From MAILER-DAEMON Wed Dec 5 19:35:20 2007 Date: 05 Dec 2007 19:35:20 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1196897720@mta.ca> X-IMAP: 1185998020 0000000026 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 Wed Aug 1 16:48:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Aug 2007 16:48:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IGK3O-0003EB-UU for categories-list@mta.ca; Wed, 01 Aug 2007 16:40:19 -0300 Date: Wed, 1 Aug 2007 12:45:06 +0100 (BST) Subject: categories: last PhD thesis from Bangor From: tporter@informatics.bangor.ac.uk To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=3Diso-8859-1 Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 1 The very last PhD thesis from Bangor (for the foreseeable future) is now available on the website. In it Richard Lewis looks at the problem of the interpretation of the formal maps to a crossed module introduced by Porter and Turaev and using a simplicial analogue of etale spaces gets a representation in terms of locally constant stacks. The links is http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/0= 7/cathom07.html#07.09 All for now, Tim PS. The abstract follows Stacks and formal maps of crossed modules Abstract: If X is a topological space then there is an equivalence between the category \pi_1(X)-Set, of actions of the fundamental group of X on sets, and the category of covering spaces on X. Moreover the latter is also equivalent to the category of locally constant sheaves on X. Grothendieck has conjectured that this should be the 'n=3D1' case of a result which is true for all n, and it is the 'n=3D2' case we look at in this thesis. The desired generalisation should replace actions of the group \pi_1(X) (which is an algebraic model for the 1-type of X) by actions of a crossed module (i.e., by an algebraic model for the 2-type) on groupoids; 'locally constant sheaves of sets' by 'locally constant stacks of groupoids'; and 'covering space' by a locally trivial object whose fibres are groupoids. This last object we handle using the machinery of simplicial fibre bundles (twisted Cartesian products) and formal maps, building a simplicial object, Z(\lambda), where the fibre is now a (nerve of) a groupoid. To interpret Z(\lambda) as a stack, we show that just as sheaves on X are equivalent to etale spaces, we can define a notion of 2-etale space corresponding to stacks and show that from Z(\lambda) we can construct a locally constant stack on X. --=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 Cymru, Bangor. Nid yw Prifysgol Cymru, 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 Cymru, 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 the University of Wales, Bangor. The University of Wales, Bangor 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 University of Wales, Bangor Finance Office. www.bangor.ac.uk (SEECS) From rrosebru@mta.ca Wed Aug 1 16:48:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 01 Aug 2007 16:48:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IGK4G-0003Hu-7C for categories-list@mta.ca; Wed, 01 Aug 2007 16:41:12 -0300 Date: Wed, 01 Aug 2007 10:26:15 +0200 (CEST) To: categories@mta.ca Subject: categories: Answers to: definition of parsimony From: Axel Rossberg 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: 2 Dear List Members, two days ago I posted a message asking whether there is a formal definition of parsimony for fundamental scientific theories, perhaps using category theory. Here is a short summary of answers I received: Ralph Wojtowicz recommended to have a look at Part D in Volume II of Johnstone's "Sketches of an Elephant", which as he wrote contains discussions of constructions involving translations between formal systems and their semantic categories. As an example for parsimony in category theory, Eduardo Ochs suggested the have a look at the relationships between set theory, local set theories, and elementary toposes. The paper by F. Wiedijk, "Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics", Journal of Applied Logic 4, 622-645, 2006 ps.gz pdf dvi via http://www.cs.ru.nl/~freek/pubs/index.html also was recommended. Many thanks to all respondents and also to Vaughan Pratt for his refreshing critical remarks! The answer to the question appears to be more difficult than I had expected. As often in philosophy, this may be to a good extent due to difficulties in explaining what the question is. One important point which I failed to clarify is the difference between fundamental mathematical theories and fundamental scientific theories. Fundamental scientific theories may, I think, assume the fundamental mathematical theory to be given. They should not describe all possible structures but, on the contrary, pick from the mathematically given structures those that are physically realized. This difference might also put the problem of parsimony for scientific theories into a different light. But I'm not sure, of course. So long, Axel From rrosebru@mta.ca Sun Aug 5 19:45:48 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 05 Aug 2007 19:45:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IHofa-0004Rf-D0 for categories-list@mta.ca; Sun, 05 Aug 2007 19:33:54 -0300 Date: Sat, 4 Aug 2007 12:14:13 +0100 (BST) From: Bob Coecke To: categories@mta.ca Subject: categories: Coalgebraic and Categorical Quantum Logic in Oxford 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 The programs for: * COALGEBRAIC LOGIC, August 10-11, Oxford, UK * CATEGORICAL QUANTUM LOGIC, August 11-12, Oxford, UK are now available at: * http://se10.comlab.ox.ac.uk:8080/FOCS/COQL_en.html Sincerely, Bob Coecke. From rrosebru@mta.ca Mon Aug 6 18:45:25 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 06 Aug 2007 18:45:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IIAHR-0005Nh-H7 for categories-list@mta.ca; Mon, 06 Aug 2007 18:38:25 -0300 Date: Mon, 6 Aug 2007 13:00:25 GMT From: Oege.de.Moor@comlab.ox.ac.uk Subject: categories: PEPM 2008 To: 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: 4 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 Aug 7 09:26:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Aug 2007 09:26:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IINyV-0003Uw-FR for categories-list@mta.ca; Tue, 07 Aug 2007 09:15:47 -0300 Date: Tue, 7 Aug 2007 12:19:29 +0100 (BST) Subject: categories: Editorial Board of 'K-Theory' has resigned From: tporter@informatics.bangor.ac.uk To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 5 Dear Categories, Please note this open letter. Thanks, Tim Porter ------------------------------------- OPEN LETTER from the Board of Editors of the Journal of K-theory Dear fellow mathematicians, The Editorial Board of 'K-Theory' has resigned. A new journal titled 'Journal of K-theory' has been formed, with essentially the same Board of Editors. The members are A.Bak, P.Balmer, S.J.Bloch, G.E.Carlsson, A.Connes, E.Friedlander, M.Hopkins, B.Kahn, M.Karoubi, G.G.Kasparov, A.S. Merkurjev, A.Neeman, T.Porter, D.Quillen, J.Rosenberg, A.A.Suslin, G.Tang, B.Totaro, V.Voevodsky, C.Weibel, and Guoliang Yu. The new journal is to be distributed by Cambridge University Press. The price is 380 British pounds, which is significantly less than half that of the old journal. Publication will begin in January 2008. We ask for your continued support, in particular at the current time. Your submissions are welcome and may be sent to any of the editors. Board of Editors Journal of K-theory From rrosebru@mta.ca Tue Aug 7 22:37:27 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 07 Aug 2007 22:37:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IIaOF-0007mv-14 for categories-list@mta.ca; Tue, 07 Aug 2007 22:31:11 -0300 Subject: categories: Re: Editorial Board of 'K-Theory' has resigned From: Tom Leinster To: categories@mta.ca Content-Type: text/plain Date: Tue, 07 Aug 2007 14:06:08 +0100 Mime-Version: 1.0 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 Congratulations, Tim. This is something to celebrate. Mathematicians