From MAILER-DAEMON Sat Jan 24 13:12:45 2009 Date: 24 Jan 2009 13:12:45 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1232817165@mta.ca> X-IMAP: 1229380461 0000000140 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From rrosebru@mta.ca Mon Sep 1 13:38:59 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Sep 2008 13:38:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KaC4T-0005TR-J5 for categories-list@mta.ca; Mon, 01 Sep 2008 13:16:05 -0300 From: "George Janelidze" To: Subject: categories: Re: More CT2008 information will be in September Date: Mon, 1 Sep 2008 13:52:26 +0200 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: RO X-Status: X-Keywords: X-UID: 1 I apologize again: I wanted to say CT2009 of course (not CT2008), and many thanks to those who noticed that misprint and wrote to me - and I hope its meaning is that CT2009 will be as nice as CT2008! George Janelidze ----- Original Message ----- From: "George Janelidze" To: Sent: Monday, September 01, 2008 1:35 AM Subject: categories: More CT2008 information will be in September > Dear Colleagues, > > This is just to say, that there will more information on CT2008 in September > (since I have not sent in August, as I promised before). I apologize for the > delay. > > George Janelidze From rrosebru@mta.ca Tue Sep 2 13:43:22 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Sep 2008 13:43:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KaYuZ-00062n-6E for categories-list@mta.ca; Tue, 02 Sep 2008 13:39:23 -0300 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Subject: categories: logics for model checking Date: Tue, 2 Sep 2008 10:24:13 -0400 From: "Wojtowicz, Ralph" To: Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 3 I am trying to understand model-checking logics such as=20 linear temporal logic (LTL), computation tree logic (CTL), and probabilistic extensions such as probabilistic computation tree logic (PCTL). These logics seem to be useful in model- checking because of correspondences between recursive=20 definitions of formulas and recursive algorithms for computing them. Syntactic inference and any semantics other than the few (transition systems, DTMCs, MDPs, etc.) for=20 which the logics were developed seem not to be of interest,=20 however. Translation theorems such as D1.4.7, D1.4.12, and D4.3.13 of Johnstone's "Elephant" are not applied, as far as=20 I have found. Here are some initial thoughts based on my very limited=20 knowledge of categorical logic. The temporal logic operators=20 "next", "eventually", and "always" may, for example,=20 be formulated using a propositional signature. Let A be a set=20 whose elements are thought of as state labels. The set of=20 atomic propositions is {a_n | a in A, n in N}. The following=20 are Horn formulas: Next^n[a] =3D a_n Always^k[a] =3D a_0 \wedge ... \wedge a_k T^k[a, b] =3D a_0 \wedge ... \wedge a_{k-1} \wedge b_k The following are coherent formulas: Until^k[a, b] =3D T^0[a, b] \vee ... \vee T^k[a, b] Eventually^k[a] =3D a_0 \vee ... \vee a_k The following are geometric formulas: Until[a, b] =3D \vee_i T^i[a, b] Eventually[a] =3D \vee_i a_i The following is infinitary: Always[a] =3D \wedge_i a_i. Alternatively, these notions can be formulated as a=20 first-order theory with basic sorts "States" and "Paths". =20 It has function symbols start : Paths -> States shift : Paths -> Paths and a set A of of relation symbols=20 a |-> States. PCTL can be formulated as a \tau-theory. It has basic sorts=20 States and Paths as above as well as the function symbols=20 start and shift. It also has function symbols=20 Prob_{ P(States)=20 (similarly for \leq, >, and \geq) for 0 \leq p\leq 1 and Cost_{ P(S)=20 (similarly for \leq, >, and \geq) for 0 \leq c. There are certainly other presentations of these theories. =20 Finally, here are a few questions. Has anyone studied the model-checking logics using the tools of categorical logic? What interactions exist between bisimulation/probabilistic=20 bisimulation and the logics? What role could categorical logic have in developing model- checking algorithms? How can the usual semantics of these theories (e.g., DTMC=20 semantics for PCTL) be viewed categorically (e.g., in the=20 topos Set by interpreting the sorts simply as sets of states and paths)? Any other thoughts, suggestions, or references would be=20 appreciated. References: Arnold. "Finite Transition Systems". Prentice-Hall. 1994. Johnstone. "Sketches of an Elephant: A Topos Theory Compendium". =20 Oxford University Press. 2002. Rutten, Kwiatkowska, Norman, and Parker. "Mathematical Techniques for Analyzing Concurrent and Probabilistic Systems". AMS. 2004. Thanks, Ralph Wojtowicz Metron, Inc. 11911 Freedom Drive, Suite 800 Reston, VA USA wojtowicz@metsci.com From rrosebru@mta.ca Wed Sep 3 11:42:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 03 Sep 2008 11:42:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KatVT-0007iQ-6w for categories-list@mta.ca; Wed, 03 Sep 2008 11:38:51 -0300 Content-Type: text/plain; charset=US-ASCII; format=flowed To: categories@mta.ca Subject: categories: Domains IX [Final Call for Registration] Date: Tue, 2 Sep 2008 18:12:05 +0100 From: Bernhard Reus Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 4 Registration for the International Workshop ------------------ DOMAINS IX ------------------ on domain theory, denotational semantics and applications, taking place