Informatik > Robert-Hooke-Str. 5 Universit=E4t Bremen > D-28359 Bremen P.O. Box 330 440 > D-28334 Bremen > phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 > mail: Lutz.Schroeder@dfki,de > www.dfki.de/sks/staff/lschrode > ------------------------------------------------------------------ > > > ------------------------------------------------------------- > Deutsches Forschungszentrum f=FCr K=FCnstliche Intelligenz GmbH > Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern > > Gesch=E4ftsf=FChrung: > Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) > Dr. Walter Olthoff > > Vorsitzender des Aufsichtsrats: > Prof. Dr. h.c. Hans A. Aukes > > Amtsgericht Kaiserslautern, HRB 2313 > ------------------------------------------------------------- > > > > > From rrosebru@mta.ca Wed Nov 21 16:42:56 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Nov 2007 16:42:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IuwBs-0006mx-Kw for categories-list@mta.ca; Wed, 21 Nov 2007 16:28:56 -0400 From: Thomas Streicher Subject: categories: when are Lindenbaum-Tarski algebras complete? To: categories@mta.ca Date: Wed, 21 Nov 2007 15:33:32 +0100 (CET) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 51 I also suspect that Sub(1) of the free topos with nno is not complete. But countability does not suffice for refuting completeness (the ordinal \omega + 1 is an infinite countable cHa which nevertheless is complete). >From Goedel's Theorem for HAH (higher order intuit. arithmetic) it follows that Sub(1) of the free topos with nno is not atomic. But that also doesn't suffice for refuting completeness. On p.169 of Freyd, Friedman and Scedrov's paper "Lindenbaum algebras of intuitionistic theories and free categories" (APAL 35) they claim "Lindenbaum algebras are almost never complete" but don't give a proof. Thomas Streicher From rrosebru@mta.ca Wed Nov 21 19:39:35 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Nov 2007 19:39:35 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iuz4v-0005wI-Jw for categories-list@mta.ca; Wed, 21 Nov 2007 19:33:57 -0400 From: Dana Scott To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Subject: categories: Re: when are Lindenbaum-Tarski algebras complete? Mime-Version: 1.0 (Apple Message framework v915) Date: Wed, 21 Nov 2007 13:30:37 -0800 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 52 On Nov 21, 2007, at 6:33 AM, Thomas Streicher wrote: > I also suspect that Sub(1) of the free topos with nno is not complete. > But countability does not suffice for refuting completeness (the > ordinal > \omega + 1 is an infinite countable cHa which nevertheless is > complete). >> > From Goedel's Theorem for HAH (higher order intuit. arithmetic) it > follows > that Sub(1) of the free topos with nno is not atomic. But that also > doesn't > suffice for refuting completeness. > > On p.169 of Freyd, Friedman and Scedrov's paper > "Lindenbaum algebras of intuitionistic theories and free > categories" (APAL 35) > they claim "Lindenbaum algebras are almost never complete" but don't > give > a proof. Ah, but if a Heyting algebra is complete, then so is the Boolean algebra of all not-not-stable elements. Familiar example: the regular open subsets of a topological space form a complete Boolean algebra. As remarked, the Sub(1) of the free topos with nno is not atomic, and with reference again to Godel's theorem via the not-not translation, the Boolean algebra of not-not-stable elements is also non atomic. But all countable, non-atomic Boolean algebras are isomorphic to the clopen subsets of the Cantor space (or the Lindenbaum algebra of classical propositional calculus, or the free Boolean algebra on countably many generators). That algebra is not complete -- as can be seen in many ways. Q.E.D. From rrosebru@mta.ca Wed Nov 21 19:39:35 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Nov 2007 19:39:35 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iuz5g-00060Y-LA for categories-list@mta.ca; Wed, 21 Nov 2007 19:34:44 -0400 Date: Wed, 21 Nov 2007 16:52:28 -0500 (EST) From: Myles TIERNEY To: categories@mta.ca Subject: categories: address change MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 53 Just want to announce that my UQAM email address has changed. >From tierney@math.uqam.ca to tierney.myles@uqam.ca. The Rutgers address remains the same: tierney@math.rutgers.edu. Myles Tierney From rrosebru@mta.ca Fri Nov 23 16:51:45 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 23 Nov 2007 16:51:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IvfJN-00061E-03 for categories-list@mta.ca; Fri, 23 Nov 2007 16:39:41 -0400 Subject: categories: SCORE: Student COntest in SoftwaRe Engineering (ICSE 2009) To: events@fmeurope.org MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 From: events-admin@fmeurope.org Date: Fri, 23 Nov 2007 17:37:11 +0100 (CET) Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 55 (Apologies if you receive multiple copies) PRELIMINARY CALL FOR PARTICIPATION Student COntest in softwaRe Engineering SCORE 2009 http://score.elet.polimi.it an initiative of the 31st International Conference in Software Engineering (ICSE 2009) 16-24 May 2009, Vancouver, Canada ICSE 2009 in Vancouver, Canada will see the first finals of the SCORE Software Engineering Contest. Teams from all over the world will enter in a competition that is open to students from undergraduate to master=92= s level. Each team will develop a system chosen from those proposed by SCORE committee members. Teams will produce a report as their first deliverable, followed by a prototype system for those chosen as semi-finalists. Evaluation will be based on the quality of the software engineering process followed, as well as the outcome. In order to accommodate all academic calendars, the 2009 SCORE Contest will run from December 2007 to January 2009 with with team sign-up starting in February 2008 and ending in November 2008. Up to 10 finalist teams will be announced in April 2009. At least one representative from each of the finalist teams will have expenses paid to attend ICSE in Vancouver, 16-24 May 2009, for final judging. The SCORE Contest is aimed at promoting and fostering software engineering in universities worldwide. The committee consists of members of industry and academia. Each participating team will benefit from the experience of working on member selected projects, and from being part of an exciting new venture. Program Chairs: Mehdi Jazayeri, University of Lugano, Switzerland Dino Mandrioli, Politecnico di Milano, Italy Upcoming Important Date: 15 December 2007: publication of the project topics on the SCORE website For all information, including a full Call for Participation, see the website: http://score.elet.polimi.it _______________________________________________ events mailing list events@fmeurope.org http://www.fmeurope.org/mailman/listinfo/events From rrosebru@mta.ca Mon Nov 26 09:53:10 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Nov 2007 09:53:10 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IweBS-0005Al-7R for categories-list@mta.ca; Mon, 26 Nov 2007 09:39:34 -0400 From: Dana Scott To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v915) Subject: categories: countable Heyting algebras Date: Sun, 25 Nov 2007 22:03:49 -0800 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 56 Thomas Streicher recently noted: > omega + 1 is an infinite countable Ha which nevertheless > is complete True. And the dual algebra {0} + omega* with the descending copy integers is also a complete Ha (cHa). But this second example, which is non-atomic, is not fully non-atomic; that is, there are minimal gaps in the ordering. Take Q = rationals in [0,1]. This is a countable, non-complete Ha, where between any two distinct elements there is a third. Note that all these Ha's are subalgebras of the unit interval [0,1] of all reals -- which is uncountable, of course. In linearly ordered Ha's, the implication b --> c == c if b > c, and is 1 otherwise. It follows that the not-not stable elements are just 0 and 1. As I remarked once before, it is an old theorem that all countable, non-atomic Boolean algebras are isomorphic. Call this Boolean algebra F for free, for short. This isomorphism theorem is far from true for Ha's, however. Take the algebra Q x Q. This is a fully non-atomic, but it has FOUR not-not stable elements = {0,1} x {0,1}. So this is not the algebra Q. Take any other finite power, Q x Q x ... x Q. They are all different by reference to the not-not stable elements. The Ba I called F has a recursive structure (as one sees from what we know about propositional calculus) in that all the operations are recursive in terms of a simple enumeration of the elements of F. The question I want to consider is what countable Ha extend F in such a way F equals the not-not stable elements. Now, F can also be considered as the clopens of the Cantor space 2^N. The cHa of all opens is fully non-atomic, but it is uncountable. Call it C for Cantor. The not-not stable elements of C are the so- called regular open sets. They form an uncountable cBa which is the completion of F. But we want to ask if there are there interesting countable subalgebras of C? We note that elements of C are determined by the elements of F they contain. (In fact, the cHa C is isomorphic to the lattice of ideals of F.) Here is one construction. Let A be the arithmetically definable elements of C. That is, take the elements of x of C where the set {a in F | a =< x } is (first-order) arithmetically definable (in terms of the enumeration of F). I claim that A is a Ha. Of course it contains F; and, if it is a subalgebra of C, then the not-not stable elements give us an extension of F. But it is countable and the extension of F, call it G, is non-atomic; hence, the stable elements of A form a Ba G