and librarians the world over should be grateful to you and the other editors: you're both saving us money and bringing another journal back under the full control of academics. Of course, the pattern of naming now established (Topology becomes Journal of Topology, K-Theory becomes Journal of K-Theory) presents a problem: what's JPAA going to do? Tom On Tue, 2007-08-07 at 12:19 +0100, tporter@informatics.bangor.ac.uk wrote: > Dear Categories, > > Please note this open letter. > > Thanks, > > Tim Porter > > ------------------------------------- > > > OPEN LETTER from the Board of Editors of the Journal of K-theory > > Dear fellow mathematicians, > > The Editorial Board of 'K-Theory' has resigned. A new journal titled > 'Journal of K-theory' has been formed, with essentially the same Board > of Editors. The members are A.Bak, P.Balmer, S.J.Bloch, G.E.Carlsson, > A.Connes, E.Friedlander, M.Hopkins, B.Kahn, M.Karoubi, G.G.Kasparov, A.S. > Merkurjev, A.Neeman, T.Porter, D.Quillen, J.Rosenberg, A.A.Suslin, G.Tang, > B.Totaro, V.Voevodsky, C.Weibel, and Guoliang Yu. > > The new journal is to be distributed by Cambridge University Press. The > price is 380 British pounds, which is significantly less than half that of > the old journal. Publication will begin in January 2008. We ask for your > continued support, in particular at the current time. Your submissions > are welcome and may be sent to any of the editors. > > Board of Editors > Journal of K-theory > > > -- Tom Leinster From rrosebru@mta.ca Wed Aug 8 10:19:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Aug 2007 10:19:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IIlIs-0000QG-6R for categories-list@mta.ca; Wed, 08 Aug 2007 10:10:22 -0300 Date: Tue, 7 Aug 2007 22:06:55 -0400 (EDT) From: Michael Barr To: categories@mta.ca Subject: categories: Re: Editorial Board of 'K-Theory' has resigned 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: 7 Of course, it will become Pure and Applied Algebra. Not that I much care. It started as the flagship journal of category theory, but has long since escaped our grasp. At any rate, it is about time, the editors resigned. Michael On Tue, 7 Aug 2007, Tom Leinster wrote: > Congratulations, Tim. This is something to celebrate. Mathematicians > and librarians the world over should be grateful to you and the other > editors: you're both saving us money and bringing another journal back > under the full control of academics. > > Of course, the pattern of naming now established (Topology becomes > Journal of Topology, K-Theory becomes Journal of K-Theory) presents a > problem: what's JPAA going to do? > > Tom > > > On Tue, 2007-08-07 at 12:19 +0100, tporter@informatics.bangor.ac.uk > wrote: > > Dear Categories, > > > > Please note this open letter. > > > > Thanks, > > > > Tim Porter > > > > ------------------------------------- > > > > > > OPEN LETTER from the Board of Editors of the Journal of K-theory > > > > Dear fellow mathematicians, > > > > The Editorial Board of 'K-Theory' has resigned. A new journal titled > > 'Journal of K-theory' has been formed, with essentially the same Board > > of Editors. The members are A.Bak, P.Balmer, S.J.Bloch, G.E.Carlsson, > > A.Connes, E.Friedlander, M.Hopkins, B.Kahn, M.Karoubi, G.G.Kasparov, A.S. > > Merkurjev, A.Neeman, T.Porter, D.Quillen, J.Rosenberg, A.A.Suslin, G.Tang, > > B.Totaro, V.Voevodsky, C.Weibel, and Guoliang Yu. > > > > The new journal is to be distributed by Cambridge University Press. The > > price is 380 British pounds, which is significantly less than half that of > > the old journal. Publication will begin in January 2008. We ask for your > > continued support, in particular at the current time. Your submissions > > are welcome and may be sent to any of the editors. > > > > Board of Editors > > Journal of K-theory > > > > > > > From rrosebru@mta.ca Wed Aug 8 10:19:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 08 Aug 2007 10:19:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IIlJp-0000ZR-AN for categories-list@mta.ca; Wed, 08 Aug 2007 10:11:21 -0300 Date: Wed, 08 Aug 2007 08:03:41 -0400 From: jim stasheff Subject: categories: Re: Editorial Board of 'K-Theory' has resigned To: categories@mta.ca MIME-version: 1.0 (Apple Message framework v752.3) Content-type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-transfer-encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 8 PAA? jim On Aug 7, 2007, at 9:06 AM, Tom Leinster wrote: > Congratulations, Tim. This is something to celebrate. Mathematicians > and librarians the world over should be grateful to you and the other > editors: you're both saving us money and bringing another journal back > under the full control of academics. > > Of course, the pattern of naming now established (Topology becomes > Journal of Topology, K-Theory becomes Journal of K-Theory) presents a > problem: what's JPAA going to do? > > Tom > From rrosebru@mta.ca Tue Aug 14 23:21:03 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Aug 2007 23:21:03 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IL8O1-0004LC-Mc for categories-list@mta.ca; Tue, 14 Aug 2007 23:13:29 -0300 Date: Tue, 14 Aug 2007 18:36:34 +0200 From: Luigi Santocanale MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Preprint: Derived semidistributive lattices Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 9 Dear Category Theorists, a preprint of my Nice-PSSL talk on "Derived semidistributive lattices" is now available through arXiv: http://arxiv.org/abs/0708.1695 Any of your comment will be very much appreciated. Best wishes, Luigi --=20 Luigi Santocanale LIF/CMI Marseille T=E9l: 04 91 11 35 74 http://www.cmi.univ-mrs.fr/~lsantoca/ Fax: 04 91 11 36 02 =09 From rrosebru@mta.ca Tue Aug 14 23:21:03 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 14 Aug 2007 23:21:03 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IL8Q8-0004VM-8X for categories-list@mta.ca; Tue, 14 Aug 2007 23:15:40 -0300 From: "Ronnie Brown" To: Subject: categories: Topology and Groupoids , by Ronald Brown Date: Tue, 14 Aug 2007 22:23:09 +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: 10 This is now available printed in the UK through amazon.co.uk RRP = =A315.99 but currently =A310.49=20 The e-version may be bought using credit cards or other ways via Kagi at = https://store.kagi.com/cgi-bin/store.cgi?storeID=3D6FEPD_LIVE for =A35. This version has some colour and hyperref.=20 Ronnie Brown=20 ww.bangor.ac.uk/r.brown From rrosebru@mta.ca Wed Aug 15 23:52:36 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Aug 2007 23:52:36 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ILVJR-0001Ov-T3 for categories-list@mta.ca; Wed, 15 Aug 2007 23:42:17 -0300 Date: Wed, 15 Aug 2007 13:35:08 -0400 (EDT) From: James Stasheff To: categories@mta.ca Subject: categories: subdivision 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: 11 Is there a `barycentric' subdivision operator Sd on categories such that with N = nerve SdN = NSd ? and how many other notions of Sd are there? jim Jim Stasheff jds@math.upenn.edu Home page: www.math.unc.edu/Faculty/jds As of July 1, 2002, I am Professor Emeritus at UNC and I will be visiting U Penn but for hard copy the relevant address is: 146 Woodland Dr Lansdale PA 19446 (215)822-6707 From rrosebru@mta.ca Sun Aug 19 03:07:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 19 Aug 2007 03:07:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IMdny-0003sB-Mz for categories-list@mta.ca; Sun, 19 Aug 2007 02:58:30 -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: The Dedekind Reals in Abstract Stone Duality Date: Sat, 18 Aug 2007 16:17:31 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 12 The Dedekind Reals in Abstract Stone Duality www.PaulTaylor.EU/ASD Andrej Bauer is a co-author of this paper, but not of this posting about it. We construct something that was supposed to be impossible: a model of (topology containing) the real line that satisfies the Heine--Borel theorem (the closed interval is compact in the "finite