From MAILER-DAEMON Sat Nov 24 12:24:33 2007 Date: 24 Nov 2007 12:24:33 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1195921473@mta.ca> X-IMAP: 1188823147 0000000066 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From rrosebru@mta.ca Mon Sep 3 09:38:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Sep 2007 09:38:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISB3G-00015h-Ik for categories-list@mta.ca; Mon, 03 Sep 2007 09:29:10 -0300 Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable From: Francois Lamarche Subject: categories: PSSL86: Last announcement Date: Mon, 3 Sep 2007 14:22:37 +0200 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 1 We remind you that The 86th edition of the Peripatetic Seminar on Sheaves and Logic will =20= be held at the Institut =C9lie Cartan (IECN) on the Universit=E9 Henri = =20 Poincar=E9 campus in Nancy, France, on the weekend of September 8-9 = 2007. http://www.loria.fr/~lamarche/psslHomeEN.html The preliminary schedule is now available at http://www.loria.fr/~lamarche/schedule.html In the PSSL tradition, people arriving not too late Friday are =20 welcome at Le Petit Cuny, 97 Grande Rue, where some of us will hang =20= around starting at 7pm, to have drinks and sample the hearty local =20 (well, Alsatian) cuisine, in particular Flammekuche, our very own =20 answer to pizza. There will also be a banquet on Saturday evening for =20= which those who want to attend will be asked a moderate sum. Getting to the IECN Take the tramway in the "CHU Brabois" direction and get off at the =20 "Callot" stop. If you are not directly next to it, you should find =20 Jacques Callot street behind the McDonald's. It leads to the entrance =20= of the campus. There are two glass-and-steel structures in a similar =20 style, and the IECN is the one that will be to your right (the other, =20= to your left, is the Loria , the computer science lab). Notice that =20 the entrance to the IECN itself is on the side of the building, =20 inside a small glass-enclosed lobby. Details to keep in mind * The IECN, where PSSL will be held, is normally closed on =20 Saturdays. The head of the IECN has allowed the doors to be =20 automatically unlocked for the following periods: 8-10 and 12-14 on =20 Saturday, and 8-10 on Sunday. If you want to enter the premises =20 outside these hours, please make sure someone is inside to open the =20 door for you. * Saturday lunch will be on the premises. You will be given more =20= information on the Saturday banquet Saturday. * You are reminded that there is only one tram every 20 minutes =20 on Sundays. Please look up the schedule before you hit the road =20 Sunday morning. From rrosebru@mta.ca Mon Sep 3 20:56:53 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 03 Sep 2007 20:56:53 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISLhu-00076a-HP for categories-list@mta.ca; Mon, 03 Sep 2007 20:51:50 -0300 From: "George Janelidze" To: Subject: categories: Max Kelly Conference in Cape Town Date: Tue, 4 Sep 2007 00:37:53 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 2 MAX KELLY CONFERENCE IN CAPE TOWN FIRST ANNOUCEMENT This International Conference in Category Theory is dedicated to the memo= ry of Gregory Maxwell Kelly, a great mathematician, a great category-theoris= t, and the Founder of the Australian School of Category Theory. Its main week is from Sunday January 20 to Sunday January 27, 2008, but t= he participants are welcome to stay also for the next week with a lot of mathematics and some tourism expected. Since the time is short, the organizers urgently need to estimate the num= ber of participants. Please inform us by writing to me ( George.Janelidze@uct.ac.za ) before 15th of September 2007 if you serious= ly intend to participate and please include estimated dates of your arrival = and departure. ORGANIZING COMMITTEE is large and involves professors and graduate studen= ts from three South African universities: University of Cape Town (UCT), University of Stellenbosch (US), and University of KwaZulu Natal (UKZN). = In particular this includes our senior colleagues Keith Hardie and Guillaume Br=FCmmer. SCIENTIFIC COMMITTEE: Martin Hyland George Janelidze Michael Johnson Peter Johnstone Stephen Lack Ross Street Walter Tholen Richard Wood LOCATION/POSTAL ADDRESS: Department of Mathematics and Applied Mathematics University of Cape Town Rondebosch 7701 Cape Town South Africa George Janelidze George.Janelidze@uct.ac.za From rrosebru@mta.ca Tue Sep 4 21:45:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Sep 2007 21:45:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISis4-0004mc-Gf for categories-list@mta.ca; Tue, 04 Sep 2007 21:35:52 -0300 Date: Mon, 3 Sep 2007 18:13:45 +0200 (CEST) From: Peter Schuster To: Categories Subject: categories: CfP: 3WFTop proceedings 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: 3 ------------------------------------------------------------------------- Call for Papers: Third Workshop on Formal Topology ------------------------------------------------------------------------- Special Issue of Annals of Pure and Applied Logic ------------------------------------------------------------------------- The Third Workshop on Formal Topology was held in Padua in May 2007: www.3wftop.math.unipd.it The proceedings of this workshop will be published as a special issue of the Annals of Pure and Applied Logic, with the following guest editors: Andrej Bauer, Thierry Coquand, Giovanni Sambin, Peter Schuster. These proceedings are open for high-level research papers on topics from or closely related to formal topology: that is, from constructive and/or point-free topology including applications. ------------------------------------------------------------------------- Deadline for submissions: Sunday, 13 January 2008 ------------------------------------------------------------------------- Submissions by email to: Andrej.Bauer@andrej.com ------------------------------------------------------------------------- From rrosebru@mta.ca Tue Sep 4 21:45:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Sep 2007 21:45:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISiv3-00058Q-So for categories-list@mta.ca; Tue, 04 Sep 2007 21:38:58 -0300 Mime-Version: 1.0 (Apple Message framework v752.3) To: categories@mta.ca From: "Michael A. Warren" Subject: categories: Paper available Date: Tue, 4 Sep 2007 13:51:59 -0400 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: O X-Status: X-Keywords: X-UID: 4 Dear categorists, The following paper, in which perhaps some of you might have an interest, is now available: "Homotopy theoretic models of identity types" by Steve Awodey and Michael A. Warren Abstract: "This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory." The paper may be found on the arXiv as 0709.0248v1 (math.LO): Best regards, Michael Warren From rrosebru@mta.ca Tue Sep 4 21:45:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Sep 2007 21:45:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISisZ-0004tB-Vl for categories-list@mta.ca; Tue, 04 Sep 2007 21:36:24 -0300 Date: Tue, 04 Sep 2007 16:52:37 +0100 From: Maria Manuel Clementino MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Annoucement of a position in Computer Science Content-Type: text/plain; charset=ISO-8859-1; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 5 ------------------------------------------- Please distribute this announcement ------------------------------------------- The Computer Science Department of the Faculty of Science at the University of Porto is seeking candidates for faculty positions at assistant professor level or higher. Candidates must hold a PhD degree in Computer Science, or equivalent. We expect selected candidates to learn the Portuguese language in order to be able to teach undergraduate courses in Portuguese, within a year. The salary for an assistant professor is around 45,000 euros per year, before taxes. Interested applicants must send their full CV together with a short research and teaching statement (one page each) to the address below. The deadline for applications is the 30th of September 2007. The Department is responsible (or co-responsible) for the following degre= es: -- BSc in Computer Science (3 years) -- MSc in Networks and Systems Engineering (5 years; BSc+MSc) -- MSc in Computer Science (2 years) -- MSc in Medical Informatics (2 years, jointly with Medicine Fac.) -- Doctoral Programme in Computer Science, jointly with the Universities of Minho, Aveiro and Porto (MAP) -- Doctoral Programme in Telecommunications, jointly with the Universities of Minho, Aveiro and Porto (MAP) The department's faculty carries out their research in the Laboratory for Artificial Intelligence and Computer Science (LIACC), the Telecommunications Institute (IT) and INESC-Porto. Department's web page: http://www.dcc.fc.up.pt/ Contact person: V=EDtor Santos Costa or Ricardo Rocha (dcc.rec @ dcc.fc.u= p.pt) Postal address: Departamento de Ci=EAncia de Computadores, Universidade do Porto, Rua Campo Alegre, 1021 4169-007 Porto Portugal Tel: +351+220402958(9) (secretariat) From rrosebru@mta.ca Tue Sep 4 21:45:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 04 Sep 2007 21:45:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ISiuA-00053K-HH for categories-list@mta.ca; Tue, 04 Sep 2007 21:38:02 -0300 Date: Tue, 4 Sep 2007 12:30:00 -0400 (EDT) From: Jeff Egger Subject: categories: Re: Teaching Category Theory 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: 6 --- Michael Shulman wrote: > My guess would be that it's because for non-category theorists, many > (perhaps most) categories which arise in practice are enriched (over > something more exotic than Set), while few are internal (to something > more exotic than Set).=20 I'm not sure I agree with that: internal groupoids, at the very least, show up in a variety of situations which non-category theorists can be, and are, interested in. Perhaps one of the reasons why some people try to deal with groupoids as if they weren't a special case of categories is because they never thought of categories in any other way than as a=20 mass of hom-sets. > Even when working over Set, I think it's fair > to say that the vast majority of categories arising in mathematical > practice are locally small. Now I do think there is a good reason for this, which is the fact=20 that in functorial semantics (by which I don't just mean the original,=20 universal-algebraic, case), the domain category is typically small. Raising to a small power does not destroy local smallness. > Since in general, neither enriched nor internal category theory is a > special case of the other, it doesn't seem justified to me to consider > either one as "more primitive".=20 I agree with this entirely, of course. It follows that, in a first=20 course on category theory, one should present both styles of definition=20 as soon as possible. This, in turn, suggests (but does not prove) that=20 one should not sweep size distinctions under the carpet. =20 > Actually, currently my favorite level of generality is something I > call a "monoidal fibration". Roughly, the idea is that you have two > different "base" categories, S and V, such that the object-of-objects > comes from S while the object-of-morphisms comes from V. When S and V > are the same, you get internal categories, and when S=3DSet, you get > classical enriched categories. This could be regarded as "explaining" > the coincidence of internal and enriched categories for V=3DSet. I > wrote a bit about this at the end of "Framed Bicategories and Monoidal > Fibrations" (arXiv:0706.1286), but I intend to say more in a > forthcoming paper. I look forward to it! Cheers, Jeff. From rrosebru@mta.ca Wed Sep 5 20:25:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Sep 2007 20:25:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IT4Ay-0003xR-B1 for categories-list@mta.ca; Wed, 05 Sep 2007 20:20:48 -0300 Date: Wed, 05 Sep 2007 15:09:12 +0200 From: Michal Przybylek Subject: categories: Re: Teaching Category Theory To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 7 "Jeff Egger" wrote: > --- Michael Shulman wrote: [I don't see the original post, so I'm responding here] > Actually, currently my favorite level of generality is something I > call a "monoidal fibration". Roughly, the idea is that you have two > different "base" categories, S and V, such that the object-of-objects > comes from S while the object-of-morphisms comes from V. When S and V > are the same, you get internal categories, and when S=Set, you get > classical enriched categories. This could be regarded as "explaining" > the coincidence of internal and enriched categories for V=Set. Heh... I'm studying the same problem as a part of my Master Thesis (under supervision of prof. Andrzej Tarlecki), but fortunately :-) in a bit different framework. The chief concept of my work is a definition of a category ("elementary category") in a fibred monoidal category (i.e. each fibre is monoidal and reindexing functors preserve the monoidal structure) over a base category with binary products. Than, roughly speaking, for a category C with finite limits, C-enriched categories are just "Fam : Fam(C) -> Set"-elementary categories, and C-internal categories are just "Cod : C^{->} -> C"-elementary categories. It turns out (if I didn't make mistakes :-)), that when C has Set-indexed coproducts, than there is an adjunction between the global section functor C(1, -) : Cod -> Fam and the "coproduct functor" \coprod_{-}(1) : Fam -> Cod. Furthermore, if the coproducts are universal, than these functors are fibred and preserves the monoidal structures, and if additionally all global sections in C are disjoint (i.e. the pullback of two different global section is an initial object) than this adjunction is an equivalence of categories (these results give us approximations C-internal categories ---> C-enriched categories and in the other direction). Best regards, Michal R. Przybylek From rrosebru@mta.ca Wed Sep 5 20:25:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Sep 2007 20:25:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IT4CJ-00046j-Dx for categories-list@mta.ca; Wed, 05 Sep 2007 20:22:11 -0300 Date: Thu, 06 Sep 2007 00:36:21 +0200 From: Michal Przybylek Subject: categories: Re: Teaching Category Theory To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=ISO-8859-1 Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 8 "Jeff Egger" wrote: > --- Michael Shulman wrote: > My guess would be that it's because for non-category theorists, many > (perhaps most) categories which arise in practice are enriched (over > something more exotic than Set), while few are internal (to something > more exotic than Set). I think it's not that. It's just because the concept of internal category is in some sense subsumed by the concept of fibration. And in practical situations we prefer to work with fibrations rather than internal categories. > Actually, currently my favorite level of generality is something I > call a "monoidal fibration". Roughly, the idea is that you have two > different "base" categories, S and V, such that the object-of-objects > comes from S while the object-of-morphisms comes from V. When S and V > are the same, you get internal categories, and when S=Set, you get > classical enriched categories. What do you mean by "you get internal/enriched categories" ? Do you have a 2-equivalence between the 2-category of all S-internal (resp. enriched) categories (S-internal functors, S-internal natural transformations) and the 2-category of your categories ? I'm asking because I have encountered some difficulties here (i.e. in my framework some diagrams are not willing to commute "on the nose"). Best regards, Michal R. Przybylek From rrosebru@mta.ca Wed Sep 5 20:25:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Sep 2007 20:25:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IT4AL-0003sk-Ak for categories-list@mta.ca; Wed, 05 Sep 2007 20:20:09 -0300 Date: Wed, 5 Sep 2007 22:00:48 +0100 (BST) Subject: categories: JKT statement From: tporter@informatics.bangor.ac.uk To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 9 STATEMENT OF THE EDITORS OF THE "JOURNAL OF K-THEORY" After several public statements and news articles regarding the Springer journal "K-Theory" (KT), and the new "Journal of K-Theory" (JKT) to be distributed by Cambridge University Press (CUP), the mathematical community has become aware of ongoing changes. On behalf of the entire Editorial Board of the new JKT, we want to give as precise a picture of the situation as we can at the moment, especially to the authors. It is very important to us that the authors should not suffer as a result of the transition. Those authors who submitted papers to KT before August 2007, regardless of whether the paper has already been accepted or is just awaiting review, have three choices: 1) Choose another journal. 2) Maintain submission with KT for final review if necessary and publication if accepted. 