isomorphic to F. To check the subalgebra part we note for x,y in C and a in F: a =< x /\ y <==> a =< x & a =< y; a =< x \/ y <==> (for some b,c in F)[ a = b \/ c & b =< x & c =< y ]. Hence, if x,y are in A, then so are x /\ y and x \/ y. At first I had thought the recursively enumerable opens would be enough, but there is trouble with negation. We have for x in C and a in F, and with -x notating negation in C: a =< -x <==> (for all b in F)[ b =< x ==> a /\ b = 0 ]. The universal quantifier seems needed, and so {a in F | a =< -x } does not seem to be r.e. in general. But, if x is in A, then so is -x. We need to generalize this remark to implication in C, so for a in F and x,y in C: a =< x --> y <==> (for all b in F)[ b =< x ==> a /\ b =< y ]. Hence, if x,y are in A, then so is x --> y. And A is a sub-Ha of C. Note that in A, if an element x is not 0, than the interval [0,x] in A is in itself a non-Boolean cHa (a homomorphic image of A). Consider next the algebra A x F. This algebra is a countable, fully non-atomic Ha. The not-not stable elements of A come out to be G x F, which is isomorphic to F (and G). But, the element x = (0,1) has the interval [0,x] in A x F isomorphic to F. And that is Boolean. So, A and A x F are NOT isomorphic. Note, however, that A x F and (A x F) x F ARE isomorphic. Also, as I forgot to mention, A and A x A are isomorphic. Anyway, we have at least two non-isomorphic, non-Boolean Ha's with an isomorph of F as the not-not stables. And they are fully non-atomic. Are there other, non-isomorphic extensions of F? At this moment I do not see a knock-down argument. One idea, is to call the above algebra A_1 (for first-order definable opens) and consider the algebras A_n of n-th order definable opens. I think we get the same conclusions about A_n and A_n x F not being isomorphic, BUT is every A_n somehow isomorphic to A_1 (with a highly non-recursive automorphism of F)? Is that possible, or have I missed an obvious point? If the A_n are all not isomorphic, then we can probably get an uncountable collection of mutually non-isomorphic extensions of F. And, how would these relate to Sub(1) Ha's of countable topoi? Or are these considerations a red herring? From rrosebru@mta.ca Mon Nov 26 13:51:56 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Nov 2007 13:51:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iwi34-0005yo-Nf for categories-list@mta.ca; Mon, 26 Nov 2007 13:47:10 -0400 Date: Mon, 26 Nov 2007 15:58:10 +0100 From: Joachim Kock Subject: categories: HOCAT 2008 To: categories@mta.ca, algtop-l@lists.lehigh.edu MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 57 This is the second announcement of the conference HOCAT 2008 Homotopy Structures in Geometry and Algebra;=20 Derived Categories, Higher Categories June 30 to July 5, 2008 Centre de Recerca Matem=E0tica Bellaterra (Barcelona) an event within the CRM thematic year on Homotopy Theory and Higher Categories http://www.crm.cat/hocat/ The following have agreed to speak at the conference: John Baez (University of California at Riverside) Paul Balmer (University of California at Los Angeles) David Benson (University of Aberdeen) Julia Bergner (Kansas State University) Tom Bridgeland (University of Sheffield) S=F8ren Galatius (Stanford University) Ezra Getzler (Northwestern University) Mikhail Kapranov (Yale University) (to be confirmed) Ralf Meyer (Georg-August Universit=E4t G=F6ttingen) (to be confirmed) Charles Rezk (University of Illinois at Urbana) Bertrand To=EBn (Universit=E9 Paul Sabatier, Toulouse) Michel Van den Bergh (Hasselt University) A limited number of slots are available for contributed talks. Prospective speakers should submit an abstract to any of the=20 organisers before March 31 (and will be notified before April 15). For registration (deadline May 30) and further information=20 about the conference, see http://www.crm.cat/HOCAT2008/ We look forward to seeing you in Barcelona. The organisers, Carles Casacuberta Andr=E9 Joyal Joachim Kock Amnon Neeman Frank Neumann From rrosebru@mta.ca Mon Nov 26 17:11:26 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Nov 2007 17:11:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iwl99-0006xa-3e for categories-list@mta.ca; Mon, 26 Nov 2007 17:05:52 -0400 From: Dana Scott To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v915) Subject: categories: Re: countable Heyting algebras Date: Mon, 26 Nov 2007 10:09:27 -0800 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 58 Woke up in the middle of the night and realized I said something obviously wrong. (Not the first time, and won't be the last, I'm afraid.) On Nov 25, 2007, at 10:03 PM, Dana Scott wrote: > Now, F can also be considered as the clopens of the Cantor space > 2^N. The cHa of all opens is fully non-atomic, but it is uncountable. > Call it C for Cantor. The not-not stable elements of C are the so- > called regular open sets. They form an