open sub-cover" sense) but in which all terms are computable. This fails in the "obvious" model of recursive analysis, ie the subset of real numbers that are representable by programs, roughly because we can enumerate them, and cover them with intervals of length m.2^-k for any chosen m (say 1/4). Then recursive [0,1] is covered by total length 2m (=1/2), but no finite subcover is enough. The clearest account of this counterexample that I have seen is in Bridges & Richman, "Varieties of Constructive Mathematics", LMS Lecture Notes 97, CUP, 1987, Section 3.4. The reason why the Heine--Borel theorem holds in ASD is that, like locale theory and formal topology, it is an account of topology based on open subspaces and not points. But, unlike them, ASD is presented as a lambda calculus that makes it LOOK like a point-based theory, and therefore shows up exactly what the difference is. This paper appeared in the proceedings of "Computability and Complexity in Analysis" (Kyoto, August 2005), but since then it has been completely rewritten, and almost every lemma has been re-thought -- so it's worth re-reading. It now provides the best available introduction to the construction of topology in ASD, and also compares the construction and properties of the real line in ASD with those in locale theory, formal topology and constructive analysis. SS 11-13 of the paper also consider real arithmetic and the relationship between Dedekind completeness and the Archimedean axiom. It has just been submitted to a journal, so we would still welcome comments. Below I shall explain the category theoretic idea behind this construction. For further discussion of constructive analysis in ASD, you should see SS 1 & 15 of the paper itself, and "A Lambda Calculus for Real Analysis", which was also presented at CCA 2005, and has been revised a lot, but is not yet ready to go to a journal. --- Before I describe this work, I feel that the time has come to make a disclaimer: ABSTRACT STONE DUALITY IS *NOT* ESCARDO'S SYNTHETIC TOPOLOGY Martin Escardo's paper "Synthetic topology of data types and classical spaces" (ENTCS 87 (2004) 21--156 or www.cs.bham.ac.uk/~mhe) is a very interesting collection of material drawn from the literature on the boundaries of topology and recursion theory. If it had been written by somebody ten years earlier, as it could and perhaps should have been, it would have provided a very useful background to my work on ASD. However, the main use to which I would have put an earlier version of that paper would have been to say WHAT ASD IS NOT. In particular, Martin describes the failure of compactness of Cantor space and of the closed real interval in recursive analysis. The theory that he then discusses is a form of "relative computability", which bolts recursion theory on to classical topology. The best developed form of this approach is Klaus Weihrauch's "Type Two Effectivity", on which Vasco Brattka (www.cca-net.de/vasco) is the most productive researcher. In ASD, Cantor space and [0,1] are compact, but this is not achieved by bolting recursion theory on to classical topology. ASD is a direct axiomatisation of the ideas of general topology, not based on set theory or recursion theory. The features that are common to Martin's work and mine are that - the topology on a space X is seen as the exponential Sigma^X, where Sigma is the Sierpinski space; I would attribute this to Dana Scott's "Continuous Lattices" (LNM 274 (1972) 97-136); and - compactness of a space K is captured by a "universal quantifier" all_K : Sigma^K -> Sigma; I picked up this idea from 1970s locale theory and categorical logic. I took these ideas for granted a long time ago, they appeared in the first ASD paper in 2000, and I have since developed them into a complete axiomatisation of computably based locally compact spaces. If Martin had developed his own calculus for topology based on similar ideas, it would have been possible to write (the paper and) this posting as an ordinary comparison of scientific theories, just as we do with locale theory, etc: "the Heine--Borel theorem holds in ASD but not in Escardo's calculus because of such-and-such". Unfortunately, there is no "Escardo's calculus". Indeed, I recently put it to him that he could eliminate the friction between us either by working with me on ASD, or by proposing some other axiom system, but he declined. He doesn't want to take a view on whether Heine--Borel is true or false. Therefore no scientific comparison is possible, and I have to make this disclaimer instead. However, the reason why I consider it necessary to make such a statement is not actually because of Martin himself. His research is not my business, and this is not an issue of priority or plagiarism. My complaint is against those who go around saying that Martin presents MY work better than I do myself, whereas in fact the fundamental ideas of ASD (see below) get no mention in his account. I encourage Martin to clarify (on "categories") both which specific results he claims as HIS OWN, and more generally what the message of "Synthetic Topology" is, as distinct from ASD. On the other hand, I would ask other people to read my papers properly. ---- THE MONADIC PRINCIPLE As a research programme, ASD is an example of a maxim due to Paul Dirac that Ronnie Brown quoted on "categories" on 8 June 2007: >> One should allow oneself to be led in the direction which the >> mathematics suggests... one must follow up a mathematical idea >> and see what its consequences are, even though one gets led to >> a domain which is completely foreign to what one started >> with.... Mathematics can lead us in a direction we would not >> take if we only followed up physical ideas by themselves. The mathematical idea that I followed in this way was a standard categorical one, and the following account is intended for categorists. It explains how the application of this idea led "directly" to important fundamental principles in real analysis. The idea (in 1993) was that, not only is the topology on X given by the exponential Sigma^X as above, but also the self-adjunction Sigma^(-) -| Sigma^(-) is monadic. I still don't know where I picked up the idea that adjunctions "ought" to be monadic. Maybe it was just that lots of interesting categories turn up that way. The main inspiration was Bob Pare's "Colimits in Topoi" (Bull AMS 80 (1974) 556), where he shows that any elementary topos has this property, with Sigma=Omega. A famous theorem of Jon Beck characterises monadic adjunctions F -| U by two clauses: - U reflects invertibility, and - U preserves U-split coequalisers. ---- SOBRIETY In the context of ASD, the first clause is equivalent to saying that every object X is SOBER, ie it is the equaliser of the diagram eta_Sigma^2 X Sigma^ Sigma^X eta_X Sigma^X ------------> Sigma X >-----> Sigma ------------> Sigma Sigma^2 eta_X When we apply this principle to the two most important ground types of mathematics (in the context of suitable extra structure), we find that - sobriety for N is equivalent to definition by description (Sober Spaces and Continuations, SS 9-10) - sobriety for R is equivalent to Dedekind completeness (Dedekind Reals in ASD, S 14). Sobriety is slightly stronger than the property of repleteness that Martin Hyland introduced in synthetic domain theory in 1991. Essentially, all of these ideas say HOW LOGIC IMPACTS ON NUMERICAL COMPUTATION. They can be formulated as "introduction rules" in a type-theoretic style in which the elimination rules are given by equality, arithmetic order or lambda-application. To put this another way, sobriety completes the square of types of operations: results: numerical logical arguments: numerical: arithmetic (+ - x /) relations (= < > !