3) Transfer their article to the new JKT. All authors who have not yet done so should please notify Professor Bak on the one hand, Professors Lueck and Ranicki on the other hand, about their choice, as soon as possible. For those who opt for choice #2, Professors Lueck and Ranicki have promised to take over the remaining editorial duties. We can guarantee that the authors who choose option (3) will have a smooth transition, with their articles progressing as if there has been no change. We will also do everything we can to help those who choose options (1) and (2). In particular, if the authors instruct us, we will be happy to forward to the journals of their choice the full information regarding the status of their articles. In 2004, because of growing dissatisfaction with Springer, the editorial board of KT authorized Prof. Anthony Bak, the Editor in Chief, to begin negotiations with other publishers. The editorial board was unhappy with the poor quality of the work done by Springer, for example the huge number of misprints in the published version of the articles, the long delay in publication and the high prices Springer was charging. The negotiations came to a conclusion in 2007. A new journal, entitled "Journal of K-theory" (JKT) will commence publication in late 2007. It will be printed by Cambridge University Press. Papers will appear earlier online, as 'forthcoming articles'. The title of JKT is currently owned by a private company. This situation is only meant as a temporary solution to restart publication of K-theory articles as soon as possible. It is the Board's intention to create a non-profit academic foundation and to transfer ownership of JKT to this foundation, as soon as possible, but no later than by the end of 2009, a delay justified by many practical considerations. This shift towards more academic control of journals is not new. We follow here a path opened by Compositio Mathematica, Commentarii Mathematici Helvetici, and others (see for instance the interesting paper of Gerard van der Geer which appeared in the Notices of the AMS in May 2004). We believe that such changes can help keep prices low. We trust in Prof. Bak's leadership for the launching of JKT and forming, together with the editorial board, the foundation to house the Journal. The statutes of the foundation will provide democratic rules governing the future course and development of the journal, including the election of the managing team. We hope to have provided a fair picture of the current situation, and we plan to issue another public statement when new developments come up. In case of further questions, please contact any of the signatories. Let us conclude from a broader perspective: The editorial board is committed to secure the journal's quality and long-term sustainability. Signatures A. Bak P. Balmer S. Bloch G. Carlsson A. Connes E. Friedlander M. Hopkins B. Kahn M. Karoubi G. Kasparov A. Merkurjev A. Neeman T. Porter J. Rosenberg A. Suslin G. Tang B. Totaro V. Voevodsky C. Weibel Guoliang Yu From rrosebru@mta.ca Sun Sep 9 09:59:03 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Sep 2007 09:59:03 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IUMGj-00068X-ON for categories-list@mta.ca; Sun, 09 Sep 2007 09:52:05 -0300 From: "Ronnie Brown" To: Subject: categories: Re: Teaching Category Theory Date: Sun, 9 Sep 2007 12:40:38 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 10 One would like to leave students with a very positive attitude. The following quotation , from the Stanford Encyclopedia of Philosophy, might help: " Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. ......." http://plato.stanford.edu/entries/category-theory/ I am writing as someone who has come into category theory from algebraic topology, and been struck by the utility of the language and results for what I needed over the years (and by the welcome). A separate matter, but intriguingly related, is categories and groupoids as sources of useful algebraic structures. This seems connected with the notion of partial operations, and so my notion of `higher dimensional group theory' and `higher dimensional algebra' is that of studying algebraic structures with partial operations defined under geometric conditions. This concurs with the vision in Higgins, Philip J. Algebras with a scheme of operators. Math. Nachr. 27 1963 115--132. In this view, the objects of a category play a key role. This has covered the developments I had in mind for modelling some underlying structures in homotopy theory, and which were developed with Philip Higgins, and later with Loday. Relevant was Philip's reporting of the view of Philip Hall that one should study the algebra that arises naturally from the geometry without trying to force the algebra into a preconceived mould. In the late 1960s, when Bill Cockcroft and I received notes from Saunders of lectures on category theory for our comments, Bill and I replied that what we really wanted was `Categories for the working mathematician'. I still hold to that. To me this means general theory with specific examples which show how the general theory makes life easier, even controls the calculations. Eilenberg insisted a construction should be defined, and its properties developed, in terms of the universal property, which should also explain existence. So when dealing with structures at various levels it is very useful to know left adjoints commute with colimits, right adjoints commute with limits, and this can tell one how to compute colimits and limits. This also leads to induced constructions (change of base). I have recently found uses (to me!) of fibrations of categories: the inclusion of a fibre preserves connected colimits. Simple examples of the use of this are: Ob: Groupoids \to Sets, forget: (groupoid modules) \to groupoids; forget: (2-Cat) \to Cat and compositions of these. Of course it was generalisations of the van Kampen theorem to higher dimensions, and the (previously rare) use in homotopy theory of colimits of algebraic structures, that made it useful to do such computations. I have only recently really understood the notion of dense subcategory, and its use for representing an object as a coend. What I have not done is use the theory of monads. Is this ignorance on my part? I am happy to be enlightened! One of the points of a course for the students might be `need to know'. Hence the need for explicit and varied examples. How to balance this with theory? Ronnie www.bangor.ac.uk/r.brown ----- Original Message ----- From: "Jeff Egger" To: Sent: Tuesday, September 04, 2007 5:30 PM Subject: categories: Re: Teaching Category Theory --- Michael Shulman wrote: > My guess would be that it's because for non-category theorists, many > (perhaps most) categories which arise in practice are enriched (over > something more exotic than Set), while few are internal (to something > more exotic than Set). I'm not sure I agree with that: internal groupoids, at the very least, show up in a variety of situations which non-category theorists can be, and are, interested in. Perhaps one of the reasons why some people try to deal with groupoids as if they weren't a special case of categories is because they never thought of categories in any other way than as a mass of hom-sets. > Even when working over Set, I think it's fair > to say that the vast majority of categories arising in mathematical > practice are locally small. Now I do think there is a good reason for this, which is the fact that in functorial semantics (by which I don't just mean the original, universal-algebraic, case), the domain category is typically small. Raising to a small power does not destroy local smallness. > Since in general, neither enriched nor internal category theory is a > special case of the other, it doesn't seem justified to me to consider > either one as "more primitive". I agree with this entirely, of course. It follows that, in a first course on category theory, one should present both styles of definition as soon as possible. This, in turn, suggests (but does not prove) that one should not sweep size distinctions under the carpet. > Actually, currently my favorite level of generality is something I > call a "monoidal fibration". Roughly, the idea is that you have two > different "base" categories, S and V, such that the object-of-objects > comes from S while the object-of-morphisms comes from V. When S and V > are the same, you get internal categories, and when S=Set, you get > classical enriched categories. This could be regarded as "explaining" > the coincidence of internal and enriched categories for V=Set. I > wrote a bit about this at the end of "Framed Bicategories and Monoidal > Fibrations" (arXiv:0706.1286), but I intend to say more in a > forthcoming paper. I look forward to it! Cheers, Jeff. -- No virus found in this incoming message. Checked by AVG Free Edition. Version: 7.5.484 / Virus Database: 269.13.2 - Release Date: 01/09/2007 00:00 From rrosebru@mta.ca Sun Sep 9 09:59:03 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Sep 2007 09:59:03 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IUMDA-0005vr-6f for categories-list@mta.ca; Sun, 09 Sep 2007 09:48:24 -0300 Date: Sun, 09 Sep 2007 13:18:32 +0200 From: "I. Moerdijk" MIME-Version: 1.0 To: categories Subject: categories: Workshop on Lie groupoids and Lie algebroids, October 25-27 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: 11 Dear categorists, Some of you may be interested in a Workshop on Lie groupoids and Lie algebroids, which is to take place near Sheffield, October 25-27. It is organised in conjunction with a regional LMS meeting on the preceding Wednesday afternoon, with lectures by Breen and Cattaneo. Further information can be found on . The conference facilities allow only a limited number of participants, so if you are interested in coming, please contact Kirill Mackenzie or me as soon as possible. Ieke Moerdijk. From rrosebru@mta.ca Sun Sep 9 21:23:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Sep 2007 21:23:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IUWyx-0002Lc-Sn for categories-list@mta.ca; Sun, 09 Sep 2007 21:18:27 -0300 From: "Michael Shulman" Subject: categories: Re: Teaching Category Theory MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Date: Sep 2, 2007 1:36 AM To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 12 [Note from moderator: a response to this item from Jeff Egger was posted earlier, but the original was not... ] On 8/31/07, Jeff Egger wrote: > Actually, what I find intriguing is that it is the definition of > enriched category which seems to have priority over the definition > of internal category. There are, I suppose, historical reasons for > this (pre-1960 the focus tended to be on AbGp-enriched categories) > ---but I think it fair to say that (for as long as I can remember, > which obviously isn't that long from a "historical" perspective) > the majority of category theorists tend to adopt the internal > category style of definition (of category) as more primitive. My guess would be that it's because for non-category theorists, many (perhaps most) categories which arise in practice are enriched (over something more exotic than Set), while few are internal (to something more exotic than Set). Even when working over Set, I think it's fair to say that the vast majority of categories arising in mathematical practice are locally small. Since in general, neither enriched nor internal category theory is a special case of the other, it doesn't seem justified to me to consider either one as "more primitive". However, it's worth pointing out that both are a special case of categories enriched in a bicategory, or in a double category. Actually, currently my favorite level of generality is something I call a "monoidal fibration". Roughly, the idea is that you have two different "base" categories, S and V, such that the object-of-objects comes from S while the object-of-morphisms comes from V. When S and V are the same, you get internal categories, and when S=Set, you get classical enriched categories. This could be regarded as "explaining" the coincidence of internal and enriched categories for V=Set. I wrote a bit about this at the end of "Framed Bicategories and Monoidal Fibrations" (arXiv:0706.1286), but I intend to say more in a forthcoming paper. Mike From rrosebru@mta.ca Tue Sep 11 14:59:08 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 11 Sep 2007 14:59:08 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IV9tn-0002SG-Nm for categories-list@mta.ca; Tue, 11 Sep 2007 14:51:43 -0300 Date: Tue, 11 Sep 2007 12:01:04 GMT From: Oege.de.Moor@comlab.ox.ac.uk To: Subject: categories: PEPM 2008: abstracts due Oct 12 MIME-Version: 1.0 Content-Type: text/plain Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 13 PEPM 2008 ACM SIGPLAN Workshop on Partial Evaluation and Program Manipulation January 7-8, 2008, San Francisco Keynotes by Ras Bodik (Berkeley) and Monica Lam (Stanford) Co-located with POPL http://www.program-transformation.org/PEPM08/WebHome PEPM is a leading venue for the presentation of cutting-edge research in program analysis, program generation and program transformation. Its proceedings are published by ACM Press; full details of the scope, submission process, and program committee can be found at the above URL. The program committee would particularly welcome submissions from category theorists on any topic relating to categorical justification of program fusion rules Abstracts are due on October 12, and the deadline for full paper submission is October 17. Prospective authors are welcome to contact the program chairs, Robert Glueck (glueck@acm.org) and Oege de Moor (oege@comlab.ox.ac.uk) with any queries they might have. From rrosebru@mta.ca Tue Sep 11 14:59:08 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 11 Sep 2007 14:59:08 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IV9sL-0002Du-N2 for categories-list@mta.ca; Tue, 11 Sep 2007 14:50:13 -0300 Date: Tue, 11 Sep 2007 00:42:25 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Stupid question: what space was Euclid working in? 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: 14 If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance. Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry. Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry. If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it? Vaughan From rrosebru@mta.ca Wed Sep 12 16:00:40 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Sep 2007 16:00:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVXLi-0002zU-O8 for categories-list@mta.ca; Wed, 12 Sep 2007 15:54:06 -0300 Date: Wed, 12 Sep 2007 16:46:20 +0100 (BST) From: Marcelo Fiore To: categories@mta.ca Subject: categories: slides and papers 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: 15 Dear All, A transcript of the slides for my CT200t invited talk (with minor revisions and added references) is now available from . The papers on `Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic' and `Equational Systems and Free Constructions (with Chung-Kil Hur)' therein might also be of interest. Best regards, Marcelo. From rrosebru@mta.ca Wed Sep 12 16:00:40 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Sep 2007 16:00:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVXJQ-0002fk-2u for categories-list@mta.ca; Wed, 12 Sep 2007 15:51:44 -0300 Date: Wed, 12 Sep 2007 11:35:13 +0200 From: France Dacar MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Stupid question: what space was Euclid working in? Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 16 Funny you should ask this just as I was trying to find a satisfying answer to this question. Now we are so accustomed that an Euclidean space has some orthonormal system plonked down somewhere in it, or that it has at least a fixed origin and the scalar product that defines lengths and angles. But, like Eve and Adam were created without navels, Euclid's space was created without an origin; also it was created completely flat (with curvature 0 in modern lingo), so that there is no 'canonical' unit of length; and it was created withot a priory orientation. But you can measure one line segment with another in it, you can also mesure angles, and you can _choose_ one of the two possible orientations and call it positive. To capture all this I cooked up some structure; you decide if it does the job. First, an Euclid's space (let's call it that, to avoid confusing it with an Euclidean space) is a real affine space (V,P), where V is a real vector space of _vectors_ in the space, and P is the set of its _points_. There are also two operations +: P >< V -> P and -: P >< P -> V; the first is an action of the additive group of V on P, so it satisfies a + 0 = a and (a + u) + v = a + (u + v) for all points a and all vectors u, v. Moreover, the two operations are 'local inverses' of each other in the sense that (a + u) - a = u and a + (b - a) = b for all points a, b and all vectors u. This definition eliminates the need for an a priory origin. To make life easier, assume V is finite dimensional. Morphism (V,P) -> (U,Q) of affine spaces is a pair of maps h: V -> U and f: P -> Q, where h is a linear map and f(a + v) = f(a) + h(v) for all points a in P and all vectors v in V. Now, the metric of the Euclid's space. There are positive-definite (symmetric bilinear) forms on V. Call two such forms similar if they differ by a constant positive factor; a similarity class is therefore a ray (open half-line with the endpoint the zero form) in the space of all symmetric bilinear forms on V. Now consider the structure E = (V, P, m), where (V,P) is a real affine space and m is a similarity class of positive-definite forms on V: this is my proposed structure of Euclid's space. Let S be the group of all automorphisms (h,f) of the affine space (V,P) that preserve m: for every g in m, g(h(u),h(v)) = c g(u,v) for some positive constant c (depending on h, not depending on u and v) and all vectors u, v. These are the similarity trasformations of E, defining the similarity geometry of E. If A is any structure built from vectors and points (and lines etc) in E, then the orbit of A under S is the object studied in this geometry. Let R be the group of all automorphisms (h,f) of the affine space (V,P) that preserve some form in m, and therefore preserve _every_ form in m: g(h(u),h(v)) = g(u,v) for all g in m and all vectors u, v. This is the group of rigid motions of E. Orientation: if dim V = n, then there is a one-dimensional space of alternating n-linear forms on V; choosing one of the two rays (half-lines with the endpoint at the zero form) in this space chooses the orientation. -- France > If similarity geometry means similarity-invariant geometry, what are its > objects? Google has a lot to say about similarity spaces, none of it > relevant to similarity invariance. > > Sticking to finite dimensions, a Euclidean space is standardly defined > as an inner product space over the reals. As such it has predicates > recognizing those line segments of unit length, and those lines passing > through the origin, neither of which I remember from Euclid, who seems > rather to be writing about similarity geometry. > > Euclid does however talk about right triangles and line segments of > equal length (and equal angles but that follows from the other two). So > clearly Euclid is doing more than just affine geometry. > > If Euclid's plane is not the Euclidean plane, what is it? If Euclid was > doing similarity geometry it should be called a similarity space > (certainly not a vector space or a Euclidean space or an affine space or > a projective space or a metric space or a topological space). If it's > called something else what is it? If Euclid was doing some other kind > of geometry than similarity geometry what kind was it? > > Vaughan -- Dr. France Dacar Email: france.dacar@ijs.si Intelligent Systems Department Phone: +386 1 477-3813 Jozef Stefan Institute Fax: +386 1 425-3131 Jamova 39, 1000 Ljubljana, Slovenia From rrosebru@mta.ca Wed Sep 12 16:00:40 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Sep 2007 16:00:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVXIJ-0002Uo-Re for categories-list@mta.ca; Wed, 12 Sep 2007 15:50:35 -0300 Date: Tue, 11 Sep 2007 23:11:01 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? 