uncountable cBa which is the > completion of F. But we want to ask if there are there interesting > countable subalgebras of C? We note that elements of C are determined > by the elements of F they contain. (In fact, the cHa C is isomorphic > to the lattice of ideals of F.) Well, it is true that the stable elements of C form the completion of F. And, every element of C is a sup of elements of F, so C is atomless in the sense of having no minimal non-zero elements. And, the stables of C form an atomless cBa. BUT -- and here is my oversight -- C does have gaps, and so the cHa is NOT fully non-atomic. Think of the Cantor set as a subspace T of the unit interval. There is a blank from 1/3 to 2/3, if we make the construction via the middle-third process. This means that [0,1/3) meet T is open in T, but [0,1/3] meet T = [0,2/3) meet T is both open and closed. This gives a gap between two opens in the cHa C. So C is not fully gapless. This gap also exists in the subalgebra A of arithmetically definable opens. Aarrgghh. Is there a fix? Can we take a quotient in the category of Ha's that closes the gaps? Maybe, and maybe not. In countable Ba's, dividing by the ideal generated by the atoms can result in a quotient algebra that still has atoms. Aarrgghh! I will have to think further. Rats! From rrosebru@mta.ca Thu Nov 29 12:19:21 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Nov 2007 12:19:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IxltO-0001Ko-Bm for categories-list@mta.ca; Thu, 29 Nov 2007 12:05:34 -0400 From: Dana Scott To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v915) Subject: categories: Re: countable Heyting algebras (third try) Date: Wed, 28 Nov 2007 21:01:14 -0800 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 59 On Nov 26, 2007, at 10:09 AM, Dana Scott wrote, recall in an earlier posting: > Now, F can also be considered as the clopens of the Cantor space > 2^N. The cHa of all opens is fully non-atomic, but it is uncountable. > Call it C for Cantor. The not-not stable elements of C are the so- > called regular open sets. They form an uncountable cBa which is the > completion of F. But we want to ask if there are there interesting > countable subalgebras of C? We note that elements of C are determined > by the elements of F they contain. (In fact, the cHa C is isomorphic > to the lattice of ideals of F.) The phrase "fully non-atomic" was not well chosen. Let us use "atomless" to mean "no minimal non-zero elements", and let us say a Ha is "gapless" if we cannot have elements a < b with [a, b] = {a, b} (i.e. nothing strictly between). An atomless Ba is always gapless. Now the cHa C above, as I pointed out in the previous message, DOES have gaps, even though it is atomless. (The Ba F is atomless, and it generates C by taking unions of clopen sets to make opens. Thus, no open set could be an atom in C.) Take any point t in the Cantor set 2^N. Let b = 2^N and let a = 2^N \ {t}. Clearly, a is a dense open set and [a, b] is a gap. By removing one pont at a time, we can have a whole sequence of dense open sets a_0 < a_1 < ... < a_n with each [a_i, a_(i+1)] being a gap. Note that negation in C gives --a_i = b, since in topological lattices double negation is interior-of-closure. In general, in any Ha which F generates, if a < b and [a, b] is a gap, then b =< --a. Because if not, then b /\ -a is non-zero. By the generation, there must be a non-zero e in F with e =< b /\ -a. Thus, e /\ a = 0. Because F is atomless, we can write e = f \/ g, with two disjoint, non-zero elements of F. But then c = a \/ f is an element strictly between a and b. This comment shows that gaps, if they exist are somewhat limited. But, C has many gaps, and in general an interval [a, --a] might be quite large. Remember, assuming a = --a for all a makes the Ha Boolean. So, I have not really made much progress in answering how F might generate a countable, non-boolean Ha. I am guessing there are many non-isomorphic ways this can happen. From rrosebru@mta.ca Thu Nov 29 21:39:57 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Nov 2007 21:39:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IxulN-00024s-3a for categories-list@mta.ca; Thu, 29 Nov 2007 21:33:53 -0400 Date: Thu, 29 Nov 2007 18:51:32 +0100 Mime-Version: 1.0 (Apple Message framework v553) Subject: categories: positions in Louvain-la-Neuve From: Enrico Vitale To: categories@mta.ca Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 60 Dear Colleagues, Two full-time academic positions at the Department of Mathematics in Louvain-la-Neuve are available. One position is in algebra and the other one in analysis. The deadline for applications is January 18, 2008. For further information please see at http://www.uclouvain.be/97693.html (algebra) http://www.uclouvain.be/38444.html (analysis) or contact Michel Willem, dean of the department, at michel.willem@uclouvain.be Best regards, Enrico Vitale