=) logical: sobriety (cut descr) logic (and or exists all) There are normalisation theorems that say that any numerical term built from these four sets of operations is equivalent to a purely logical one, with numerical variables and constants on the inside and a single sobriety operator on the outside. Computation is then by logic programming. (See "Interval analysis without intervals".) ----- MONADICITY Turning to the more difficult half of Beck's theorem, this yields the Heine--Borel theorem, and also compactness of Cantor space, 2^N. In fact it was already known that (Dedekind completeness and) the Heine--Borel theorem are consequences of the view that the algebra of open sets is fundamental in topology, and not the set of points. Fourman & Hyland showed this for locales (LNM 753 (1979) 280-301) and Negri for formal topology (LNCS 1158 (1996)). Because of the monadic principle, ASD is also an "open sets" account of topology. We say that a subspace i : X >---> Y is SIGMA-SPLIT if there's a map I : Sigma^X >---> Sigma^Y such that Sigma^i . I = id. Monadicity provides for the existence of Sigma-split subspaces. The representation of reals as Dedekind cuts defines a subspace i : R >---> Sigma^Q x Sigma^Q The crucial observation (in S 2 of "Dedekind reals in ASD") is that this subspace is Sigma-split, classically, using the fact that [0,1] is compact. (This result is already significant in interval analysis: it says that interval computations can be made arbitarily accurate by making the interval arguments narrow enough.) The idempotent composite E = I . Sigma^i can be expressed entirely in terms of the rationals, not the reals. This is the key idea of the construction of R in ASD (S 8), and the same formula E turns up again in the definitions of the quantifiers that make [0,1] compact and R overt (SS 9-10): all x:[d,u].phi x = I phi (lambda x.x Sigma, ie a term of one higher type, surely you should be willing to accept an inter-definable term of another higher type (I:Sigma^X >---> Sigma^Y) as the categorical explanation of the Heine--Borel theorem. This term also exists in the "relative computability" models: what ASD does is to abstract the computable, topological ideas away from proofs that rely on non-topological, non-computable set-theoretic hypotheses. Sigma-split subspaces also provide a simple categorical characterisation of local compactness: X is locally compact iff it can be expressed as a Sigma-split subspace of Sigma^N. ("Computably based locally compact spaces".) ---- THE MONADIC LAMBDA CALCULUS What sets ASD apart from locale theory and formal topology is that it LOOKS as if it's based on "points", whereas locale theory looks like lattice theory, and formal topology works with basic open sets and coverage relations. How does ASD manage this? We now have 40 years experience of the interpretation of lambda calculi (predicate calculus, type theories, etc) in categories. We know, in particular, that the introduction, elimination, beta and eta rules for a logical connective correspond to various features of an adjunction between functors (see S 7.2 of my book). Now that we have seen this correspondence for numerous logical operations, we can turn it around: given an adjunction between functors, we might be able to design a formal calculus that corresponds to it. That is what "Subspaces in ASD" does with the monadic property. It allows one to work in a natural way with the given structure of the category (in particular N and the lattice Sigma), but then to introduce Sigma-split subspaces given certain "subspace data" called a NUCLEUS. This word was appropriated from locale theory because nuclei serve the same essential purpose in both cases, but the definitions are slightly different. It is important to understand that the "subspaces" in this calculus are not subsets. Putting this back in categorical language, SS 4--6 of that paper constructs the monadic category by formally adjoining NEW Sigma-split equalisers. ---- THE EUCLIDEAN & PHOA PRINCIPLES I want to stick to one categorical idea in this posting, but I should also point out that the thing that made monadicity look like topology rather than just category theory or lambda calculus was the Phoa principle. The "Frobenius laws" for open and proper maps follow from this. (These things are not in "Synthetic Topology" either.) This says that open and closed subspaces of X are in bijection, not because of set-theoretic complementation, but because they each correspond bijectively to maps X --> Sigma. Whereas locale theory has infinite joins but finite meets, and there are double negations all over the place in intuitionistic analysis, ASD has a very strong duality between open and closed ideas. This is applied to connectedness and (approximate forms of) the intermediate value theorem in "Lambda calculus for real analysis". The "miracle" that led to my working full time on ASD was the relationship between the Euclidean principle and dominances, and the fact that overt discrete spaces form a pretopos (See "Geometric and higher order logic"). --- DIFFICULTIES WITH THE ASD PROGRAMME I want to cross-examine the charge that I cannot present my own work. It has turned out that the nucleus that we use to construct R is the simplest non-trivial example, and the one that is best motivated by other considerations. It is a pity that I did not know this ten years ago, and that I was so reluctant to do real analysis. Instead, I spent a lot of time developing more general locally compact spaces, and also overt discrete ones. However, the construction of the Dedekind reals involves topological and combinatorial arguments that we can now present with confidence ONLY BECAUSE I had previously done a thorough study of them. Working with ASD nuclei is extremely difficult. For each one, several sections of a paper are needed simply to show that some bizarre formula satisfies another bizarre equation. The proofs are long and intricate because the axioms are very weak and the techniques are novel. The reason for this difficulty is that I am working with monads and Beck's theorem. Let me point out some history. Although this result appears in many textbooks (including mine), Jon Beck never wrote it up himself. He might have taught us about non-Abelian cohomology, and what "exactness" means for general algebras, except that his thesis advisers told him that homological algebra had to be done with Abelian groups. In that subject, there is a black art of finding contraction maps. Beautiful though monadic adjunctions are, they don't compose. Altogether, Beck's theorem was a very awkward beast long before I had even heard of category theory. I feel that, with the monadic lambda calculus, I have begun to tame this beast. (Moggi's calculus did so too, but that worked with the Kleisli category, not the Eilenberg--Moore one that is crucial to Heine--Borel.) Whereas locale theory, formal topology, topos theory and algebraic geometry stuck to the abstract algebra, ASD turns it into topological notation. Constructing new spaces with nuclei may still be a black art, for the reasons that I have mentioned, but the continuous functions between them are handled very naturally. ---- BEYOND LOCAL COMPACTNESS The awkwardness of working with (the monad and) nuclei is directly related to the difficulty in generalising them beyond locally compact spaces. The point is that we want general equalisers and function spaces without sacrificing the principle of using the algebra of open subspaces. Actually, I have plenty of proposals for more general AXIOMS, and for theorems that we might prove with them. The problem with Beck's theorem lies in the restricted form of the second clause. There are several interesting cases of monads where the forgetful functor U : A=Algebras --> C=Carriers PRESERVES ALL REFLEXIVE COEQUALISERS. - Pare's theorem, for any topos C; - Richard Wood's A=CCD, C=CCC^op; - A=Frm, C=Dcpo; - C=Set, A= algebras for any single-sorted finitary algebraic theory. Unfortunately, I have no idea how to construct such a category for topology in the ASD formulation. There are plenty of cartesian closed supercategories of topological spaces, but they are point-based. Reinhold Heckmann constructed a CCC in which the category of locales is reflective (MSCS 16 (2006) 