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: 17 Some helpful discussion with Dana Scott and Fred Linton got me to the point where I felt I could say reasonably succinctly exactly what a Euclidean space is (or ought to be). The following summarizes my present understanding, which is a lot clearer than 24 hours ago. I've seen only two essentially different approaches so far, bottom-up (adding structure) and top-down (forgetting structure). Bottom-up: A Euclidean space is an affine space transformable only by similarities. Since affine spaces are already understood to be transformable only by affinities, the further restriction to similarities is equivalent to the preservation of circles, or of right isosceles triangles. Affine spaces are in turn definable as an affine-closed subspace of a projective space transformable only by affinities. Constraining the morphisms in this way is morally equivalent to adding structure (of some unspecified kind), whence "bottom-up". Top-down: A Euclidean space is a torsor for a Euclidean inner product space E. A torsor, or principal homogeneous space, is (in this instance) a generalized metric space with vector distances in place of real distances so as to make the triangle inequality an equality, with d(x,y) = -d(y,x) and d(x,y) = 0 iff x = y. Like metric spaces torsors have no origin. E supplies the distances. I put "Euclidean" as a modifier for "inner product space" to connote the liberalization of morphisms thereof to preserving inner product only up to a constant factor (as opposed to the presumed default of on the nose). This liberalization accommodates scaling, and can be considered as forgetting the scale. That and torsors for forgetting the origin makes this approach "top-down." I've used both approaches in software for things like surveying and computer graphics, using respectively top-down and bottom-up. Top-down is simple at a low level, is primarily vector oriented, involves little or no multiplication, and works fine with single-precision floating point or even fixed point arithmetic. Bottom-up is simple at a high level (if you don't try to follow the individual arithmetic steps when debugging), uses an extra dimension to express projective geometry (needed to represent the part at infinity dropped when passing to affine space), is primarily matrix oriented, involves lots of multiplication, and benefits from double precision. All high-end video cards today use bottom-up exclusively, understandable for rendering which needs projective rather than Euclidean geometry, but its benefits for rigid solid modeling over top-down are less clear to me, though for articulated solid modeling bottom-up seems preferable. What other approaches to defining Euclidean space have been proposed? Vaughan From rrosebru@mta.ca Wed Sep 12 16:00:40 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 12 Sep 2007 16:00:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVXKA-0002l7-Bh for categories-list@mta.ca; Wed, 12 Sep 2007 15:52:30 -0300 Subject: categories: Re: Stupid question: what space was Euclid working in? From: Eduardo Dubuc Date: Wed, 12 Sep 2007 10:42:19 -0300 (ART) To: categories@mta.ca (categories list) MIME-Version: 1.0 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: 18 Euclides was doing Euclidean Geometry ! > > If similarity geometry means similarity-invariant geometry, what are its > objects? Google has a lot to say about similarity spaces, none of it > relevant to similarity invariance. > > Sticking to finite dimensions, a Euclidean space is standardly defined > as an inner product space over the reals. As such it has predicates > recognizing those line segments of unit length, and those lines passing > through the origin, neither of which I remember from Euclid, who seems > rather to be writing about similarity geometry. > > Euclid does however talk about right triangles and line segments of > equal length (and equal angles but that follows from the other two). So > clearly Euclid is doing more than just affine geometry. > > If Euclid's plane is not the Euclidean plane, what is it? If Euclid was > doing similarity geometry it should be called a similarity space > (certainly not a vector space or a Euclidean space or an affine space or > a projective space or a metric space or a topological space). If it's > called something else what is it? If Euclid was doing some other kind > of geometry than similarity geometry what kind was it? > > Vaughan > > From rrosebru@mta.ca Thu Sep 13 13:48:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Sep 2007 13:48:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVrnX-0006OX-5x for categories-list@mta.ca; Thu, 13 Sep 2007 13:44:11 -0300 Date: Thu, 13 Sep 2007 15:25:43 +0100 From: Steve Vickers MIME-Version: 1.0 To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? 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: 19 Vaughan Pratt wrote: > Top-down: > > A Euclidean space is > a torsor for a Euclidean inner product space E. > > A torsor, or principal homogeneous space, is (in this instance) a > generalized metric space with vector distances in place of real > distances so as to make the triangle inequality an equality, with d(x,y) > = -d(y,x) and d(x,y) = 0 iff x = y. Like metric spaces torsors have no > origin. E supplies the distances. I put "Euclidean" as a modifier for > "inner product space" to connote the liberalization of morphisms thereof > to preserving inner product only up to a constant factor (as opposed to > the presumed default of on the nose). This liberalization accommodates > scaling, and can be considered as forgetting the scale. That and > torsors for forgetting the origin makes this approach "top-down." Dear Vaughan, I too thought about torsors, but couldn't see round the problem that they fix a unit length. (Suppose E is an inner product space and X a torsor for it, then for any x, y in X there is a unique v in E taking x to y, and so the length of v gives the distance from x to y.) Does "liberalizing the morphisms" in the way you suggest really do the trick? That seems to require a new notion of torsor, and I can't see how it would work technically. Regards, Steve. From rrosebru@mta.ca Thu Sep 13 13:48:26 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Sep 2007 13:48:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVrm9-0006Co-AW for categories-list@mta.ca; Thu, 13 Sep 2007 13:42:45 -0300 Date: Wed, 12 Sep 2007 13:44:50 -0700 From: "Greg Meredith" To: categories@mta.ca Subject: categories: "prime" monads? 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: 20 Categorically-minded, Is there a notion of prime monad where the notion of (de-)composition is adjoint situation? For example, are there monads such that the only adjoint situations giving rise to them are the Kleisli and Eilenberg-Moore algebras? Best wishes, --greg -- L.G. Meredith Managing Partner Biosimilarity LLC 505 N 72nd St Seattle, WA 98103 +1 206.650.3740 http://biosimilarity.blogspot.com From rrosebru@mta.ca Thu Sep 13 13:48:27 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 13 Sep 2007 13:48:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IVrmm-0006Ht-Hl for categories-list@mta.ca; Thu, 13 Sep 2007 13:43:24 -0300 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Wed, 12 Sep 2007 16:29:41 -0700 To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 21 Vaughan Pratt writes: > > I've seen only two essentially different approaches so far, bottom-up > (adding structure) and top-down (forgetting structure). > Where does Euclid's approach fit in this scheme? Starting from say Hilbert's formulation of Euclidean geometry it is certainly possible to get to an inner product space and all the rest, but it's certainly not direct and not easy. -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Fri Sep 14 12:11:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 12:11:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWCgR-0004iV-It for categories-list@mta.ca; Fri, 14 Sep 2007 12:02:15 -0300 Subject: categories: Re: Stupid question: what space was Euclid working in? Date: Thu, 13 Sep 2007 16:57:02 -0400 From: wlawvere@buffalo.edu To: categories list MIME-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 22 Dear Colleagues, The artificial choice of unit of length=20 should indeed be avoided in fundamental considerations since, among other things, it trivializes the relation between length & area, etc. Over a given rig R, a Euclidean space E seems to be 1) a torsor over an R-module V where V is equipped with=20 2) an isomorphism=20 V->Hom(V, L) 3) where L is an invertible R-module. The fact that L itself has no given rig structure can be compared with the general idea of metric (TAC Reprints 1) as valued in a monoidal category (which has one covariant binary operation, NOT two.) The R-modules R, L, L@L, etc, may be=20 non-isomorphic, and there may even be=20 other invertibles corresponding to time, force, etc. However, this Picard group will have all its elements of order two (i.e.,each invertible module will have its own R-valued pairing) if R is real in the sense that a sum of several squares is invertible if one of them=20 is (as shown a few years ago by Steve Schanuel). That result is for the category of abstract sets, destroying my hope that the free abelian group on three generators might occur for a suitable spatial topos of affine algebraic geometry. Of course, a unit of M, where M is an invertible=20 module, should be an isomorphism R->M, but=20 in general such isomorphisms exist only locally. To see nontrivial invertible modules, look at classical arithmetic or complex analysis, or in the present geometric vein, look at rigs and R-modules not in abstract sets, but in more cohesive or variable toposes. Garrett Birkhoff recommended the topos of G-sets for a certain group G of homogeneities,=20 obtaining (in a rather tautological way) the result=20 that the group of dimensional analysis occurs as a=20 Picard group. Looking forward to your thoughts Bill Quoting Steve Vickers : > Vaughan Pratt wrote: > > Top-down: > > > > A Euclidean space is > > a torsor for a Euclidean inner product space E. > > > > A torsor, or principal homogeneous space, is (in this instance) a > > generalized metric space with vector distances in place of real > > distances so as to make the triangle inequality an equality, with > d(x,y) > > =3D -d(y,x) and d(x,y) =3D 0 iff x =3D y. Like metric spaces torsors > have no > > origin. E supplies the distances. I put "Euclidean" as a modifier > for > > "inner product space" to connote the liberalization of morphisms > thereof > > to preserving inner product only up to a constant factor (as > opposed to > > the presumed default of on the nose). This liberalization > accommodates > > scaling, and can be considered as forgetting the scale. That and > > torsors for forgetting the origin makes this approach "top-down." >=20 > Dear Vaughan, >=20 > I too thought about torsors, but couldn't see round the problem that > they fix a unit length. (Suppose E is an inner product space and X a > torsor for it, then for any x, y in X there is a unique v in E taking > x > to y, and so the length of v gives the distance from x to y.) Does > "liberalizing the morphisms" in the way you suggest really do the > trick? > That seems to require a new notion of torsor, and I can't see how it > would work technically. >=20 > Regards, >=20 > Steve. >=20 >=20 >=20 >=20 From rrosebru@mta.ca Fri Sep 14 12:11:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 12:11:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWCjO-00058S-1K for categories-list@mta.ca; Fri, 14 Sep 2007 12:05:18 -0300 Date: Thu, 13 Sep 2007 16:06:30 -0700 From: "Greg Meredith" Subject: categories: Re: "prime" monads? To: categories@mta.ca 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: 23 Fred, Thanks. My 'intuition' is that there are some adjoint decompositions that genuinely reveal internal structure of the monad and others that ... make no such disclosure ;-). But, perhaps i can disabuse myself of this with some application to calculation. Best wishes, --greg On 9/13/07, Fred E.J. Linton wrote: > > Greg Meredith asks, > > > ... are there monads such that the only adjoint > > situations giving rise to them are the Kleisli and Eilenberg-Moore > algebras? > > NOt even the identity monad on SETS has this property, as it is > the adjunction monad also for the adjoint pair > > [underlying pointset]: [topological spaces] --> SETS , > [discrete topology on]: SETS --> [topological spaces] . > > There ARE a few monads for which the Kleisli and E-M categories > "coincide," however, beyond the identity monads. First example > coming to mind is the FreeVectorSpace monad on SETS. I'm sure > other Categories-readers will point out more. > > -- Fred From rrosebru@mta.ca Fri Sep 14 12:11:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 12:11:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWCjx-0005DO-By for categories-list@mta.ca; Fri, 14 Sep 2007 12:05:53 -0300 Subject: categories: Re: Stupid question: what space was Euclid working in? From: Eduardo Dubuc Date: Thu, 13 Sep 2007 22:17:01 -0300 (ART) To: categories@mta.ca (categories list) MIME-Version: 1.0 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: 24 the sense (a joke) of my posting was lost because some how the title of it was lost in the way my posting had the following title: "an stupid answer" > > > Euclides was doing Euclidean Geometry ! > > > > > If similarity geometry means similarity-invariant geometry, what are its > > objects? Google has a lot to say about similarity spaces, none of it > > relevant to similarity invariance. > > > > Sticking to finite dimensions, a Euclidean space is standardly defined > > as an inner product space over the reals. As such it has predicates > > recognizing those line segments of unit length, and those lines passing > > through the origin, neither of which I remember from Euclid, who seems > > rather to be writing about similarity geometry. > > > > Euclid does however talk about right triangles and line segments of > > equal length (and equal angles but that follows from the other two). So > > clearly Euclid is doing more than just affine geometry. > > > > If Euclid's plane is not the Euclidean plane, what is it? If Euclid was > > doing similarity geometry it should be called a similarity space > > (certainly not a vector space or a Euclidean space or an affine space or > > a projective space or a metric space or a topological space). If it's > > called something else what is it? If Euclid was doing some other kind > > of geometry than similarity geometry what kind was it? > > > > Vaughan > > > > > > > From rrosebru@mta.ca Fri Sep 14 12:11:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 12:11:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWChE-0004q2-Ho for categories-list@mta.ca; Fri, 14 Sep 2007 12:03:04 -0300 Date: Thu, 13 Sep 2007 18:50:19 -0400 From: "Fred E.J. Linton" To: Subject: categories: Re: "prime" monads? 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: 25 Greg Meredith asks, > ... are there monads such that the only adjoint > situations giving rise to them are the Kleisli and Eilenberg-Moore algebras? NOt even the identity monad on SETS has this property, as it is the adjunction monad also for the adjoint pair = [underlying pointset]: [topological spaces] --> SETS , [discrete topology on]: SETS --> [topological spaces] . There ARE a few monads for which the Kleisli and E-M categories "coincide," however, beyond the identity monads. First example coming to mind is the FreeVectorSpace monad on SETS. I'm sure other Categories-readers will point out more. -- Fred From rrosebru@mta.ca Fri Sep 14 21:02:20 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 21:02:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWL1K-0004ss-Mi for categories-list@mta.ca; Fri, 14 Sep 2007 20:56:22 -0300 Date: Thu, 13 Sep 2007 23:21:39 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? 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: 26 From: Steve Vickers > That seems to require a new notion of torsor, and I can't see how it > would work technically. I don't know if it's a new notion, but the following construction of what I'll call Aff(C), the affinitization (affination?) of C, should work for any category C concrete over Ab via a faithful functor U: C --> Ab. (Ab(Z,-): Ab --> Set automatically makes Ab concrete over Set.) The subcategories of Vct_k we've been talking about, in particular Euc with origin but liberal morphisms, are all instances of such a C. If this is different from torsors I'd appreciate some insight into the difference. Definition: Take the objects of Aff(C) to be those of C. A morphism in Aff(C) from c to d is a pair (h, m) where m: c --> d is a morphism of C and h: U(c) --> U(d) is any group homomorphism such that h(x) - h(y) = U(m)(x - y) for all x, y in U(c). End definition. The effect of this construction is to make the objects of Aff(c) generalized metric spaces. Reading between the lines of the above, an implicit metric D(x, y) is given by the group structure in the form x + D(x, y) = y, or equivalently D(x, y) = y - x; there is no need to make D explicit with its own symbol. All four Frechet axioms follow after modifying them to make the triangle inequality an equality and D(x, y) = -D(y, x) (symmetry becomes antisymmetry). The morphism (h, m) maps the points of the metric space via h subject to coherently (over the whole space) maintaining the distance with m. That should do it for Euc. France Dacar did things in the other order: torsor first, liberate from scale second. I guess they commute ... but: To be sure we've been talking about the same category Euc of what France called Euclid's spaces, for the Euc I have in mind, Euc(E_m,E_n) has (m+1)n - (m+1 choose 2) + 1 degrees of freedom when 1 <= m <= n. Outside that range Euc(E_m,E_n) has n degrees of freedom, with rigidity obliging E_m for m > n to collapse to E_0. When m = n > 0 this reduces to 1 + n + (n choose 2), corresponding to the one dilatation, n translations, and rotation group of order (n choose 2) = n(n-1)/2 in E_n. Here's a table. . 0 1 2 3 4 5 6 7 8 9 ----------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 1 | 0 2 4 6 8 10 12 14 16 18 2 | 0 1 4 7 10 13 16 19 22 25 3 | 0 1 2 7 11 15 19 23 27 31 4 | 0 1 2 3 11 16 21 26 31 36 5 | 0 1 2 3 4 16 22 28 34 40 6 | 0 1 2 3 4 5 22 29 36 43 7 | 0 1 2 3 4 5 6 29 37 45 8 | 0 1 2 3 4 5 6 7 37 46 9 | 0 1 2 3 4 5 6 7 8 46 In contrast Aff_R(A_m,A_n) (affine spaces) has (m+1)n degrees of freedom, while Vct_R(V_m,V_n) (vector spaces) has mn. If Euler invented affine geometry as being somehow cleaner than Euclidean geometry he was going in the right direction, with vector spaces making things simpler yet via familiar mxn matrices. Regarding Bob Knighten's question on approaches to synthetic Euclidean geometry such as Hilbert's axiomatization, has anyone axiomatized more than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean space needs to account for all finite dimensions. Vaughan From rrosebru@mta.ca Fri Sep 14 21:02:20 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 21:02:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWL2v-0004zm-KD for categories-list@mta.ca; Fri, 14 Sep 2007 20:58:01 -0300 Date: Fri, 14 Sep 2007 10:24:02 +0200 From: France Dacar MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Stupid question: what space was Euclid working in? Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 27 Bourbaki in his Algebra, and MacLane & Birkhoff in their Algebra, have similar definitions. I do not know the one you cite. However, the definition I have given describes an affine space as an honest equationally presented algebraic structure; usually they just assert that given any two points a and b, there exists a unique vector v such that a + v = b. With the difference of points an explicitly given operation of the structure, the axiom a + (b - a) = b gives the existence, and the axiom (a + v) - a = v the uniqueness. Using an n-dimensional real space instead of beloved R^n gets rid of the `canonical' basis. Observing vectors as translations of the space of points does away with an a priory origin. My contribution was that to do without an a priory unit of length, let the metric of the Euclid's space be given by a class of mutually proportinal positive definite forms, instead of a single form. > "An affine space is, roughly speaking, a vector space, > but without a particular vector being chosen as zero." Or shall we say that an affine space is a perfect geometric communism where all points are equal, while in a vector space one vector is more equal than others. -- France Ellis D. Cooper wrote: > FYI, Hassler Whitney, "Geometric Information Theory", 1957, Appendix > I, Section 10, presents the notion of "affine space" essentially as > you describe: "An affine space is, roughly speaking, a vector space, > but without a particular vector being chosen as zero." > >>Funny you should ask this just as I was trying to find a satisfying >>answer to this question. Now we are so accustomed that an Euclidean >>space has some orthonormal system plonked down somewhere in it, >>or that it has at least a fixed origin and the scalar product that >>defines lengths and angles. But, like Eve and Adam were created >>without navels, Euclid's space was created without an origin; also >>it was created completely flat (with curvature 0 in modern lingo), >>so that there is no 'canonical' unit of length; and it was created >>without a priory orientation. But you can measure one line segment >>with another in it, you can also mesure angles, and you can _choose_ >>one of the two possible orientations and call it positive. >>To capture all this I cooked up some structure; you decide if it >>does the job. >> >>First, an Euclid's space (let's call it that, to avoid confusing it >>with an Euclidean space) is a real affine space (V,P), where >>V is a real vector space of _vectors_ in the space, and P is the set >>of its _points_. There are also two operations +: P >< V -> P and >>-: P >< P -> V; the first is an action of the additive group of V on P, >>so it satisfies a + 0 = a and (a + u) + v = a + (u + v) for all >>points a and all vectors u, v. Moreover, the two operations are >>'local inverses' of each other in the sense that (a + u) - a = u and >>a + (b - a) = b for all points a, b and all vectors u. This definition >>eliminates the need for an a priory origin. To make life easier, >>assume V is finite dimensional. >> >>Morphism (V,P) -> (U,Q) of affine spaces is a pair of maps >>h: V -> U and f: P -> Q, where h is a linear map and >>f(a + v) = f(a) + h(v) for all points a in P and all vectors v in V. >> >>Now, the metric of the Euclid's space. There are positive-definite >>(symmetric bilinear) forms on V. Call two such forms similar if they >>differ by a constant positive factor; a similarity class is therefore >>a ray (open half-line with the endpoint the zero form) in the space >>of all symmetric bilinear forms on V. Now consider the structure >>E = (V, P, m), where (V,P) is a real affine space and m is a similarity >>class of positive-definite forms on V: this is my proposed structure >>of Euclid's space. >> >>Let S be the group of all automorphisms (h,f) of the affine space >>(V,P) that preserve m: for every g in m, g(h(u),h(v)) = c g(u,v) >>for some positive constant c (depending on h, not depending on u and v >>and the choice of g in m) >>and all vectors u, v. These are the similarity trasformations of E, >>defining the similarity geometry of E. If A is any structure built >>from vectors and points (and lines etc) in E, then the orbit of A >>under S is the object studied in this geometry. >> >>Let R be the group of all automorphisms (h,f) of the affine space >>(V,P) that preserve some form in m, and therefore preserve >>_every_ form in m: g(h(u),h(v)) = g(u,v) for all g in m and all >>vectors u, v. This is the group of rigid motions of E. >> >>Orientation: if dim V = n, then there is a one-dimensional >>space of alternating n-linear forms on V; choosing one of the >>two rays (half-lines with the endpoint at the zero form) in this >>space chooses the orientation. >> >>-- France -- Dr. France Dacar Email: france.dacar@ijs.si Intelligent Systems Department Phone: +386 1 477-3813 Jozef Stefan Institute Fax: +386 1 425-3131 Jamova 39, 1000 Ljubljana, Slovenia From rrosebru@mta.ca Fri Sep 14 21:02:20 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 21:02:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWL61-0005BI-KX for categories-list@mta.ca; Fri, 14 Sep 2007 21:01:13 -0300 Message-ID: <004f01c7f6ec$a1a68cb0$0b00000a@C3> From: "George Janelidze" To: Subject: categories: Re: "prime" monads? Date: Fri, 14 Sep 2007 18:30:50 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 28 Dear Fred, If we are talking about funny things - all right, let us talk about funny things: I claim that there is exactly one pair (C,T) in which: 1. C is a category, and T is a monad on C; 2. the only adjoint situations giving rise to T are determined by the Kleisli and the Eilenberg-Moore categories. Even though "unfortunately", in this case the Kleisli and the Eilenberg-Moore categories coincide. In this unique pair C is the empty category. Indeed, if C is non-empty, then take any category A with zero object and look at the projection AxAlg(T) ---> Alg(T) composed with the forgetful functor Alg(T) ---> C. This composite together with its obvious left adjoint will give rise to T. Just coincidence of the Kleisli and the Eilenberg-Moore categories is another story of course... George Janelidze ----- Original Message ----- From: "Fred E.J. Linton" To: Sent: Friday, September 14, 2007 12:50 AM Subject: categories: Re: "prime" monads? Greg Meredith asks, > ... are there monads such that the only adjoint > situations giving rise to them are the Kleisli and Eilenberg-Moore algebras? NOt even the identity monad on SETS has this property, as it is the adjunction monad also for the adjoint pair [underlying pointset]: [topological spaces] --> SETS , [discrete topology on]: SETS --> [topological spaces] . There ARE a few monads for which the Kleisli and E-M categories "coincide," however, beyond the identity monads. First example coming to mind is the FreeVectorSpace monad on SETS. I'm sure other Categories-readers will point out more. -- Fred From rrosebru@mta.ca Fri Sep 14 21:02:20 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 21:02:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWL3l-00052x-AM for categories-list@mta.ca; Fri, 14 Sep 2007 20:58:53 -0300 Date: Fri, 14 Sep 2007 08:34:33 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Homomorphisms on Z^n 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: 29 Many years ago (at least 45) Harrison mentioned to me that for any n (including infinite cardinals), Hom(Z^n,Z) = n.Z, in other words the Z-dual of the product is the sum. This is obviously a very special property of Z, almost the negation of injectivity. Has anyone on this list ever seen this before and can give me a reference? Michael From rrosebru@mta.ca Fri Sep 14 21:03:53 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Sep 2007 21:03:53 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWL7w-0005Jm-8R for categories-list@mta.ca; Fri, 14 Sep 2007 21:03:12 -0300 Date: Fri, 14 Sep 2007 10:40:02 -0700 From: Toby Bartels To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 30 Vaughan wrote in part: >[A "Euclidean space"] has predicates >recognizing those line segments of unit length, and those lines passing >through the origin, neither of which I remember from Euclid, [...]. >Euclid does however talk about right triangles and line segments of >equal length (and equal angles but that follows from the other two). You seem to have answered your main question, but here are two side points: I have seen a definition of "Euclidean space" better than the above (as a real inner product space): a real affine space with a compatible metric and the parallelogram identity. This is equivalent to a torsor of a real inner product space, or to a real affine space modulo length-preserving transformations. I agree that your definition is correct and this is wrong, since Euclid had only relative lengths, not absolute length. But it's worth knowing that the term "Euclidean space" has varied meanings. I can try to track down the reference if you wish. I also think that it's enlightening to look at Tarski's axioms for geometry. These specify a set of points equipped with a ternary and a quaternary relation (betweenness: A lies on the line segment BC; and AB is congruent to CD). Most of the axioms are independent of dimension, and the betweenness- and congruence-preserving bijections are precisely the affine similarities, as you want. (Indeed, preserving betweenness makes it affine; preserving congruence makes it a similarity.) But one final axiom sets the dimension; this states (in effect) that there exist n + 1 points, affinely independent and affinely spanning the space. If you label the first n points (1,0,0,...) through (...,0,0,1) and the last point the origin, then every point has a unique label, and the space becomes \R^n. Thus the difference between the bad "Euclidean space" as \R^n (down to having a specific basis, with no nontrivial automorphisms) and the good Euclid's space as you want to define it is precisely whether the isomorphisms also fix these n + 1 points. (Allowing the isomorphisms to permute the n non-origin points gives us the common "Euclidean space" as a real inner product space; allowing them to permute all of the n + 1 points gives us "Euclidean space" as a torsor of a real inner product space, as in the first section of this post.) --Toby From rrosebru@mta.ca Sat Sep 15 10:29:02 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Sep 2007 10:29:02 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWXe7-0002ik-4P for categories-list@mta.ca; Sat, 15 Sep 2007 10:25:15 -0300 Date: Fri, 14 Sep 2007 18:45:41 -0700 From: Toby Bartels To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 31 I wrote in part: >I also think that it's enlightening to look at Tarski's axioms for geometry. >[...] >But one final axiom sets the dimension; >this states (in effect) that there exist n + 1 points, >affinely independent and affinely spanning the space. >If you label the first n points (1,0,0,...) through (...,0,0,1) >and the last point the origin, then every point has a unique label, >and the space becomes \R^n. I just realised (reading my post again on the mailing list) that Tarski's axiom doesn't give the relative distances between these points, so these labels (and the later remarks about fixing them) are invalid. Of course, given Tarski's n + 1 points, you can find my n + 1 points, and even do this algorithmically (using the Gram-Schmidt process), so my other remarks hold only if you actually do this (so that preserving the points means preserving the results only after the Gram-Schmidt process has been applied). Thus, the situation is a little less elegant than I implied (but maybe that's Tarski's fault for phrasing his axiom so liberally ^_^). --Toby From rrosebru@mta.ca Sat Sep 15 10:29:02 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Sep 2007 10:29:02 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWXdL-0002gu-Tc for categories-list@mta.ca; Sat, 15 Sep 2007 10:24:27 -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: Re: "prime" monads? Date: Sat, 15 Sep 2007 11:40:23 +1000 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 32 Has someone already mentioned that Eilenberg-Moore algebra categories and Kleisli categories "coincide" for all idempotent monads? The forgetful is full. Ross > There ARE a few monads for which the Kleisli and E-M categories > "coincide," however, beyond the identity monads. First example > coming to mind is the FreeVectorSpace monad on SETS. I'm sure > other Categories-readers will point out more. From rrosebru@mta.ca Sat Sep 15 15:46:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Sep 2007 15:46:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWcXd-0007SB-Tq for categories-list@mta.ca; Sat, 15 Sep 2007 15:38:53 -0300 From: "Ronnie Brown" To: "\"Categories list\"" Subject: categories: Re: Homomorphisms on Z^n Date: Sat, 15 Sep 2007 10:02:50 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 33 Michael, I think Christopher Zeemann did something on this in or around the 1950s but cannot at the moment access mathscinet to check. In the 1970s several of us extended Pontrjagin duality: 15. (with P.J. HIGGINS and S.A. MORRIS), ``Countable products of lines and circles: their closed subgroups, quotients and duality properties'', {\em Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32. in particular defining `strong duality'. Ronnie ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Friday, September 14, 2007 1:34 PM Subject: categories: Homomorphisms on Z^n > Many years ago (at least 45) Harrison mentioned to me that for any n > (including infinite cardinals), Hom(Z^n,Z) = n.Z, in other words the > Z-dual of the product is the sum. This is obviously a very special > property of Z, almost the negation of injectivity. Has anyone on this > list ever seen this before and can give me a reference? > > Michael > > > > > > -- > No virus found in this incoming message. > Checked by AVG Free Edition. > Version: 7.5.487 / Virus Database: 269.13.18/1007 - Release Date: > 13/09/2007 21:48 > From rrosebru@mta.ca Sat Sep 15 15:46:15 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 15 Sep 2007 15:46:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWcWR-0007MD-8X for categories-list@mta.ca; Sat, 15 Sep 2007 15:37:39 -0300 Date: Sat, 15 Sep 2007 01:02:37 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: "prime" monads? 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: 34 Fred E.J. Linton wrote: > There ARE a few monads for which the Kleisli and E-M categories > "coincide," however, beyond the identity monads. First example > coming to mind is the FreeVectorSpace monad on SETS. I'm sure > other Categories-readers will point out more. Fred, your numbering makes the example preceding your "first example" (the identity monad) your zeroth example, but I'll number them 1 and 2 anyway. Example 2 should work for all Archimedean fields (and presumably all ordered fields). Does it also work for fields not of characteristic 0, or for the complex number field? For more examples, all on Set, how about the following? Example 3 extends example 1 (i.e. the identity monad is a submonad of example 3), the rest (examples 4-9) are submonads of example 2. This should bring the crop of examples of Kl=EM up to 6 if I'm not mistaken (a big if), with examples 7-9 not contributing. All of 1-9 are finitary monads, unlike for example the covariant power set monad whose signature is a proper class. 3. The lift monad 1+X on Set. Obviously Kl=EM in this case. But not for 1+1+X: the free bipointed sets are in a natural bijection with the non-free ones, the latter having the two points identified--this must be about the simplest nontrivial instance of obtaining an algebra as a quotient of a free algebra, not a bad introductory example when explaining how to get algebras from free algebras. 4. The FreeAffineSpace monad on Set as the submonad of example 2 consisting of those operations whose coefficients sum to 1, i.e. barycentric weightings where the individual weights can be any real. I noticed this the other day and assumed it must be well known (is it?). It gives an alternative to France Dacar's equations yesterday showing that the affine spaces are algebraic over Set. The Kleisli-EM identification certainly holds for finite-dimensional affine spaces (any quotient is a space of lower dimension, but all spaces of a given finite dimension are dense and therefore isomorphic); is there any reason why the proof (with Choice) for infinite-dimensional vector spaces would not go through for the affine case? By way of insight into this monad, if you take those operations whose coefficients sum to 3 say (still as a "submonad" of example 2), the multiplication produces exactly the operations whose coefficients sum to 9, so that would-be monoid isn't closed under multiplication. If you think to fix this by taking the operations whose coefficients sum to 0 (remember we're allowing negative coefficients) then that attempt doesn't have a unit. That just leaves the barycentric operations as the only possible submonad of example 2 consisting of operations of a fixed total weight. My guess would be that Kl=EM for the next two, by density (cf. examples 7 and 8). Do statisticians know about these monads? 5. Conservative statistics. The submonad of example 4 where the coefficients must be positive (and hence in (0,1]). The next example shows the sense in which this is "conservative." 6. Lossy statistics. The submonad of example 2 where both the individual coefficients *and* their sum (over the coefficients of an operation) are in (0,1]. Lossy statistics permits its sample spaces to leak out through a wormhole. The following two are examples in this vein where EM > Kl. 7. The lattice monad (JAMS-friendly lattices, not that quaint order-theoretic stuff that went out with the Bauhaus, http://www.math.rutgers.edu/~zeilberg/Opinion81.html). The submonad of example 2 restricted to integer coefficients. The free lattice on n generators is Z^n; for n <= 2 I think all non-free quotients must be a circle, cylinder, or torus, but for n = 3, (x,y,z) |--> (2x+z,2y+z) produces the 2D checkerboard as an infinite quotient of Z^3 with no cycles, pointing up the role of density in 4 above. 8. Affine lattices. The intersection of examples 4 and 7 (still allowing negative coefficients). This is the appropriate monad for studying crystalline structure, which has no origin. The free lattices are the cubical crystals and all other crystalline structures are non-free. Unless I'm missing something the quotient (x,y,z) |--> (2x+z,2y+z) works here too to show EM > Kl. 9. The intersection of examples 5 and 7 (same as 6 and 7). Isn't this just example 1? The chances of all my guesses being right are pretty slim, but if they are we'd now be up to six examples of Kl=EM, namely 1 to 6 above. What is the structure of the submonads of a monad on Set? Anything like algebraic lattices? Vaughan From rrosebru@mta.ca Sun Sep 16 12:14:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Sep 2007 12:14:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWvhG-0001dB-IB for categories-list@mta.ca; Sun, 16 Sep 2007 12:06:06 -0300 Date: Sun, 16 Sep 2007 01:12:37 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: "prime" monads? 