from September 22nd - 24th at the University of Sussex is about to close now. Registration forms must be received by Friday, September 5th, 6am BST. More information on the workshop and how to register can be found at: http://www.informatics.sussex.ac.uk/events/domains9/index.htm Regards, Bernhard From rrosebru@mta.ca Wed Sep 3 11:42:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 03 Sep 2008 11:42:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KatWd-00000N-8J for categories-list@mta.ca; Wed, 03 Sep 2008 11:40:03 -0300 Date: Tue, 2 Sep 2008 15:00:39 -0700 From: John Baez To: categories Subject: categories: Re: KT Chen's smooth CCC Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 5 Bill Lawvere wrote: >By urging the study of the good geometrical ideas and constructions of >Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier, >Steenrod, I am of course not advocating the preferential resurrection of >the particular categories they tentatively devised to contain the >constructions. I chose Chen's framework when Urs Schreiber and I were doing some work in mathematical physics and we needed a "convenient category" of smooth spaces. I decided to choose one that was easy to explain to people brainwashed by the "default paradigm", in which spaces are sets equipped with extra structure. Later I realized I needed to write a paper establishing some properties of Chen's framework. By doing that I guess I'm guilty of reinforcing the default paradigm, and for that I apologize. If I understand correctly, one can actually separate the objections to continuing to develop Chen's theory of "differentiable spaces" into two layers. Let me remind everyone of Chen's 1977 definition. He didn't state it this way, but it's equivalent: There's a category S whose objects are convex subsets C of R^n (n = 0,1,2,...) and whose maps are smooth maps between these. This category admits a Grothendieck pretopology where a cover is an open cover in the usual sense. A differentiable spaces is then a sheaf X on S. We think of X as a smooth space, and X(C) as the set of smooth maps from C to X. But the way Chen sets it up, differentiable spaces are not all the sheaves on S: just the "concrete" ones. These are defined using the terminal object 1 in S. Any convex set C has an underlying set of points hom(1,C). Any sheaf X on S has an underlying set of points X(1). Thanks to these, any element of X(C) has an underlying function from hom(1,C) to X(1). We say X is "concrete" if for all C, the map sending elements of X(C) to their underlying functions is 1-1. The supposed advantage of concrete sheaves is that the underlying set functor X |-> X(1) is faithful on these. So, we can think of them as sets with extra structure. But this advantage is largely illusory. The concreteness condition is not very important in practice, and the concrete sheaves form not a topos, but only a quasitopos. That's one layer of objections. Of course, *these* objections can be answered by working with the topos of *all* sheaves on S. This topos contains some useful non-concrete objects: for example, an object F such that F^X is the 1-forms on X. But now comes a second layer of objections. This topos of sheaves still lacks other key features of synthetic differential geometry. Most importantly, it lacks the "infinitesimal arrow" object D such that X^D is the tangent bundle of X. The problem is that all the objects of S are ordinary "non-infinitesimal" spaces. There should only be one smooth map from any such space to D. So as a sheaf on S, D would be indistinguishable from the 1-point space. So I guess the real problem is that the site S is concrete: that is, the functor assigning to any convex set C its set of points hom(1,C) is faithful. I could be jumping to conclusions, but it seems to me that that sheaves on a concrete site can never serve as a framework for differential geometry with infinitesimals. Best, jb From rrosebru@mta.ca Wed Sep 3 11:42:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 03 Sep 2008 11:42:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KatWF-0007mP-A9 for categories-list@mta.ca; Wed, 03 Sep 2008 11:39:39 -0300 MIME-version: 1.0 Content-transfer-encoding: 7BIT Content-type: text/plain; charset=ISO-8859-1; format=flowed Date: Tue, 02 Sep 2008 14:07:31 -0400 From: Claudio Hermida To: categories@mta.ca Subject: categories: Re: logics for model checking Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 Wojtowicz, Ralph wrote: > Finally, here are a few questions. > > Has anyone studied the model-checking logics using the tools > of categorical logic? > > Here's a little note I wrote about relational modalities from a categorical logic viewpoint; it includes a few further references which may be relevant to your question http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.3240 Claudio Hermida From rrosebru@mta.ca Wed Sep 3 18:59:51 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 03 Sep 2008 18:59:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kb0L3-0006Kp-5i for categories-list@mta.ca; Wed, 03 Sep 2008 18:56:33 -0300 Date: Wed, 03 Sep 2008 22:59:53 +0200 From: Luigi Santocanale MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: logics for model checking Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 7 Hi, > Any other thoughts, suggestions, or references would be > appreciated. In L. S. Completions of \mu-algebras. APAL, 