231--253), but this relies on the Axiom of Collection, and I don't think that it yields what I want either. Another possibility is a sheaf topos over the effective topos, which provides a model of ASD that has all exponentials and finite limits. However, whilst it satisfies the Heine--Borel theorem, there will be higher-type analogues of the failure in recursive analysis. Maybe we should look for models that just satisfy specific instances of this property. The underlying principle seems to be this: given a parallel pair (u,v) of homomorphisms of algebras and a function f with f.u = f.v, under what circumstances does f factor through a homomorphism w with w.u = w.v? Heine--Borel, monadicity, injectivity of Sigma, and the Beck--Chevalley condition for open and proper maps seem to be examples of this principle. Can we find models based on traditional topology, locale theory, formal topology or recursion theory that satisfy it in more cases? ---- CONCLUSION Synthetic Topology interprets ideas of general topology in cartesian closed categories and lambda calculus. Abstract Stone Duality shows that, in order to make the topology work PROPERLY (with the Heine--Borel theorem etc), we need to postulate additional higher type lambda terms associated with subspaces. I have just stated what I think the challenge is regarding the construction of more general models. Separate from this are other challenges in real analysis and computation. I have taken the intellectual and personal risks and done much of the groundwork to demonstrate the viability of ASD as an approach to constructive topology and analysis. It seems to me that these challenges are worthy of the attention of the best categorical minds. However, it will only be possible to achieve these things if my colleagues are willing to make a professional study and assessment of what has already been done in ASD. Paul Taylor pt07 @ PaulTaylor.EU www.PaulTaylor.EU/ASD From rrosebru@mta.ca Fri Aug 24 00:03:17 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 24 Aug 2007 00:03:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IOPJ3-0004PL-Hm for categories-list@mta.ca; Thu, 23 Aug 2007 23:53:53 -0300 From: Gaucher Philippe Subject: categories: preprint : "Globular realization and cubical underlying homotopy type of time flow of process algebra" Date: Thu, 23 Aug 2007 18:43:27 +0200 MIME-Version: 1.0 To: categories@mta.ca 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: 13 Dear all, Here is a new preprint. Best regards. pg. Title: Globular realization and cubical underlying homotopy type of time flow of process algebra Abstract: We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying homotopy type is naturally isomorphic to the homotopy type of the standard cubical complex associated with the precubical set. Comments: 31 pages URL: http://www.pps.jussieu.fr/~gaucher/prepubli.html From rrosebru@mta.ca Sat Aug 25 15:04:15 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Aug 2007 15:04:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IOzpr-000730-HY for categories-list@mta.ca; Sat, 25 Aug 2007 14:54:11 -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: subdivision Date: Sat, 25 Aug 2007 17:27:49 +1000 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 14 Dear Jim The answer is No, I believe, but the only thinking I have done about the question is as follows. In his paper on adjoint functors, Kan constructed a category %A for each category A. He used this to construct what we now call Kan extensions. It provides the domain of a diagram whose limit gives the end of a functor T : A^op x A --> X. At Bowdoin College in 1969, Mac Lane called %A the Kan subdivision category. It is a bit like barycentric subdivision in that each arrow f (edge) of A becomes an object (vertex) [f] of %A; the only non-identity arrows of %A are formally adjoined as in the situation [a] --> [f] <-- [b] where f : a --> b and a and b are identified with their identity arrows. There are not many composable pairs in %A so it seems that N%A is not the barycenric subdivision of the nerve of A. At this point I dug out my old copy of Kan's 1957 paper "On c.s.s. complexes" on which I scribbled some notes back in the late 1960s. If S is the category of finite sets and P : S --> Ord is the covariant powerset functor into ordered sets, we can define a functor D : S --> Simp into the category of simplicial sets (= css complexes) by (Ds)_q = Ord([q] , Ps). Kan's functor "Delta prime" is the restriction of D to the (topologists') simplicial category. Then Sd : Simp --> Simp is the extension along the Yoneda embedding of "Delta prime" to a colimit preserving functor. This yields the formula Sd(X)_q = coend^[n] X_n x Ord([q] , P[n]), but I see no way of using this when X is the nerve of a category A. Maybe there is more chance in the case of groupoids. (The left adjoint to Sd is Kan's functor Ex which starts a simplicial set on its way to becoming a Kan complex.) What made you suspect the existence of such a subdivision of categories? Ross On 16/08/2007, at 3:35 AM, James Stasheff wrote: > Is there a `barycentric' subdivision operator Sd on categories > such that with N = nerve > SdN = NSd > ? From rrosebru@mta.ca Sun Aug 26 11:59:25 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 26 Aug 2007 11:59:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IPJUf-0000WC-4h for categories-list@mta.ca; Sun, 26 Aug 2007 11:53:37 -0300 To: categories@mta.ca Subject: categories: Ph.D. scholarships at the IT University of Copenhagen From: Lars Birkedal Date: Sun, 26 Aug 2007 09:40:15 +0200 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: 15 A number of Ph.D. scholarships are available at the IT University of Copenhagen, including some in the areas of the Programming, Logic and Semantics (PLS) Group with research in programming languages, automated reasoning, logical frameworks, type theory, semantics, category theory, domain theory, distributed and mobile computing, business processes, concurrency theory, electronic voting. Please let potential students know. Deadline for application is October 22. See http://www1.itu.dk/sw66047.asp for the official announcement. Best wishes, Lars Birkedal Head of the PLS group. From rrosebru@mta.ca Mon Aug 27 09:52:36 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Aug 2007 09:52:36 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IPdxS-0001Nw-LD for categories-list@mta.ca; Mon, 27 Aug 2007 09:44:42 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Transfer-Encoding: quoted-printable From: Marco Grandis Subject: categories: Preprint: Cubical cospans and higher cobordisms Date: Mon, 27 Aug 2007 11:58:00 +0200 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 16 The following preprint is available from my web page: M. Grandis Cubical cospans and higher cobordisms (Cospans in Algebraic Topology, =20= III) Dip. Mat. Univ. Genova, Preprint 559 (2007). http://www.dima.unige.it/~grandis/wCub3.pdf http://www.dima.unige.it/~grandis/wCub3.ps Abstract. After two papers on weak cubical categories and collarable =20 cospans, respectively, we put things together and construct a weak =20 cubical category of cubical collared cospans of topological spaces. =20 We also build a second structure, called a quasi cubical category, =20 formed of arbitrary cubical cospans concatenated by homotopy =20 pushouts. This structure, simpler but weaker, has lax identities. It =20 contains a similar framework for cobordisms of manifolds with corners =20= and could therefore be the basis to extend the study of TQFT's of =20 Part II to higher cubical degree. The previous two papers of the same series are available at: http://www.tac.mta.ca/tac/volumes/18/12/18-12.dvi (ps, pdf) http://www.dima.unige.it/~grandis/wCub2.pdf Marco Grandis Dipartimento di Matematica Universit=E0 di Genova Via Dodecaneso, 35 16146 Genova Italy e-mail: grandis@dima.unige.it tel: +39 010 353 6805 http://www.dima.unige.it/~grandis/= From rrosebru@mta.ca Mon Aug 27 10:00:21 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Aug 