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: 35 From: Fred Linton > I was relying on the Axiom of Choice (AC) to know that every vector space > (scalars from an arbitrary field) has a basis. Paraphrasing: every > vector space is free. Thinking (loosely) of the Kleisli category as > the full subcategory of the EM category consisting of the free algebras, > this means the K and the EM categories "coincide" (to within equivalence > of categories). The exact nature of the field is irrelevant. But AC (or > something else strong enough to guarantee every vector space has a basis) > is crucial. This seems right for ordered fields. Is it also true for finite fields and other non-ordered fields like the complex numbers? If every Archimedean field k gives rise to a Klem (Kleisli ~ EM) monad Vct_k, that's uncountably many Klem monads right there, all in the lattice of submonads of Vct_R. (The submonads of a monad must form a complete lattice, more precisely a complete semilattice, if they're closed under arbitrary intersection; is it always an algebraic semilattice, as for algebras?) k doesn't even need to be a field or an additive group, any subrig of R will do, as Bill was hinting at. (The coefficients have to form a rig for T to be functorial.) Call this Method A for forming a submonad of Vct_R. Method B is to limit the operations to those whose weight (sum of coefficients) is drawn from a submonoid of the monoid R under multiplication. (It has to be a submonoid for the multiplication mu and unit eta to remain defined.) The smallest such monoid is {1}, which gave rise to the affine spaces as per my previous post. R has plenty of other multiplicative submonoids, such as the set {c^i} for any real c > 0 and i ranging over any submonoid of Z under addition, or for any real c and i ranging over any submonoid of N under addition. This can be combined with Method A (coefficients from a subrig of R) to get even more submonads. Questions: 1. Do all submonads of Vct_R arise as above (coefficients limited to a subrig of R, weights limited to a multiplicative submonoid of R)? 2. Among the submonads of Vct_R, are the Klem ones exactly those for which the set of coefficients is dense in some open interval of R? >> But not for 1+1+X: the free bipointed sets are in a natural bijection >> with the non-free ones, the latter having the two points >> identified > > I'm not entirely sure what's "natural" here. Certainly the free bipointed > sets are in bijective correspondence with the (just plain) sets: that's a > "consequence," if you like, of the canonical functor from SETS to any > Kleisli category over SETS being a bijection on objects. So the non-free > bipointed sets, coinciding as they do with the (singly) pointed sets, are > likewise in bijective correspondence with the (just plain) sets. But the > same is true for the free algebras over ANY monad on sets. So? "This"? > >> ... --this must be about the simplest nontrivial instance of >> obtaining an algebra as a quotient of a free algebra, not a bad >> introductory example when explaining how to get algebras from free > algebras. Point taken. I was trying to say that identifying the two constants of a free bipointed set was the canonical way of producing any nonfree bipointed set. What's a slicker way to say this? Vaughan From rrosebru@mta.ca Sun Sep 16 12:14:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Sep 2007 12:14:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IWvge-0001aH-FW for categories-list@mta.ca; Sun, 16 Sep 2007 12:05:28 -0300 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Sat, 15 Sep 2007 14:06:55 -0700 To: categories list Subject: categories: Re: Stupid question: what space was Euclid working in? Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 36 Vaughan Pratt writes: > > Regarding Bob Knighten's question on approaches to synthetic Euclidean > geometry such as Hilbert's axiomatization, has anyone axiomatized more > than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean > space needs to account for all finite dimensions. > Postings crossing in the ether make this a bit confusing, but Toby Bartels effectively answered this with his reference to Tarski's axioms so my only additional contribution is a reference where the details for all finite dimensions are discussed: Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," Bulletin of Symbolic Logic 5: 175-214. -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Sun Sep 16 20:39:01 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 16 Sep 2007 20:39:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IX3ZJ-0000cu-BT for categories-list@mta.ca; Sun, 16 Sep 2007 20:30:25 -0300 From: Colin McLarty To: categories@mta.ca Date: Sun, 16 Sep 2007 12:33:17 -0400 MIME-Version: 1.0 Subject: categories: Re: Stupid question: what space was Euclid working in? Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 37 A lot of people long before Tarski, or Hilbert, knew how to extend this kind of axiomatization to any finite dimension. But is it "satisfactory"? Specifically, what about (n-dimensional) volumes. Euclid 11--13 does not well axiomatize the method of exhaustion he uses for the volume of solids. And Dehn's theorem that already for polyhedra the theory of volume will not reduce to equidecomposition. I should probably already know this, but what is known about the theory of n-dimensional volume in axiomatic n-dimensional Euclidean geometry? Colin ----- Original Message ----- From: Robert L Knighten Date: Sunday, September 16, 2007 11:26 am Subject: categories: Re: Stupid question: what space was Euclid working in? To: categories list > Vaughan Pratt writes: > > > > Regarding Bob Knighten's question on approaches to synthetic > Euclidean > geometry such as Hilbert's axiomatization, has anyone > axiomatized more > > than E_3 (Euclid Books 11-13)? A satisfactory definition of > Euclidean > space needs to account for all finite dimensions. > > > > Postings crossing in the ether make this a bit confusing, but Toby > Bartelseffectively answered this with his reference to Tarski's > axioms so my only > additional contribution is a reference where the details for all > finitedimensions are discussed: > > Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," > Bulletin of Symbolic Logic 5: 175-214. > > -- Bob > > -- > Robert L. Knighten > RLK@knighten.org > > > From rrosebru@mta.ca Mon Sep 17 20:07:56 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Sep 2007 20:07:56 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXPcL-0007dr-3u for categories-list@mta.ca; Mon, 17 Sep 2007 20:03:01 -0300 From: "Categorical Methods" To: Subject: categories: J Adamek and W Tholen's 60th Date: Mon, 17 Sep 2007 14:32:16 +0100 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: 38 CATEGORICAL METHODS IN ALGEBRA, TOPOLOGY AND COMPUTER SCIENCE WORKSHOP IN HONOUR OF JIRI ADAMEK AND WALTER THOLEN =3D=3DReminder=3D=3D Deadline for Registration: September 20, 2007 Deadline for Abstract Submission: September 30, 2007 URL: http://www.mat.uc.pt/~cmatcs/ From rrosebru@mta.ca Mon Sep 17 20:07:56 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Sep 2007 20:07:56 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXPdE-0007jB-6D for categories-list@mta.ca; Mon, 17 Sep 2007 20:03:56 -0300 From: "Lurdes Sousa" To: Subject: categories: Volume dedicated to J Adamek Date: Mon, 17 Sep 2007 14:17:21 +0100 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: 39 SPECIAL VOLUME IN HONOUR OF JIRI ADAMEK ON THE OCCASION OF HIS SIXTIETH BIRTHDAY Jiri Adamek will celebrate his 60th birthday this October, and there will be a Special Issue of the journal Cahiers de Topologie et = G=E9om=E9trie Diff=E9rentielle Cat=E9goriques dedicated to this occasion. Submission of papers on areas where Jiri Adamek has worked and published are encouraged. The deadline for submission is 15 March 2008. All papers will be carefully refereed following the standards of Cahiers de Topologie et G=E9om=E9trie Diff=E9rentielle Cat=E9goriques. For more information, please visit the web page http://www.mat.uc.pt/~cmatcs/adamek/JAdamek_volume.html The Guest Editors, Francis Borceux Achim Jung Jir=ED Rosicky Lurdes Sousa From rrosebru@mta.ca Mon Sep 17 20:07:56 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Sep 2007 20:07:56 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXPaI-0007SG-O5 for categories-list@mta.ca; Mon, 17 Sep 2007 20:00:54 -0300 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Sun, 16 Sep 2007 23:56:27 -0700 To: categories@mta.ca Subject: categories: Re: Stupid question: what space was Euclid working in? Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 40 Colin McLarty writes: > A lot of people long before Tarski, or Hilbert, knew how to extend this > kind of axiomatization to any finite dimension. But is > it "satisfactory"? > > Specifically, what about (n-dimensional) volumes. Euclid 11--13 does > not well axiomatize the method of exhaustion he uses for the volume of > solids. And Dehn's theorem that already for polyhedra the theory of > volume will not reduce to equidecomposition. > > I should probably already know this, but what is known about the theory > of n-dimensional volume in axiomatic n-dimensional Euclidean geometry? > . . . > > Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," > > Bulletin of Symbolic Logic 5: 175-214. > > Of course Tarski's axiomatization had precursors going all the way back to Euclid, and as is discussed in the Tarski and Givant paper none of his axioms are new with Tarski. There is some discussion of the actual development of Euclidean geometry from the axioms in that article, but I believe (though I do not actually have the book) the detailed development was presented in Schwabhauser, W. and Szmielew, W. and Tarski, A., Metamathematische Methoden in der Geometrie, Springer-Verlag, 1983 But the development is in showing how to go from the axioms to analytic geometry not in developing a direct extension of Euclid's methods. -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Mon Sep 17 20:07:56 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Sep 2007 20:07:56 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXPbA-0007WJ-1O for categories-list@mta.ca; Mon, 17 Sep 2007 20:01:48 -0300 Date: Mon, 17 Sep 2007 13:10:07 +0100 (BST) Subject: categories: Re: Stupid question: what space was Euclid working in? (almost) From: tporter@informatics.bangor.ac.uk To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 41 Dear All, Although not strictly relevant to the discussion here, I recall once looking at some of the theory of the `region connection calculus' and the work of the Qualitative Spatial and Spatio-temporal Reasoning group at Leeds. This is concerned with the interrelationships between regions of a space. The logical models vary from biHeyting algebras to various multimodal logics. There was some discussion about trying to detect dimension within such RCC systems. If one assumes that `region' is a more basic notion of position than `point' a lot of Euclid still goes through but dimension seems very hard to handle. (I recall that Tarski worked on this area at one time.) Can any one tell me more as it seems of use in Geographic Information Systems= , and other models of qualitative or descriptive spatial information. I know of John Stell's work on this and he has clarified things from a somewhat categorical viewpoint. Perhaps the old puns about `pointless' arguments need revisiting! Tim --=20 Gall y neges e-bost hon, ac unrhyw atodiadau a anfonwyd gyda hi, gynnwys deunydd cyfrinachol ac wedi eu bwriadu i'w defnyddio'n unig gan y sawl y cawsant eu cyfeirio ato (atynt). Os ydych wedi derbyn y neges e-bost hon trwy gamgymeriad, rhowch wybod i'r anfonwr ar unwaith a dil=EBwch y neges. Os na fwriadwyd anfon y neges atoch chi, rhaid i chi beidio =E2 defnyddio, cadw neu ddatgelu unrhyw wybodaeth a gynhwysir ynddi. Mae unrhyw farn neu safbwynt yn eiddo i'r sawl a'i hanfonodd yn unig ac nid yw o anghenraid yn cynrychioli barn Prifysgol Bangor. Nid yw Prifysgol Bangor yn gwarantu bod y neges e-bost hon neu unrhyw atodiadau yn rhydd rhag firysau neu 100% yn ddiogel. Oni bai fod hyn wedi ei ddatgan yn uniongyrchol yn nhestun yr e-bost, nid bwriad y neges e-bost hon yw ffurfio contract rhwymol - mae rhestr o lofnodwyr awdurdodedig ar gael o Swyddfa Cyllid Prifysgol Bangor. www.bangor.ac.uk (YCYG) This email and any attachments may contain confidential material and is solely for the use of the intended recipient(s). If you have received this email in error, please notify the sender immediately and delete this email. If you are not the intended recipient(s), you must not use, retain or disclose any information contained in this email. Any views or opinions are solely those of the sender and do not necessarily represent those of Bangor University. Bangor University does not guarantee that this email or any attachments are free from viruses or 100% secure. Unless expressly stated in the body of the text of the email, this email is not intended to form a binding contract - a list of authorised signatories is available from the Bangor University Finance Office. www.bangor.ac.uk (SEECS) From rrosebru@mta.ca Mon Sep 17 20:07:56 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Sep 2007 20:07:56 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXPgR-0000DG-Ut for categories-list@mta.ca; Mon, 17 Sep 2007 20:07:15 -0300 To: From: "Maria Manuel Clementino" Subject: categories: Volume dedicated to Walter Tholen's 60th Date: Mon, 17 Sep 2007 13:10:41 +0100 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: 42 SPECIAL VOLUME IN HONOUR OF WALTER THOLEN ON THE OCCASION OF HIS SIXTIETH BIRTHDAY Walter Tholen will celebrate his 60th birthday this October,and there will be a special volume of Theory and Applications of Categories dedicated to this occasion. The deadline for submission is 15 March 2008. For more information, please visit the web page http://www.mat.uc.pt/~cmatcs/WTholen_volume.html The Guest Editors, Maria Manuel Clementino, George Janelidze, Robert Rosebrugh, Jir=ED = Rosicky From rrosebru@mta.ca Mon Sep 17 20:11:21 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Sep 2007 20:11:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXPkB-0000aE-Ex for categories-list@mta.ca; Mon, 17 Sep 2007 20:11:07 -0300 Date: Mon, 17 Sep 2007 01:54:30 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: "prime" monads? 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: 43 (This supersedes my previous post, responding to Fred Linton, which was written too hastily in retrospect. The following is hopefully more accurate.) In my response to Fred, "method A" imposed the condition on the coefficients of the linear combinations constituting the operations of Vct_R that they come from a subrig C of R, in line with Bill Lawvere's mention of rigs. Earlier I had pointed out that affine spaces arose from a submonad of vector spaces obtained by requiring the weight of every operation (sum of its coefficients) to be 1, i.e. the condition W = {1} where W is the set of permitted weights. The latter construction evidently generalizes to all submonads produced by method A. I then wrote, > Method B is to limit the operations to those whose weight (sum of > coefficients) is drawn from a submonoid of the monoid R under > multiplication. Sorry, that attempt at generalizing method A doesn't produce submonads after all (the \mu of Vct_R produces operations with weights outside W, as I'll make clearer shortly). Worse, I can't see what the right fix should be. In fact I can't think of any other submonads T of "Vct_C", namely those for which T(1) is the underlying set of a subrig C of R, such that C contains a coefficient greater than 1, other than those submonads with W = R or W = {1}. Conjecture: there are none. Monads probably aren't the most convenient framework for talking about submonads of Vct_R, certainly for the typical reader of the American Math Monthly, who would surely understand them better in terms of ordinary finite matrices and their multiplication since the submonads all have the same equational theory forming the basis for matrix multiplication, just restricted to fewer operations. To make the connection between matrix algebra and the Kleisli category KT for any submonad T of Vct_R, the object n of KT for n a finite set is (from the standpoint of the morphisms of KT) a free algebra A_n on a set that can be viewed schizophrenically as a set of n variables and a basis for A_n. (Semantics qua EM doesn't specify a basis but syntax qua Kleisli does.) KT(1,m) = T(m) is schizophrenically the m-ary operations and the points of A_m understood as column vectors. KT(n,1) = T(1)^n is the set of permitted row vectors aka dual points. More generally KT(n,m) is the set of permitted mxn matrices over T(1) representing the morphisms from n to m in the Kleisli category of this particular submonad T of Vct_R. It follows that submonads can only constrain rows by constraining T(1), which amounts to constraining the coefficients; what I've been calling the set C of permitted coefficients is T(1). Submonads can constrain columns in more general ways, but not too general. Restating my conjecture above in this mixed monad-matrix language, given any submonad T of the monad for Vct_R for which T(1) contains a coefficient c > 1, the only proper submonad of T that leaves the permitted matrix entries (the set T(1)) unchanged is that for which every column of the permitted matrices sums to 1. This is almost how affine geometry is standardly implemented in computational geometry and computer graphics (CG), the difference being that the last row of each matrix is constrained to be a unit vector with its 1 in the "translation" column (projective geometry projected from infinity - the translation column specifies the vector translating the transformed point). The following is the generic affine transformation of a generic point (x,y) in the plane in the CG approach. ( a b s ) ( x ) = ( ax + by + s) ( c d t ) ( y ) = ( cx + dy + t) ( 0 0 1 ) ( 1 ) = ( 1 ) The 2x2 matrix at the top left performs an ordinary linear transformation of the plane. The "translation column" is at the right, and acts by translating the plane right s and up t. This constraint on the last row, with no constraints on the columns, seems in total violation of the above. However the violation is not an essential one as the matrix ( a b s ) A = ( c d t ) ( 0 0 1 ) is similar (http://en.wikipedia.org/wiki/Similar_matrix) to -1 ( a+s b+s s ) P A P = ( c+t d+t t ) ( 1-(a+s+c+t) 1-(b+s+d+t) 1-(s+t) ) where ( 1 0 0 ) P = ( 0 1 0 ) ( 1 1 1 ) (Similarity is the matrix algebra counterpart of functoriality in category theory. That is, the evident nxn counterparts of P for all n exhibit an equivalence, in fact an isomorphism, between the CG representation of the category for affine spaces and its CT counterpart as presented by its Kleisli category.) It is easily seen that after applying this functor the column sums are now 1, while a counting argument for degrees of freedom (6) shows that the rows are no longer at all constrained. The determinant remains ad-bc while the trace remains a+d+1, and eigenvalues also remain unchanged. (I only noticed this isomorphism of the two categories today. In retrospect it is a routine application of similarity, but is CG aware of this particular functor?) Other conditions on the morphisms, such as that they be orthogonal matrices, will not in general produce submonads because they tamper with the rows in what is likely to be an essential way that destroys algebraicity. The matrix viewpoint makes it a lot easier to see where my method B breaks down: the only constraints on column sums that seem to work when C (aka T(1)) = R are W = R and W = {1} (the conjecture above). So what restrictions *can* we impose on the columns that preserve algebraicity, specifically that preserve the equational theory of Vct_R? The sets I = [0,1] and I- = (0,1] look like they could work independently for each of C and W. However for n > 1 we need 0 for the unit vectors (\eta in the monad), ruling out C = I- and leaving C = I as the only possibility here. This still leaves the two possibilities W = I and W = I-, the latter corresponding to disallowing zero column vectors. The free algebras A_n on n generators for these two submonads can be visualized geometrically as follows. C = W = I: Unit simplexes with n+1 vertices, namely the origin and the n unit vectors. These are closed, containing their (n+1 choose d+1) d-dimensional subfaces. So n generators create ordinary n-dimensional linear algebra confined to a simplex. C = I, W = I- : Ditto less one point, namely the origin. In the latter case, if the one zeroary operation (constant) 0 in T(0) is understood as having weight 0, W = I- rules it out whence that constant disappears. So the 0-dimensional space A_0 is empty while A_1 is (0,1], A_2 is a triangle less the origin, etc. To summarize, we have as submonads of Vct_R the following: 1. Vct_S obtained by subsetting T(1) to a subrig S of R, and independently requiring either W = S (no constraint on W) or W = {1} (the strongest possible constraint on W). S could be the natural numbers for example, with or without 0, but not the integers mod n because that would not be a subrig of R (it would introduce new equations we're trying to avoid). 2. The pair of monads C = W = I and C = I, W = I-, intersected with S as per 1 and so giving rise to lots of variants of the pair. Does Vct_R have any other submonads? Keep in mind the conditions that 1 belong to C and W (consider \eta) and that the rows be T(1)^n. Vaughan From rrosebru@mta.ca Tue Sep 18 20:33:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Sep 2007 20:33:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXmPF-0005VM-VX for categories-list@mta.ca; Tue, 18 Sep 2007 20:23:02 -0300 Date: Tue, 18 Sep 2007 09:36:10 -0300 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Stupid question: what space was Euclid working in? (almost) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 44 tporter@informatics.bangor.ac.uk wrote: > If one assumes that `region' is a more basic notion of position than > `point' a lot of Euclid still goes through but dimension seems very hard > to handle. The topological form of Helly's theorem might be a place to start. However, defining dimension in terms of _convex_ structure is very tricky once you get into general spaces. For instance, I showed in my thesis (in a section eventualLy rewritten for _Cahiers_) that if you define a "convex set" on S^1 to be an arc shorter than an open semicircle, you get the obvious homology. However, if closed semicircles and their intersections are convex, the homology becomes that of the 2-sphere. -Robert From rrosebru@mta.ca Tue Sep 18 20:33:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Sep 2007 20:33:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXmQT-0005bd-II for categories-list@mta.ca; Tue, 18 Sep 2007 20:24:17 -0300 Date: Tue, 18 Sep 2007 12:53:26 -0400 (EDT) Subject: categories: Re: Homomorphisms on Z^n From: "Stephen Urban Chase" 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: 45 Zeeman proved the assertion in 1955 for non-measurable cardinals n. Ther= e are more general results which comprise the theory of slender (abelian) groups; see, e.g., Chapter XIII, Sections 94-95 of [L. Fuchs, Infinite Abelian Groups vol. 2, Academic Press, 1973], especially Corollary 94.6 o= n p 162. I haven't thought much about abelian groups since the early 1960's, but i= t is well known that infinite direct products have very interesting properties, both as abstract and as topological groups. For example, a closed subgroup of a direct product of countably many copies of Z is also a direct product, but not so for uncountable products (see [R.J. Nunke, On direct products of infinite cyclic groups, Proc. Amer. Math. Soc. 13 (1962), pp 66-71]). In fact, Zeeman's result implies that a countable free group is a closed subgroup of a direct product with uncountably many factors. A generalization of Nunke's theorem and some related results are contained in my old paper [Function topologies on abelian groups, Ill. J. Math. 7 (1963), pp 593-608]. Steve Chase ---------------------------- Original Message ---------------------------= - Subject: categories: Homomorphisms on Z^n From: "Michael Barr" Date: Fri, September 14, 2007 8:34 am To: "Categories list" -------------------------------------------------------------------------= - Many years ago (at least 45) Harrison mentioned to me that for any n (including infinite cardinals), Hom(Z^n,Z) =3D n.Z, in other words the Z-dual of the product is the sum. This is obviously a very special property of Z, almost the negation of injectivity. Has anyone on this list ever seen this before and can give me a reference? Michael From rrosebru@mta.ca Tue Sep 18 20:33:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 18 Sep 2007 20:33:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IXmRz-0005jd-J9 for categories-list@mta.ca; Tue, 18 Sep 2007 20:25:51 -0300 Date: Tue, 18 Sep 2007 18:06:18 +0200 From: Andree Ehresmann To: categories@mta.ca Subject: categories: A multidisciplinary book using categories MIME-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 46 I call to your attention our new book: "Memory Evolutive Systems: =20 Hierarchy, Emergence, Cognition", by Andr=E9e Ehresmann and Jean-Paul =20 Vanbremeersch, published in Elsevier's Series on Multidisciplinarity =20 (Volume 4, 2007). This book, based on a 20 years long series of papers, develops the =20 theory of Memory Evolutive Systems which we have introduced as a =20 mathemati-cal model, based on category theory, for complex natural =20 systems, such as biological, social or cognitive systems. It shows how =20 well-known categorical operations (in particular (co)limits, partial =20 completions of categories and fibrations) give an approach to the =20 problems of hierarchy, emergence/reductionism, self-organization and =20 learning. The main tools are exposed in Part A (Hierarchy and emergence), the =20 global theory in Part B (Memory Evolutive Systems), and Part C =20 (Applications to cognition and consciousness) is devoted to the case =20 of cognitive systems, studying the formation of a procedural and a =20 semantic memory allowing for the emergence of higher cognitive =20 processes up to consciousness. The book is written for a multidisciplinary audience, with many =20 illustrative examples in the most varied domains, but also with short =20 proofs of the main mathematical results. An animation can be found on =20 the internet site: http://perso.wanadoo.fr/vbm-ehr/ Comments and critics will be most welcomed. Andree Ehresmann From rrosebru@mta.ca Wed Sep 19 13:45:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 19 Sep 2007 13:45:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IY2Yt-00005o-9V for categories-list@mta.ca; Wed, 19 Sep 2007 13:38:03 -0300 Date: Tue, 18 Sep 2007 20:58:46 -0400 (EDT) From: Josh Nichols-Barrer To: categories@mta.ca Subject: categories: unital weak functor? 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: 47 Hi everyone, Is there a name for a weak functor between bicategories which takes identity 1-morphisms to identity 1-morphisms? "Unital weak functor" would seem an apt name, but if there is another with more precedent I'll just use that instead. Best, Josh From rrosebru@mta.ca Thu Sep 20 18:07:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Sep 2007 18:07:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYT6D-0006sS-BH for categories-list@mta.ca; Thu, 20 Sep 2007 17:58:13 -0300 From: Juergen Koslowski Subject: categories: CT07 pictures To: categories@mta.ca Date: Thu, 20 Sep 2007 13:14:50 +0200 (CEST) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: A X-Keywords: X-UID: 48 Dear CT07 Participants, Pictures of the conference are also available on my home page http://www.iti.cs.tu-bs.de/~koslowj/cgi-bin/pE333?CT07-0/idx&256 http://www.iti.cs.tu-bs.de/~koslowj/PHOTOS/CT07 (group picture) As usual, I'm missing some names. So please send me a mail if you can identify someone with a question mark instead of a name. However, I'm presently travelling in Vietnam and Cambodia, so I will only be able do fix the names after my return (October 14). Best regards, -- Juergen -- Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj@iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR) From rrosebru@mta.ca Thu Sep 20 18:07:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Sep 2007 18:07:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYT78-0006y9-RV for categories-list@mta.ca; Thu, 20 Sep 2007 17:59:10 -0300 Date: Thu, 20 Sep 2007 14:53:54 -0300 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: cat group Subject: categories: Commutative diagrams go public! 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: 49 Members of the Cat Group may be interested to see an example of a commutative diagram in public signage: http://cs.stmarys.ca/~dawson/CommDiag.html Not exactly cutting-edge mathematics, but it's a start! -Robert Dawson From rrosebru@mta.ca Thu Sep 20 18:07:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Sep 2007 18:07:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYT4W-0006jF-JN for categories-list@mta.ca; Thu, 20 Sep 2007 17:56:28 -0300 Date: Wed, 19 Sep 2007 20:30:11 +0100 From: Robin Houston To: categories@mta.ca Subject: categories: Re: unital weak functor? Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 50 On Tue, Sep 18, 2007 at 08:58:46PM -0400, Josh Nichols-Barrer wrote: > Is there a name for a weak functor between bicategories which takes > identity 1-morphisms to identity 1-morphisms? They are usually called 'normal', at least by the Sydney school. Robin From rrosebru@mta.ca Thu Sep 20 18:07:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Sep 2007 18:07:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYT6d-0006up-Ox for categories-list@mta.ca; Thu, 20 Sep 2007 17:58:39 -0300 Date: Thu, 20 Sep 2007 09:01:21 -0700 From: John Baez MIME-Version: 1.0 To: categories@mta.ca Subject: categories: unital weak functor? 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: 51 Josh Nichols-Barrer wrote: > Hi everyone, > > Is there a name for a weak functor between bicategories which takes > identity 1-morphisms to identity 1-morphisms? "Unital weak functor" > would > seem an apt name, but if there is another with more precedent I'll just > use that instead. I don't think there's a standard term. But, I like the term "normalized", since in certain circumstances weak functors between bicategories are described by cocycles in group cohomology, and the cocycle is then said to be "normalized" when the weak functor preserves identity 1-morphisms. It's an old fact that every cocycle is equivalent to a normalized one, and this is related to the fact that every weak functor is isomorphic to a normalized one. For more information on this, see: Andre Joyal and Ross Street, Braided monoidal categories, Macquarie Mathematics Report No. 860081, November 1986. Also available at http://rutherglen.ics.mq.edu.au/~street/JS86.pdf or for a pedagogical treatment, try section 8.3, "Classifying 2-groups using group cohomology", of this: John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-Groups, Theory and Applications of Categories 12 (2004), 423-491. Also available at http://arxiv.org/abs/math.QA/0307200 Best, jb From rrosebru@mta.ca Thu Sep 20 18:07:45 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Sep 2007 18:07:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYT5c-0006pN-3i for categories-list@mta.ca; Thu, 20 Sep 2007 17:57:36 -0300 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Subject: categories: RE: unital weak functor? Date: Thu, 20 Sep 2007 07:03:07 +1000 From: "Stephen Lack" To: Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 52 Dear Josh, These are often called normal lax functors (or normal morphisms of = bicategories). Regards, Steve. -----Original Message----- From: cat-dist@mta.ca on behalf of Josh Nichols-Barrer Sent: Wed 9/19/2007 10:58 AM To: categories@mta.ca Subject: categories: unital weak functor? =20 Hi everyone, Is there a name for a weak functor between bicategories which takes identity 1-morphisms to identity 1-morphisms? "Unital weak functor" = would seem an apt name, but if there is another with more precedent I'll just use that instead. Best, Josh From rrosebru@mta.ca Fri Sep 21 14:21:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 21 Sep 2007 14:21:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYm6J-00011D-Lj for categories-list@mta.ca; Fri, 21 Sep 2007 14:15:35 -0300 Date: Thu, 20 Sep 2007 14:20:36 -0700 From: John Baez To: categories@mta.ca Subject: categories: unital weak functor? Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 53 On Thu, Sep 20, 2007 at 07:03:07AM +1000, Stephen Lack wrote: > These are often called normal lax functors > (or normal morphisms of bicategories). I suggested "normalized", but "normal" is clearly better: you normalize something to make it normal. On a wholly different note - I hope people take a look at the new videos by the Catsters. They're using YouTube in an interesting new way: to explain monads, adjunctions and the like. For more info: http://golem.ph.utexas.edu/category/2007/09/the_catsters_latest_hit_adjunc.html Best, jb From rrosebru@mta.ca Fri Sep 21 14:21:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 21 Sep 2007 14:21:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYm6y-00017J-54 for categories-list@mta.ca; Fri, 21 Sep 2007 14:16:16 -0300 Date: Fri, 21 Sep 2007 01:23:42 +0200 From: Joachim Kock Subject: categories: Re: unital weak functor? To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 54 >Josh Nichols-Barrer wrote: > >>Is there a name for a weak functor between bicategories which takes >>identity 1-morphisms to identity 1-morphisms? The term 'normalised' (already mentioned in the replies) goes back at least to Grothendieck, SGA1, Exp VI: he calls a cleavage for a fibred category E -> B normalised if the cartesian lift of each identity arrow is an identity arrow -- this is exactly the condition for the correponding pseudo-functor B^op -> Cat to preserve identity arrows strictly. Joachim. From rrosebru@mta.ca Fri Sep 21 14:21:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 21 Sep 2007 14:21:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IYm7N-0001Ay-Hk for categories-list@mta.ca; Fri, 21 Sep 2007 14:16:41 -0300 To: LICS List From: Kreutzer + Schweikardt Subject: categories: LICS Newsletter 111 Date: Fri, 21 Sep 2007 15:21:30 +0200 (CEST) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 55 Newsletter 111 September 10, 2007 ******************************************************************* * Past issues of the newsletter are available at http://www.informatik.hu-berlin.de/lics/newsletters/ * Instructions for submitting an announcement to the newsletter can be found at http://www.informatik.hu-berlin.de/lics/newsletters/inst.html * To unsubscribe, send an email with "unsubscribe" in the subject line to lics@informatik.hu-berlin.de ******************************************************************* TABLE OF CONTENTS * BOOK ANNOUNCEMENTS Reactive Systems - Aceto, Ingolfsdottir, Larsen, and Srba * CONFERENCES AND WORKSHOPS LICS 2008 - Call for Workshop Proposals M4M-5 ICALP 2008 - Second Call for Workshop Proposals PODS 2008 - Call for Papers SAT 2008 - Call for Papers REACTIVE SYSTEMS: MODELLING, SPECIFICATION AND VERIFICATION by Luca Aceto, Anna Ingolfsdottir, Kim G. Larsen and Jiri Srba Cambridge University Press, ISBN-13: 9780521875462. * A textbook providing a balanced introduction to the theory and practice of concurrency for advanced undergraduates and graduate students. It describes various approaches for the modelling, specification and verification of reactive systems, their strengths and weaknesses, and when they are best used. The book has arisen from various courses taught in Iceland, Denmark and elsewhere, and is designed to give students a broad introduction to the area, with exercises throughout. * Preface; Part I. A Classic Theory of Reactive Systems: 1. Introduction; 2. The language CCS; 3. Behavioural equivalences; 4. Theory of fixed points and bisimulation equivalence; 5. Hennessy-Milner logic; 6. Hennessy-Milner logic with recursive definitions; 7. Modelling and analysis of mutual exclusion algorithms; Part II. A Theory of Real-Time Systems: 8. Introduction; 9. CCS with time delays; 10. Timed automata; 11. Timed behavioural equivalences; 12. Hennessy-Milner logic with time; 13. Modelling and analysis of Fischer's algorithm; Appendix; Bibliography; Index. * For more information, please visit: http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=3D9780521875462 http://www.cs.aau.dk/rsbook/ IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS 2008) Call for Workshop Proposals 24th=E2=80=9327th June 2008, Pittsburgh, Pennsylvania, USA * The Twenty-Third IEEE Symposium on Logic In Computer Science (LICS 2008) (see http://www2.informatik.hu-berlin.de/lics/lics08 ) will be held in Pittsburgh, Pennsylvania (USA) June 24--27, 2008. It will be colocated with CSF (IEEE Computer Security Foundations). * Possible dates for workshops are June 21-23 (i.e. up to three days before LICS). * Researchers and practitioners are invited to submit proposals for workshops on topics relating logic - broadly construed - to computer science or related fields. Typically, LICS workshops feature a number of invited speakers and a smaller number of contributed presentations. LICS workshops do not produce formal proceedings. However, in the past there have been special issues of journals based in part on certain LICS workshops. * Proposals should include: - A short scientific summary and justification of the proposed topic. This should include a discussion of the particular benefits of the topic to the LICS community. - A discussion of the proposed format and agenda. - The proposed duration, which is typically one to two days, and your preferred dates. (important!) - Let us know if you would like your workshop to be a joint workshop with CSF. - Procedures for selecting participants and papers. - Expected number of participants (this is important!) - Potential invited speakers. - Plans for dissemination (for example, special issues of journals). * Proposals are due Nov. 20, 2007 and should be submitted electronically to both of us: Adriana Compagnoni abc@cs.stevens.edu Philip Scott phil@site.uottawa.ca Workshops Chairs, LICS 2008 * Please specify if you wish to have a 1 or 2 day workshop. And your preferred dates. * Workshops will be chosen by a committee consisting of the LICS General Chair, LICS Workshop Chairs, LICS 2008 PC Chair and LICS 2008 Conference Chair. 