2007 10:00:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IPeBE-0002rU-D6 for categories-list@mta.ca; Mon, 27 Aug 2007 09:58:56 -0300 Date: Mon, 27 Aug 2007 02:58:42 +0100 (BST) Subject: categories: Teaching Category Theory From: "Tom Leinster" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: 8bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 17 Dear all, Glasgow is just now introducing a Masters-level mathematics programme, and I'm teaching the Category Theory course. I'm looking for suggestions on a particular aspect of teaching it. It's a question of "size". Most of the times I've taught category theory previously were at Cambridge, where the students are exposed to ZFC-style set theory as undergraduates. Every year there'd be a few people who'd really worry about the set-theoretic validity of category theory: "doesn't Russell's paradox forbid a category of sets?", etc. I'd tell them, essentially, not to think about it; one can make a distinction between "small" and "large" collections, and experience shows that this suffices. Not a profound answer, but there you are. At Glasgow I'm going to have the opposite problem. Undergraduates here do no set theory of any kind. So, for instance, there's no reason why they should have heard of ZFC, or that there are collections "too big to be sets". Be careful what you wish for: after years of telling Cambridge students to forget their set theory, I now have students with no set theory to forget. And the question I'm having trouble answering is this: what do I need to tell them about sets? I can't tell them nothing, as far as I can see. For instance, I want them to know that the category of groups has "all" limits; but of course, Grp doesn't really have all limits, only small limits, so they'll need to know what "small" means. Later, I'll want to teach the Adjoint Functor Theorems. A rough and ready solution would be to tell them that there is a distinction between "small" and "large" collections, otherwise known as "sets" and "proper classes". This would necessitate giving them an example of a large collection, and I guess the obvious choice is the class in Russell's Paradox. But then I'd have to tell them that this is exactly the kind of thing that they shouldn't be thinking about! It's hardly satisfactory. There's probably a better solution involving an axiomatization of the category of sets (along the lines of the Lawvere-Rosebrugh book), or at least a listing of some its properties. I have two difficulties here. One - which readers of the list may be able to help me with - is that I haven't figured out how this would work in practice: for instance, how it would feed into the statement above on the completeness of Grp. Does anyone have experience of this? The other is that I haven't got room to be too radical, as the syllabus is already set (categories, functors, transformations; adjunctions, representables, limits; monads and/or monoidal categories). In a way this is an ideal situation: a classful of minds innocent of ZFC, able to come at set theory in a completely fresh way. I'd very much appreciate suggestions on how best to use this freedom. Tom From rrosebru@mta.ca Tue Aug 28 12:34:57 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Aug 2007 12:34:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IQ2wN-0001sc-H0 for categories-list@mta.ca; Tue, 28 Aug 2007 12:25:15 -0300 Date: Tue, 28 Aug 2007 14:57:43 +0200 From: "zoran skoda" To: categories@mta.ca Subject: categories: Categories in geometry MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 18 Dear colleagues, the webpage of the conference Categories in geometry and in mathematical physics, Split, Croatia, Sep 24(23)-28, 2007 is at http://www.irb.hr/korisnici/zskoda/catconf.html Questions should be addressed via email zskoda (AT) irb.hr Zoran Skoda From rrosebru@mta.ca Tue Aug 28 12:34:57 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Aug 2007 12:34:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IQ2xM-00021j-4d for categories-list@mta.ca; Tue, 28 Aug 2007 12:26:16 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable From: Francois Lamarche Subject: categories: PSSL86: Getting nearer Date: Tue, 28 Aug 2007 15:05:11 +0200 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 19 Less than two weeks are left before the 86th Peripatetic Seminar on =20 Sheaves and Logic, http://www.loria.fr/~lamarche/psslHomeEN.html to be held at the Institut Elie Cartan in Nancy on September 8-9. A list of participants is now available at http://www.loria.fr/~lamarche/listPart.html It is incomplete: some people who have told me they were coming have =20 not registered yet. As you can see we still can accomodate a small =20 number of talks. 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 about 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. Looking forward to seeing you in Nancy Fran=E7ois Lamarche From rrosebru@mta.ca Tue Aug 28 12:34:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 28 Aug 2007 12:34:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IQ2vc-0001kk-6J for categories-list@mta.ca; Tue, 28 Aug 2007 12:24:28 -0300 Date: Mon, 27 Aug 2007 18:04:59 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Teaching Category Theory 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: 20 > A rough and ready solution would be to tell them that there is a > distinction between "small" and "large" collections, otherwise known as > "sets" and "proper classes". This would necessitate giving them an > example of a large collection, and I guess the obvious choice is the class > in Russell's Paradox. [...] > > There's probably a better solution involving an axiomatization of the > category of sets (along the lines of the Lawvere-Rosebrugh book), or at > least a listing of some its properties. The question is more when than whether to bring up this distinction. Although the Lawvere-Rosebrugh book doesn't define "large" it does define "small relative to Set" in the antepenultimate paragraph of the book (p.250), namely as "can be parameterized by an object of Set". Near the midpoint of the book (p.130) is the statement of Cantor's theorem X < 2^X with the consequence that Set cannot be parameterized by any of its objects (and hence by the definition on p.250 cannot be small relative to itself). In contrast Borceux in his 3-volume series gets the size issue out of the way on pages 1-4 of Volume 1. CTWM is in between, addressing it on p.22 after covering categories, natural transformations, monics, epis, and zeros. One benefit of getting the distinction out of the way near the beginning is that the students won't feel so mystified when they run across it while reading other category-relevant material (as the better students will). The combinatorics of sets (n^m functions from a set of m elements to a set of n elements, etc., which they definitely should know) in no way prepares one for the possibility of an object larger than any set, for which Cantor's theorem is very helpful. Of the above three positionings, I like CTWM's best: early on, yet not so early as to exaggerate its importance relative to the fundamental concepts of CT. Not to say that CTWM starts out ideally. Spending three pages defining "metacategory" and then defining "category" as "any interpretation of the category axioms within set theory" is impenetrably idiosyncratic for students used to more conventional introductions in their other pure maths courses. Once past the size issue Borceux is much more conventional and direct, except for the relatively mild criticism that his definition of "category" is actually the definition of "locally small category." But that's not at all the stumbling block to understanding that "metacategory" presents, in fact if anything it is helpful not to be distracted at the outset by the prospect of large homobjects. While on the topic of size of homobjects, what drawbacks are there to regarding both the objects and homobjects of any n-category as all lying within the n-th Grothendieck universe for a suitable hierarchy U_1 < U_2 < ... of such (with U_0 = 1)? Although admittedly idiosyncratic, it seems very natural to account for large homobjects in a category C with the explanation that C is really a 2-category, whether or not one is making use of the 2-cells. I can see a methodological objection, namely that there is no logical connection between size and existence of n-cells for a given n. Does it create any actual difficulty somewhere? Vaughan From rrosebru@mta.ca Wed Aug 29 10:02:00 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 29 Aug 2007 10:02:00 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IQN5b-0004gB-Bg for categories-list@mta.ca; Wed, 29 Aug 2007 09:56:07 -0300 Date: Wed, 29 Aug 2007 13:35:34 +0100 (BST) Subject: categories: Maxwell Institute Colloquium on Khovanov Homology From: paul@ma.hw.