5th WORKSHOP ON "METHODS FOR MODALITIES" (M4M-5) November 29-30, 2007, Cachan, France http://m4m.loria.fr/M4M5 * The workshop ``Methods for Modalities'' (M4M) aims to bring together researchers interested in developing algorithms, verification methods and tools based on modal logics. Here the term ``modal logics'' is conceived broadly, including temporal logic, description logic, guarded fragments, conditional logic, temporal and hybrid logic, etc. To stimulate interaction and transfer of expertise, M4M will feature a number of invited talks by leading scientists, research presentations aimed at highlighting new developments, and submissions of system demonstrations. We strongly encourage young researchers and students to submit papers and posters, especially for experimental and prototypical software tools which are related to modal logics. * Invited speakers: Ahmed Bouajjani (University of Paris 7), Patricia Bouyer (OUCL, Oxford - LSV, ENS Cachan), Balder ten Cate (University of Amsterdam), Koen Claessen (Chalmers University of Technology), Wiebe van der Hoek (University of Liverpool) * Important dates: Deadline for submissions: September 7th, 2007 Notification: October 10, 2007 Camera ready versions: November 5, 2007 Workshop dates: November 29-30, 2007 * Paper Submissions: Authors are invited to submit papers in the following three categories. - Regular papers up to 15 pages, describing original results, work in progress, or future directions of research. - System descriptions of up to 12 pages, describing new systems or significant upgrades of existing ones. - Presentation-only papers, describing work recently published or submitted (no page limit). These will not be included in the proceedings, but pre-prints or post-prints can be made available to participants. * Final versions of accepted papers will be published online in an Elsevier ENTCS volume. A preliminary version of the proceedings will also be available at the workshop. 35TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING Second Call for Affiliated Workshops ICALP 2008 Reykjavik - Iceland * Conference Dates: July 7-11, 2008 Affiliated Workshop Dates: July 6, 12, and 13, 2008 * Researchers and practitioners are invited to submit proposals for workshops on topics related to the conference tracks, namely: Algorithms, Automata, Complexity and Games (track A); Logic, Semantics, and Theory of Programming (track B); and Security and Cryptography Foundations (track C). * The purpose of the workshops is to provide participants with a forum for presenting and discussing novel ideas in a small and interactive atmosphere. * The main responsibility of organising a workshop goes to the chairperson of the workshop. The format of each workshop is determined by its organisers. * The workshop proposals will be selected by the ICALP organization, under advice of the ICALP PC chairs and EATCS. * Registrations for the workshops affiliated with ICALP 2008 will be handled by the ICALP organizing committee. * Proposals should include: - Name and duration (from half a day to two days) of the proposed workshop. - Preference for a pre- or post-conference workshop, if any. (Note, however, that the final choice of the day when the workshop will be held and other logistic aspects will be decided by the organizing committee.) - A short scientific summary of the topic, including a discussion on the relation with the ICALP topics. - A description of past versions of the workshop, if any, including dates, organisers, submission and acceptance counts, attendance. - Procedures for selecting participants and papers, and expected number of participants. - Plans for dissemination (for example, published proceedings or special issues of journals). * IMPORTANT DATES: October 31, 2007: Deadline for submitting workshop proposals November 21, 2007: Notification of acceptance * Workshop proposals must be submitted in plain text, PDF or Postscript format by e-mail to icalp08@ru.is * For further information consult the ICALP 2008 web site: http://www.ru.is/icalp08 or contact the ICALP 2008 Workshops Chairs: - Bjarni V. Halldorsson - MohammadReza Mousavi 27TH ACM SIGMOD-SIGACT-SIGART SYMPOSIUM ON PRINCIPLES OF DATABASE SYSTEMS PODS 2008 Call for Papers June 9-11, 2008, Vancouver, Canada http://www.sigmod08.org/ * The PODS symposium series, held in conjunction with the SIGMOD conference series, provides a premier annual forum for the communication of new advances in the theoretical foundation of database systems. For the 27th edition, original research papers providing new insights in the specification, design, or implementation of data management tools are called for. * Topics that fit the interests of the symposium include the following (as they pertain to databases): algorithms; complexity; computational model theory; concurrency; constraints; data exchange; data integration; data mining; data modeling; data on the Web; data streams; data warehouses; distributed databases; information retrieval; knowledge bases; logic; multimedia; physical design; privacy; quantitative approaches;query languages; query optimization; real-time data; recovery; scientific data; security; semantic Web; semi-structured data; spatial data; temporal data; transactions; updates; views; Web services; workflows; XML. * Important Dates: Abstracts submission due: 28 Nov 2007; Paper submission: 5 Dec 2007; Notification: 26 Feb 2008. 11TH INTERNATIONAL CONFERENCE ON THEORY AND APPLICATIONS OF SATISFIABILITY TESTING (SAT 2008) Call for Papers May 12 - 15, 2008 Guangzhou, P. R. China http://www.upb.de/cs/SAT08 * The International Conference on Theory and Applications of Satisfiability Testing is the primary annual meeting for researchers studying the propositional satisfiability problem (SAT). SA5DT08 is the eleventh SAT conference. SAT08 features the SAT Race, the Max-SAT Evaluation, and the QBFEVAL. * SCOPE Many hard combinatorial problems can be encoded into SAT. Therefore improvements on heuristics on the practical side, as well as theoretical insights into SAT apply to a large range of real-world problems. More specifically, many important practical verification problems can be rephrased as SAT problems. This applies to verification problems in hardware and software. Thus SAT is becoming one of the most important core technologies to verify secure and dependable systems. The topics of the conference span practical and theoretical research on SAT and its applications and include but are not limited to proof systems, proof complexity, search algorithms, heuristics, analysis of algorithms, hard instances, randomized formulae, problem encodings, industrial applications, solvers, simplifiers, tools, case studies and empirical results. SAT is interpreted in a rather broad sense: besides propositional satisfiability, it includes the domain of quantified boolean formulae (QBF), constraints programming techniques (CSP) for word-level problems and their propositional encoding and particularly satisfiability modulo theories (SMT). * SUBMISSION Submissions should contain original material and can either be regular research papers up to 14 pages or short papers up to 6 pages. Double submissions including submissions as short and long papers will be rejected. Submissions should use the Springer LNCS style. All appendices, tables, figures and the bibliography must fit into the page limit. Submissions deviating from these requirements may be rejected without review. All accepted papers including short papers will be published in the proceedings of the conference. The conference proceedings will be published within Springer LNCS series. The submission page is http://www.easychair.org/SAT2008. Papers have to be submitted electronically as PDF files. * IMPORTANT DATES January 11, 2008 Abstract Submission January 18, 2008 Paper Submission February 18, 2008 Author Notification February 25, 2008 Final Version * PROGRAM CHAIRS Hans Kleine B=C3=BCning, University of Paderborn, Germany Xishun Zhao, Sun Yat-Sen University, Guangzhou, P.R. China * Further information on SAT Race, Max-SAT Evaluation, and QBFEVAL will be published also on the SAT08 conference web page. From rrosebru@mta.ca Wed Sep 26 19:27:22 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 26 Sep 2007 19:27:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iaf9J-0000FT-AW for categories-list@mta.ca; Wed, 26 Sep 2007 19:14:29 -0300 Date: Wed, 26 Sep 2007 12:56:57 -0400 From: Walter Tholen MIME-Version: 1.0 To: categories@mta.ca Subject: categories: category position Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 56 York University CATEGORY THEORY AND ITS APPLICATIONS Applications are invited for one tenure-track appointment at the Assistant Professor level in the Department of Mathematics and Statistics to commence July 1, 2008. Candidates in the area of Category Theory and its applications to mathematics, computer science or physics will be considered. The successful candidate must have a Ph.D. in hand or near completion (expected in 2008), a proven record of independent research excellence, and superior teaching ability. The successful candidate must be eligible for prompt appointment to the Faculty of Graduate Studies. Preference will be given to candidates who can strengthen existing areas of present and ongoing research activity. Applications must be received by January 15, 2008. Applicants should send resumes and arrange for three signed letters of recommendation (one of which should address teaching) to be sent directly to: Pure Mathematics Search Committee Department of Mathematics and Statistics N520 Ross, York University 4700 Keele Street Toronto, Ontario Canada M3J 1P3 E-mail: puremath.recruit@mathstat.yorku.ca , Website: www.math.yorku.ca/Hiring All positions at York are subject to budgetary approval. York University is an Affirmative Action Employer. The Affirmative Action Program can be found on York's website at www.yorku.ca/acadjobs or a copy can be obtained by calling the affirmative action office at 416-736-5713. All qualified candidates are encouraged to apply; however, Canadian citizens and Permanent Residents will be given priority. From rrosebru@mta.ca Wed Sep 26 23:49:28 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 26 Sep 2007 23:49:28 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IajMA-00014k-Pg for categories-list@mta.ca; Wed, 26 Sep 2007 23:44:02 -0300 Message-ID: <46FAD6A1.8090408@cs.stanford.edu> Date: Wed, 26 Sep 2007 15:01:05 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: The division lattice as a category: is 0 prime? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 57 Has the division lattice been organized as a category somewhere in the literature in a way that accounts somehow for 0? A simplistic construction is [FinSet^N] denoting the finitary finite-set-valued functors from the discrete category N. ("Finitary" in the sense of being zero almost everywhere.) Interpreting N as indexing the primes and coproduct as numeric multiplication, product becomes gcd and the pushout of (a,b) over its gcd is its lcm. It is simplistic by virtue of omitting (numeric) 0, which is standardly placed at the top of the division lattice. Unlike the rest of the lattice 0 is not generated from the primes by finite coproducts, suggesting it needs to feature in some sort of basis for the complete lattice. The natural thing would be to remove all but the primes and 0 from the division lattice and then try to put them back finitarily. This makes the starting point the inverted flat CPO N^* where N consists of the primes and "bottom" * is now at the top, denoting 0. (That convention makes N_* the usual flat CPO of natural numbers.) If I'm not mistaken the completion of N^* under finite coproducts has as objects two copies of the natural numbers. Below * is [FinSet^N] understood as previously. Above * is FinSet, which was created from * by completion under coproducts. This amounts to FinSet + [FinSet^N] joined at the hip with a shared initial object (numeric 1) and a shared final object (numeric 0, or *). From the Yoneda standpoint the objects are functors from N_* (the usual CPO) to FinSet. The ones below are the functors that are 0 at * and cofinitely many elements of N. The ones above are 0 at N, with * unconstrained. Yoneda's hands are a bit tied here because we are only taking coproducts. Closing under finite colimits presumably frees up Yoneda to produce FinSet x [FinSet^N] = [FinSet^{N_*}]. This might come in handy when one wants a system of pairs of numbers (m,n) for which (m,n)+(m',n') = (m+m',nxn'). Is there some abstract nonsense reason why coproducts produced the sum (actually pushout over the initial and final objects) while colimits produced the product? I arrived at all this after Steve Vickers mentioned on the univalg mailing list that ring theorists define 0 to be a prime number because then they could define n to be prime just when the ring Z/nZ extends to a field. This got me to wondering how this could be reflected in the division lattice, which has 0 at the top without however being considered a prime. I personally am too old to believe that 0 is a prime, but I can see where a younger generation could be hoodwinked. Even with the above understanding however I don't see how 0 can be understood as just another ordinary prime, any more than bottom is just another ordinary number in N_*. Vaughan Pratt From rrosebru@mta.ca Thu Sep 27 10:44:47 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 27 Sep 2007 10:44:47 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IatZJ-0007O5-VP for categories-list@mta.ca; Thu, 27 Sep 2007 10:38:18 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: continuous lattices for analysts?? Date: Thu, 27 Sep 2007 12:27:10 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 58 As you will have gathered from my previous posting to "categories", over the past three years I have been applying Abstract Stone Duality to the foundations of (constructive) analysis. This means that I have been bringing not only ASD itself but categorical ideas more generally to a new audience. However, my new friends are not very familiar with some of the ideas that I normally take for granted. Two things in particular have turned out to be difficult. One of these, not surprisingly, is my use of LAMBDA CALCULUS to define open subspaces. I have found notational ways to sugar this pill, such as defining functions WITH arguments (as in most programming languages), so avoiding lambda-abstraction unless absolutely necessary, and simply writing "..." instead of the usual name "Gamma" for a context or list of parameters. I see this as part of the programme of relating formal logical notation to the idioms of vernacular of mainstream mathematics, as in Charles Wells' "Handbook of Mathematical Discourse", and sections 1.6 and 6.5 of my own book, "Practical Foundations of Mathematics", where I explain the phrase "there exists" and the usual manipulation of finite sets. However, there is another problem that is not simply a matter of unfamiliar notation. I had understood that the theory of CONTINUOUS LATTICES had grown out of half a dozen different disciplines (represented by the authors of the "Compendium"), and in particular that on of these had been the concept of SEMI-CONTINUITY in real analysis. I had expected to find at least a basic awareness of continuous lattices amongst analysts, but I was mistaken. However, it would not be appropriate to include an introduction to them in ASD, since it does not build directly on the standard theory of continuous lattices. Instead, it abstracts ideas from them (in particular the paper "Computably based locally compact spaces" has an abstract "way below" relation), and one of its basic principles is to hide Scott continuity in the foundations. I would like to be able to cite an introduction to continuous lattices that is written for (and ideally by) real analysts. So far, my enquiries amongst the experts on continuous lattices have drawn a blank, but maybe some analyst has had occasion to use them, or maybe teach a graduate course about them. To generalise the question, is there a good account of non-Hausdorff topology apart from those written for domain theory in theoretical computer science? My specific context is to rewrite Section 7 of "A lambda calculus for real analysis", which was presented at "Computability and Complexity in Analysis" in Kyoto in August 2005. Paul Taylor www.PaulTaylor.EU From rrosebru@mta.ca Thu Sep 27 18:17:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 27 Sep 2007 18:17:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ib0eg-0003f7-0r for categories-list@mta.ca; Thu, 27 Sep 2007 18:12:18 -0300 Date: Thu, 27 Sep 2007 10:40:05 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: The division lattice as a category: is 0 prime? 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: 59 Mike Barr's response to my > Even with the above understanding however I don't see how 0 can be > understood as just another ordinary prime, any more than bottom is just > another ordinary number in N_*. was "I don't see your problem." And now that I reflect on that comment, I don't see it myself. With the benefit of sleeping on the problem combined with Mike's prod, the moral for me is clear. The Fundamental Theorem of Arithmetic is incomplete as stated. It should read as follows. Every natural number partitions uniquely as a sum of 1's, and every positive integer factors uniquely as a product of primes. The constructive proof of the theorem exhibits this ostensibly two-part structure uniformly as the completion under coproducts of the inverted flat CPO N^*. This coproduct-complete category is naturally analyzed into two components, additive upstairs and multiplicative downstairs. The components share the initial and final objects of the category, with the former manifesting as 0 in the additive component and 1 in the multiplicative, and conversely for the latter. And that's why 0 is at the top of the division lattice. The reason it (qua 1) is at the bottom of the additive component and not the top (its default location in an arbitrary category with 1) is because it generates Set (or in this case FinSet). It makes no sense to consider 0 as a prime because there is no way to define things such that 0 factors uniquely. The role of the morphisms in N^* is to prevent 0 from being an atom in the completion, instead making it final in the inverted CPO, which coproducts preserve and colimits do not. Had we completed under colimits, by Yoneda the final object would have been the constantly 1 functor. The top * of N^* would then no longer be the final object of the completion, being the unit functor for *, namely 1 at * and 0 elsewhere. With either completion the primes are the other unit functors, but only with the coproduct completion is the final object a unit functor. Vaughan From rrosebru@mta.ca Thu Sep 27 18:17:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 27 Sep 2007 18:17:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ib0fI-0003ix-95 for categories-list@mta.ca; Thu, 27 Sep 2007 18:12:56 -0300 Date: Thu, 27 Sep 2007 13:31:48 -0700 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: continuous lattices for analysts?? 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: 60 > I would like to be able to cite an introduction to continuous lattices > that is written for (and ideally by) real analysts. So far, my > enquiries amongst the experts on continuous lattices have drawn a > blank, but maybe some analyst has had occasion to use them, or > maybe teach a graduate course about them. Define "real analyst." These range from the practical cowboys to the sensitive constructivists (though as Hollywood reminds us the intersection need not be empty). This distinction persists in computational analysis, with Blum-Shub-Smale representing the cowboys and Metropolis, Rota, Edalat, Escardo, Freyd, Leinster, etc. bringing up to date the descriptive set theory program started by Borel, Baire, and Lebesgue. The Compendium came out in 1980. Maybe to those of you on the right hand side of the Atlantic it might have seemed to be addressing computer scientists, but to most of us in the Western hemisphere (pace Wand, Tennent, and a couple of others) it looked like it was written for analysts. I doubt if you're going to find a treatment written *more* for analysts than the Compendium and its updates and successors. For your purposes its three downsides might be its length, its datedness (not so dated remarkably when you consider how new the subject was then and how much has been learnt since), and its relative inaccessibility (~$100 for second-hand copies in good condition, $50 for a solitary "acceptable" copy, ~$160 for the new books). The Wikipedia article on Lattices (order) has a brief introduction to continuous and algebraic lattices that might hold the fort---if two more sentences would do the trick add them yourself, no one will stop you. Then there's the longer article on Domain Theory. It's hard to imagine any analyst who's likely to be interested in abstract domain theory not being willing to tackle the domain theory article on its own merits, recognizing the intrinsically computational aspects of constructive analysis, at least as the computing professionals see it. The modern constructive analyst is going to have to merge the paradigms of analysis and computation in order to keep up with where computer scientists have been pushing the subject. Don't pander to the retards. Vaughan From rrosebru@mta.ca Fri Sep 28 11:13:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Sep 2007 11:13:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IbGRP-0000vp-K1 for categories-list@mta.ca; Fri, 28 Sep 2007 11:03:39 -0300 Date: Thu, 27 Sep 2007 16:36:34 -0700 From: Vaughan Pratt To: categories list Subject: categories: Re: The division lattice as a category: is 0 prime? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 61 Jeff Egger wrote: > And I thought that every generation since Dedekind, Krull and Noether > knew that divisibility lattices are (in the general case) a red herring > and that it is the lattice of ideals of a ring (or its opposite, if you > prefer) which is really important. > ... > although I don't really understand your motivation. Right, I should have been clearer about the motivation. I wanted to construct the division lattice abstractly from the primes in some finitary way, analogously to how one can construct the power set 2^X as the free upper semilattice generated by the singletons of X. Putting that in terms of ideals, I'd like to be able to form all the ideals of Z from just the prime ideals. I don't know much about ring theory so I could be confused about this, but I would have thought intersecting them could only get you the square-free ideals. Starting from the prime power ideals takes care of that but what's the trick for getting all the ideals from just the prime ideals? The category Div was my suggestion for that, but if there's a more standard approach in ring theory I'd be happy to use that instead (or at least be aware of it---Div is starting to grow on me). Now that I think of it, I suppose the standard completion must be the formation of finite subdirect products (aka sums?) of the quotients Z_p = Z/pZ over the prime ideals pZ. By including Z along with the Z's, that way you reconstruct Div with the lower part consisting of Z_n = Z/nZ and the upper part n.Z (if I understand the notation). That puts the ring structure of Z back into play however, which doesn't feel quite as "pure" as simply closing a flat inverted CPO under finite coproducts. > Perhaps the answer to your original > question is to take (finite-valued) sheaves on this space of primes, Right, that (by Yoneda) was the completion under finite colimits approach at the end of my 10:40 am message this morning, which didn't "work" in the sense of not being the minimal solution and not having an obviously pleasing structure either. Completion under finite coproducts was as small as I could make it, and initially I was miffed that there was still this junk above the division lattice that I was hoping would go away. But then I decided that rather than complicate the completion process to prevent 0 from sprouting sow's ears above it, I'd try to make a silk purse out of the ears. This ended up being the two-part Fundamental Theorem of Arithmetic via the single construction. With coproducts instead of colimits it's still sheaves but with the condition that if the stalk at * is nonempty then all the other stalks must be empty. I don't know what the abstract-nonsense name for that is. Vaughan From rrosebru@mta.ca Fri Sep 28 11:13:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Sep 2007 11:13:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IbGQH-0000nc-4J for categories-list@mta.ca; Fri, 28 Sep 2007 11:02:29 -0300 Date: Thu, 27 Sep 2007 17:59:41 -0400 (EDT) From: Jeff Egger Subject: categories: Re: The division lattice as a category: is 0 prime? 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: 62 --- Vaughan Pratt wrote: > I arrived at all this after Steve Vickers mentioned on the univalg > mailing list that ring theorists define 0 to be a prime number because > then they could define n to be prime just when the ring Z/nZ extends to > a field. =20 Um, well, for arbitrary ideals I in a commutative ring R, R/I "extends to= =20 a field" (or, in more common parlance, "is an integral domain") if and on= ly=20 if I is a prime ideal; hence the previous assertion can be simplified to=20 ring theorists define 0 to be a prime number because then they could define n to be prime just when nZ is a prime ideal.=20 which doesn't seem so unreasonable. =20 > This got me to wondering how this could be reflected in the > division lattice, which has 0 at the top without however being > considered a prime. I personally am too old to believe that 0 is a > prime, but I can see where a younger generation could be hoodwinked. And I thought that every generation since Dedekind, Krull and Noether knew that divisibility lattices are (in the general case) a red herring=20 and that it is the lattice of ideals of a ring (or its opposite, if you=20 prefer) which is really important. Surely, it makes sense to fix=20 terminology according to what does work in the general case. =20 > Even with the above understanding however I don't see how 0 can be > understood as just another ordinary prime, any more than bottom is just > another ordinary number in N_*. Although 0 can be a prime (depending on the ring under consideration),=20 it is plainly never "just another ordinary prime": there is a well-known topology on the set of prime ideals of a commutative ring which clearly distinguishes 0 from its fellows. Perhaps the answer to your original=20 question is to take (finite-valued) sheaves on this space of primes,=20 although I don't really understand your motivation. =20 Cheers, Jeff Egger. Get news delivered with the All new Yahoo! Mail. Enjoy RSS feeds r= ight on your Mail page. Start today at http://mrd.mail.yahoo.com/try_beta= ?.intl=3Dca From rrosebru@mta.ca Fri Sep 28 11:13:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Sep 2007 11:13:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IbGRy-000108-UG for categories-list@mta.ca; Fri, 28 Sep 2007 11:04:15 -0300 Date: Fri, 28 Sep 2007 11:36:13 +0200 (CEST) From: Peter Schuster To: Categories Subject: categories: 10th Asian Logic Conference, Kobe, Japan, 2008 (first announcement) 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: 63 The 10th Asian Logic Conference will be held at Kobe University, Kobe, Japan during September 1-6, 2008. Several short course lectures (four hours each), plenary invited talks (one hour each) and special sessions are planned apart from contributed talks. The Asian Logic Conference has occurred every three years in Asia-Pacific region since 1981, Singapore. The purpose of the conference is to facilitate interaction between researchers interested in mathematical logic, logic in computer science, and philosophical logic. It aims at promoting activities in mathematical logic in the Asia-Pacific so that logicians both from within Asia and elsewhere would get together and exchange information and ideas. Call for papers will begin in the coming winter, and the registration in an early spring, 2008. Please write down in your notebook as 'ALC, Kobe, Sep. 1-6, 2008'. Thank you, Toshiyasu Arai Graduate School of Engineering Kobe University Rokko-dai, Nada-ku, Kobe, 657-8501, Japan From rrosebru@mta.ca Fri Sep 28 11:13:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Sep 2007 11:13:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IbGQs-0000rz-KR for categories-list@mta.ca; Fri, 28 Sep 2007 11:03:06 -0300 Date: Thu, 27 Sep 2007 17:14:15 -0500 From: "Yemon Choi" To: categories@mta.ca Subject: categories: Re: continuous lattices for analysts?? 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: 64 On 27/09/2007, Vaughan Pratt wrote: > Don't pander to the retards. Speaking as a *nonconstructive* analyst and ``retard'', who *might* like to know more, I'm not sure how to take this... Most analysts I know come to accept ``better technology'' or ``a more correct perspective'' through use, not abuse. The advantage of topological over metric arguments in *some* contexts is what sells us on topology, not because the definition of a maximal filter gives us a warm glow... Didn't Aesop have something to say about the relative merits of shouting and cajoling? Retardedly, YC (off to read Wikipedia) -- Dr. Y. Choi 519 Machray Hall Department of Mathematics University of Manitoba Winnipeg. Manitoba Canada R3T 2N2 Tel: (204)-474-8734 From rrosebru@mta.ca Fri Sep 28 17:09:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 28 Sep 2007 17:09:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IbM4K-0007Zb-AE for categories-list@mta.ca; Fri, 28 Sep 2007 17:04:12 -0300 Date: Fri, 28 Sep 2007 08:07:26 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: continuous lattices for analysts?? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 65 > Didn't Aesop have something to say about the relative merits of > shouting and cajoling? Please pardon my French. I'd have used "laggard" if I'd thought of it in time, "retard" does push the wrong button in English. For the irreconcilably thin-skinned: better that those standing shivering by the warm pool jump in than that you should have to bring the water to them. Vaughan From rrosebru@mta.ca Sun Sep 30 20:29:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 30 Sep 2007 20:29:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ic7zB-0001Xo-Ax for categories-list@mta.ca; Sun, 30 Sep 2007 20:14:05 -0300 Date: Sat, 29 Sep 2007 10:49:33 -0400 (EDT) From: Bill Lawvere To: categories list Subject: categories: Re: The division lattice as a category: is 0 prime? 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: 66 Vaughn remarks ...........I would have thought intersecting them could only get you the square-free ideals........ Indeed, as Jeff points out, we learned from Kummer and Dedekind to replace elements by ideals, but we categorists have been late in providing a clear account of this transition and, in particular, of the reason why the result is not primarily a lattice, but a monoidal closed category with colimits. Below I will elaborate on the following three points: (1) The actual "ideal number" functor itself is clear enough (though never made explicit), but why should it exist? (2) The standard account of "why" is very categorical, but does not directly address the algebraic category of rings nor the geometry of intersection theory. (3) The universal algebraists have developed a tool that might be applied to the "why", but for some reason the universality is not often applied to algebra or to geometry. In more detail: (1) The Kummer functor I goes from rigs (or K-rigs, where K is given, e.g. Z or Q) to 2-rigs, where 2 is the 2-element rig in which 1+1=1. (Yes, the rig that launched ring theory is not itself a ring). The functorality, as well as the multiplication itself, depends on the set-theoretic operation of image. The principal ideal concept is a natural transformation from M to MI where M is the underlying multiplication. (2) A rig can serve (not only as functions on a scheme but) as an abstract general whose semantically corresponding concrete general is its category of modules, which is a monoidal closed category with colimits. This 2-functor can be composed with the functor to posets that extracts from the big category of modules just the submodules of the unit object. Again, the image operation must in general be applied to the result of tensoring two submodules (because of the lack of flatness). A monoidal poset with colimits is also a 2-rig. (3) Intersecting closed subspaces of a space may give only a shadow of a description of their clash (e.g. the clash of Africa & Europe produces a bulge i.e. the Alps). Although geometric figures are in general singular, a notion of closed subspace which refines the notion of mere subset provides a useful partial record. In terms of the rigs of variable quantity on the spaces there is a corresponding refinement: The distributive lattice of radical ideals is refined to the monoidal poset of all ideals. The ideal product under discussion is a key ingredient in a construction of unions of subspaces that takes into account the clashes. As it would be desirable geometrically to see even nonsingular figures as images of maps in the category of spaces itself, it would dually be desirable to see R/ab as the result of a construction on R/a and R/b within the category of K-rigs itself, without the detour (2) through modules. At least the case where K is a ring is indeed covered in principle by a construction called the "commutator" (a misleadingly particular "general terminology", ....groups are apparently not a typical algebraic category). That this construction does reduce to a certain concatenation of limits and colimits has been shown by categorists in terms of congruence relations. But the application to rings and ideals still remains to be done. Bill ************************************************************ F. William Lawvere, Professor emeritus Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************ On Thu, 27 Sep 2007, Vaughan Pratt wrote: > Jeff Egger wrote: >> And I thought that every generation since Dedekind, Krull and Noether >> knew that divisibility lattices are (in the general case) a red herring >> and that it is the lattice of ideals of a ring (or its opposite, if you >> prefer) which is really important. >> ... >> although I don't really understand your motivation. > > Right, I should have been clearer about the motivation. I wanted to > construct the division lattice abstractly from the primes in some > finitary way, analogously to how one can construct the power set 2^X as > the free upper semilattice generated by the singletons of X. Putting > that in terms of ideals, I'd like to be able to form all the ideals of Z > from just the prime ideals. I don't know much about ring theory so I > could be confused about this, but I would have thought intersecting them > could only get you the square-free ideals. Starting from the prime > power ideals takes care of that but what's the trick for getting all the > ideals from just the prime ideals? The category Div was my suggestion > for that, but if there's a more standard approach in ring theory I'd be > happy to use that instead (or at least be aware of it---Div is starting > to grow on me). > > Now that I think of it, I suppose the standard completion must be the > formation of finite subdirect products (aka sums?) of the quotients Z_p > = Z/pZ over the prime ideals pZ. By including Z along with the Z's, > that way you reconstruct Div with the lower part consisting of Z_n = > Z/nZ and the upper part n.Z (if I understand the notation). That puts > the ring structure of Z back into play however, which doesn't feel quite > as "pure" as simply closing a flat inverted CPO under finite coproducts. > >> Perhaps the answer to your original >> question is to take (finite-valued) sheaves on this space of primes, > > Right, that (by Yoneda) was the completion under finite colimits > approach at the end of my 10:40 am message this morning, which didn't > "work" in the sense of not being the minimal solution and not having an > obviously pleasing structure either. Completion under finite coproducts > was as small as I could make it, and initially I was miffed that there > was still this junk above the division lattice that I was hoping would > go away. But then I decided that rather than complicate the completion > process to prevent 0 from sprouting sow's ears above it, I'd try to make > a silk purse out of the ears. This ended up being the two-part > Fundamental Theorem of Arithmetic via the single construction. > > With coproducts instead of colimits it's still sheaves but with the > condition that if the stalk at * is nonempty then all the other stalks > must be empty. I don't know what the abstract-nonsense name for that is. > > Vaughan > > > >