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: 21 Maxwell Institute Colloquium on Khovanov Homology Friday, 16th November, 10AM - 5.30PM ICMS, 14 India St, Edinburgh, UK As part of the series of Maxwell Institute colloquia we will be holding a one day meeting on Khovanov homology in November. The speakers are M. Khovanov (Columbia), M. Mackaay (Algarve), I. Smith (Cambridge), C. Stroppel (Glasgow/Princeton) and B. Webster (Berkeley/Princeton). The meeting is supported by the Maxwell Institute and the Glasgow Mathematical Journal Trust. We have some very limited funds available to help cover expenses of UK participants. Please email khovanovhomology@googlemail.com if you wish to attend, as space is limited. More details can be found at http://www.ma.hw.ac.uk/~paul/MIC. I. Gordon, C. Stroppel and P. Turner From rrosebru@mta.ca Thu Aug 30 17:00:15 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Aug 2007 17:00:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IQq1m-0005jw-LH for categories-list@mta.ca; Thu, 30 Aug 2007 16:50:06 -0300 Date: Thu, 30 Aug 2007 13:50:38 -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: O X-Status: X-Keywords: X-UID: 22 --- Jeff Egger wrote: > Date: Thu, 30 Aug 2007 13:48:36 -0400 (EDT) > From: Jeff Egger > Subject: Re: categories: Teaching Category Theory > To: Tom Leinster >=20 > Dear Tom, >=20 > I find that the set/class distinction is much less compelling than the=20 > type/collection distinction, so my initial reaction is that one should > develop a kind of "naive type theory" to replace "naive set theory" > ---but I don't know to what extent it is possible to do this in a=20 > pedagogically sound fashion. =20 >=20 > [Googling "naive type theory" yields some interesting-looking articles,= =20 > but I haven't really had time to look at any of them in anything > approaching a serious fashion.] >=20 > The central tenet of NTT should be the intuition that one can't compare= =20 > apples and oranges. In particular, you can't ask whether two things=20 > are equal unless they were already "of the same type", which is to=20 > say that they were chosen from the same set to begin with. =20 >=20 > [Interestingly, there exist better motivating examples than "apples=20 > and oranges". Does the speed of light equal the charge of a positron,=20 > for example? Of course one could say that the answer is yes if we=20 > measure the speeds in light-seconds per second and charges in=20 > elementary charges, or we could say that the answer is no if we use=20 > more conventional units such as km/h and coulombs. But if we try to > conceptualise physical quantities as real entities independent of a=20 > choice of unit of measurement, then we recognise the question itself > as flawed.] >=20 > Of course, elementhood should also not be a global predicate, for=20 > otherwise we could subvert the non-existence of a global equality=20 > predicate by asserting two things to be equal if they are equal=20 > in every type to which they both belong. =20 >=20 > The non-existence of a global elementhood predicate renders the=20 > extensionality axiom of conventional set theory meaningless.=20 > This, in turn, calls into question whether equality of types is a=20 > meaningful predicate. But the existence of a type of types would=20 > entail the existence of such a predicate, and thus we are led to=20 > a situation where the non-existence of a type of all types can be=20 > regarded as a feature, not as a bug. =20 >=20 > You see what I really have in mind is not so much topos theory=20 > (which you might have suspected at first), but FOLDS. [But NTT=20 > should be set up in such a way that elementary topos theory=20 > becomes a (or even, the) natural result of attempting to formalise=20 > one's naive intuitions about types. For example, one can talk about=20 > (product- and power-)type constructors in a naive way...I think. =20 > Ideally, I would hope that naturality could be adequately described=20 > in terms of polymorphic lambda-calculus---but even I wouldn't suggest=20 > springing that on an unsuspecting first-term graduate student.] =20 >=20 > Here is another helpful intuition for students: a set/class/type/ > collection should not be thought of as a "glass box", but rather=20 > as a black box with a button: when you push the button it gives=20 > you, not an element of the set, but a little receipt bearing the=20 > name of an element of the set. [Riders of the Montreal metro=20 > system may recognise the boxes from which one obtains bus transfers,=20 > which held a strange fascination over me when I was a child.] > For arbitrary collections, it is possible that an element have more=20 > than one name---and hence, when you ask for two elements, you may=20 > in fact receive two names of the same element, _and_ be left none=20 > the wiser for it. A type is (naively) a collection for which every=20 > element has a unique name. =20 >=20 > Now using NTT/FOLDS as a basis for category theory does restrict one=20 > to dealing with locally small categories (if, one regards types as=20 > necessarily "smaller" than arbitrary collections---which is not as=20 > easily justifiable as it might seem), but I would argue that's not=20 > such a great loss. [In my experience, non-category-theorists, when=20 > asked to provide a definition of category, almost uniformly supply=20 > (what amounts to) the definition of an enriched category, in the case > V=3DSet---which I find quite intriguing.] It also destroys the notion=20 > of skeletal category, which is probably a good thing too. >=20 > I hope this helps---I was originally planning to write a lot more > (and might still do so). >=20 > Cheers, > Jeff. >=20 >=20 >=20 From rrosebru@mta.ca Fri Aug 31 09:37:21 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Aug 2007 09:37:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IR5do-0003KV-1R for categories-list@mta.ca; Fri, 31 Aug 2007 09:30:24 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: categories@mta.ca Content-Transfer-Encoding: 7bit From: Steve Vickers Subject: Re: categories: Teaching Category Theory Date: Fri, 31 Aug 2007 10:55:19 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 23 Dear Tom, Here is a rationale for a "cop-out" answer. When I taught categories at Imperial I had 10 lectures in a Computer Science department, so there was a limit to what I could do. For the adjoint functor theorem, I just taught the easy direction - that a (let's say) left adjoint preserves all colimits that exist. That's already an extremely useful thing to know. For the converse (a functor from a cocomplete category is a left adjoint if ...), one can already prove it for posets as a special case. I didn't go any further because (a) I didn't have time, and (b) - like you - I didn't know how to explain the set/class issues. But in retrospect I'm completely happy with that because the result relies on classical reasoning principles. In practice there is a lot to be said for constructing a right adjoint explicitly by other means. Checking that colimits are preserved then becomes a prudent check before you spend time looking for an adjoint. You also mention the issue of the category of groups having "all" limits. It seems to me that if you show them how infinite products have the universal property, and how then other limits can be constructed, then you'll have shown them the main thing they need to know. I would keep the set/class issue as a secret aside for those who have seen it before. Regards, Steve Vickers. On 27 Aug 2007, at 02:58, Tom Leinster wrote: > Dear all, > > Glasgow is just now introducing a Masters-level mathematics > programme, and > I'm teaching the Category Theory course. I'm looking for > suggestions on a > particular aspect of teaching it. > > It's a question of "size". Most of the times I've taught category > theory > previously were at Cambridge, where the students are exposed to ZFC- > style > set theory as undergraduates. Every year there'd be a few people > who'd > really worry about the set-theoretic validity of category theory: > "doesn't > Russell's paradox forbid a category of sets?", etc. I'd tell them, > essentially, not to think about it; one can make a distinction between > "small" and "large" collections, and experience shows that this > suffices. > Not a profound answer, but there you are. > > At Glasgow I'm going to have the opposite problem. Undergraduates > here do > no set theory of any kind. So, for instance, there's no reason why > they > should have heard of ZFC, or that there are collections "too big to be > sets". Be careful what you wish for: after years of telling Cambridge > students to forget their set theory, I now have students with no set > theory to forget. And the question I'm having trouble answering is > this: > what do I need to tell them about sets? > > I can't tell them nothing, as far as I can see. For instance, I > want them > to know that the category of groups has "all" limits; but of > course, Grp > doesn't really have all limits, only small limits, so they'll need > to know > what "small" means. Later, I'll want to teach the Adjoint Functor > Theorems. > > A rough and ready solution would be to tell them that there is a > distinction between "small" and "large" collections, otherwise > known as > "sets" and "proper classes". This would necessitate giving them an > example of a large collection, and I guess the obvious choice is > the class > in Russell's Paradox. But then I'd have to tell them that this is > exactly > the kind of thing that they shouldn't be thinking about! It's hardly > satisfactory. > > There's probably a better solution involving an axiomatization of the > category of sets (along the lines of the Lawvere-Rosebrugh book), > or at > least a listing of some its properties. I have two difficulties here. > One - which readers of the list may be able to help me with - is > that I > haven't figured out how this would work in practice: for instance, > how it > would feed into the statement above on the completeness of Grp. Does > anyone have experience of this? The other is that I haven't got > room to > be too radical, as the syllabus is already set (categories, functors, > transformations; adjunctions, representables, limits; monads and/or > monoidal categories). > > In a way this is an ideal situation: a classful of minds innocent > of ZFC, > able to come at set theory in a completely fresh way. I'd very much > appreciate suggestions on how best to use this freedom. > > Tom > > > > > > > From rrosebru@mta.ca Fri Aug 31 17:02:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Aug 2007 17:02:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IRCXp-00005R-Mu for categories-list@mta.ca; Fri, 31 Aug 2007 16:52:41 -0300 Date: Fri, 31 Aug 2007 09:34:51 -0400 (EDT) From: Jeff Egger Subject: categories: Re: Teaching Category Theory To: categories@mta.ca In-Reply-To: <200708311017.17603.spitters@cs.ru.nl> 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: 24 --- Bas Spitters wrote: > > > You see what I really have in mind is not so much topos theory > > > (which you might have suspected at first), but FOLDS.=20 >=20 > Could you give me a link to more information about FOLDS? Sorry for not explaining! FOLDS is an acronym for First Order Logic=20 with Dependent Sorts, with which I knew that Tom is familiar (having=20 discussed its pros and cons with him back when we were both students). > Google was not very helpful. Googling the whole phrase does produce satisfying results,=20 but to save you the effort, I can point you (all) towards http://www.math.mcgill.ca/makkai/folds/ Cheers, Jeff. Ask a question on any topic and get answers from real people. Go to= Yahoo! Answers and share what you know at http://ca.answers.yahoo.com From rrosebru@mta.ca Fri Aug 31 17:02:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Aug 2007 17:02:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IRCYb-00009V-Oy for categories-list@mta.ca; Fri, 31 Aug 2007 16:53:29 -0300 Date: Fri, 31 Aug 2007 09:37:06 -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: O X-Status: X-Keywords: X-UID: 25 --- Peter LeFanu Lumsdaine wrote: > >> [In my experience, non-category-theorists, when asked to > >> provide a definition of category, almost uniformly supply (what amou= nts > >> to) the definition of an enriched category, in the case V=3DSet---wh= ich I > >> find quite intriguing.]=20 >=20 > Surely the intriguing thing here is not (as I understand you to be > suggesting) the set-centricity that they're imposing, but rather that t= hey're > not imposing it as far as usual?=20 Actually, what I find intriguing is that it is the definition of=20 enriched category which seems to have priority over the definition of internal category. There are, I suppose, historical reasons for=20 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,=20 which obviously isn't that long from a "historical" perspective) the majority of category theorists tend to adopt the internal=20 category style of definition (of category) as more primitive. =20 The issue at stake may seem minor: do we think of a class of arrows=20 (which can later be partitioned into homsets), or do we think of the=20 homsets first (and take their disjoint union later)? But perhaps=20 the fact that one group of people prefers one approach and everyone else the other is symptomatic of a psychological divide?=20 It's also worth noting, perhaps, how flukey it is that in the case=20 V=3DSet, V-internal and small V-enriched categories happen to coincide. Consider V=3DCat, for example. Or, note how different the requirements=20 on V are, for V-internal and V-enriched categories to be defined.=20 >When asked to define pretty much any > algebraic gadget, most mathematicians will define a model of that algeb= raic > gadget in Set (see e.g. en.wikipedia.org/wiki/Group_%28mathematics%29 )= . It is true that one would expect set-theoretic conservatives to deal with small categories (~internal categories in the case V=3DSet), and more fle= xible mathematicians to use arbitrary large categories (~internal categories, w= here=20 V is a category of "large sets", or classes). This only re-inforces the=20 points made above. Cheers, Jeff. From rrosebru@mta.ca Fri Aug 31 17:02:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 31 Aug 2007 17:02:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IRCZL-0000CP-A2 for categories-list@mta.ca; Fri, 31 Aug 2007 16:54:15 -0300 Mime-Version: 1.0 (Apple Message framework v752.3) Content-Transfer-Encoding: 7bit Message-Id: <91F61AA1-C252-4B41-9FE4-BAF9540C4AC3@pps.jussieu.fr> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: categories@mta.ca From: Pierre-Louis Curien Subject: categories: Journees Jean-Yves Girard Date: Fri, 31 Aug 2007 16:27:45 +0200 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 26 This is the last announcement for the **** Journees Jean-Yves Girard **** http://www-lipn.univ-paris13.fr/jyg60/index-fr.php 10-12 September, IHP, Paris The final programme is now available on the site. On-line (free but obligaotry) registration is open (on the site) until Tuesday, September 5. Best regards, Pierre-Louis Curien