154(1):27-50, May 2008. I studied the problem of the completeness of the modal mu-calculus from=20 an finitary algebraic point of view. In that paper some categorical=20 ideas, mainly from W. Tholen, Pro-categories and multiadjoint functors, Canad. J. Math. 36=20 (1) (1984) 144=96155. play the relevant role. The challenge is to prove that in free modal=20 \mu-algebras, the relation \mu.f =3D \bigvee_{n>=3D0} f^n(\bot) holds -- where \mu.f, the least fixpoint of f, is axiomatized by=20 equational implications and free modal \mu-algebras are not known to be=20 complete. Best, 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 Wed Sep 3 19:52:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 03 Sep 2008 19:52:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kb1A6-0001Mo-6w for categories-list@mta.ca; Wed, 03 Sep 2008 19:49:18 -0300 Date: Wed, 3 Sep 2008 17:57:21 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Octoberfest MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 8 We have received a bit of money from Concordia Univeristy towards the Octoberfest and have decided that we would like to help out graduate students who wish to attend. So any grad student who is planning to attend and would like support should write to Robert Seely (rags@math.mcgill.ca) by Sept 15 at the latest and make a request. We may or may not be able to help with travel expenses, but we should be able to pay for local hotel bills, although everything depends on how many applicants. Participants will have to submit original bills to be reimbursed. Sorry, but it will be impossible to provide money beforehand. As usual, people wishing to talk should let us know ASAP. Since we are trying to arrange a lunch on Saturday, we would also appreciate knowing who is planning to come. Again ASAP. Michael From rrosebru@mta.ca Thu Sep 4 09:05:49 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 04 Sep 2008 09:05:49 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KbDY2-0000cd-Nn for categories-list@mta.ca; Thu, 04 Sep 2008 09:02:51 -0300 Date: Thu, 04 Sep 2008 08:45:02 +0100 From: Maria Manuel Clementino To: categories@mta.ca Subject: categories: Research positions Content-Type: text/plain; charset=windows-1252; 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 5-YEAR RESEARCH POSITIONS The Centre for Mathematics of the University of Coimbra (CMUC) invites applications for five 5-year research positions in all areas of Mathemati= cs. Applicants should have a PhD in Mathematics and at least three years of post-doctoral experience, with an international-level publication record=20 and demonstrated ability to perform independent research. Preference=20 will be given to candidates working in one of the areas in which the=20 members of CMUC are currently active. A very good command of English is=20 required. The appointments will be made for three years, renewable for two more. Successful applicants are expected to develop research within the areas=20 of interest of CMUC. They should also contribute to graduate studies,=20 supervising students from the Department of Mathematics. The basic salary is equivalent to the index 195 of the Portuguese research career (in the order of 42 500 euros a year), in accordance=20 with the Ci=EAncia 2008 guidelines http://alfa.fct.mctes.pt/apoios/contratacaodoutorados/edital2008 Benefits include private health insurance and social security. Applicants should send =95 cover sheet (available from www.mat.uc.pt/=18cmuc/coversheet.pdf),=20 including names and contact information of three persons who can provide=20 a letter of recommendation =95 curriculum vitae (publication list included) =95 a 5-year research proposal (2 pages max), emphasizing their own research goals as pdf files to cmuc@mat.uc.pt. The subject of the e-mail=20 message must include the job reference C2008 FCTUC CMUC v15-v19. The deadline for applications is *September 10, 2008.* ------------------------------ More information can be found at www.mat.uc.pt From rrosebru@mta.ca Fri Sep 5 07:41:17 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 05 Sep 2008 07:41:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KbYf7-0001yj-KT for categories-list@mta.ca; Fri, 05 Sep 2008 07:35:33 -0300 Date: Thu, 4 Sep 2008 16:22:34 -0700 From: "Meredith Gregory" To: categories@mta.ca Subject: categories: language for infinitary compositions? 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: 10 Categorists, Is there a commonly accepted language for infinitary compositions? Here's the sort of thing i'm thinking about. There's a purely syntactic correspondence between a 'braced' notation and an infix notation for composition. Suppose we have a categorically-friendly notion of composition, say c. (Meaning the entities c composes can be viewed as morphisms in a category and c the categorical composition.) Then we can just as easily write - f c g -- c is merely making notationally explicit the interpretation of 'o' in f o g -- we're coloring the 'o', as it were or - {c| f, g |c} -- we've moved from infix to (not quite) prefix notation for the composition. The braced notation, however, is suggestive of a very powerful notational mechanism, comprehension notation. We could easily imagine a language allowing expressions of the form {c| pattern | predicate |c} which would denote pattern{subst_1} c pattern{subst_2} c ... where subst_i is a substition for 'variables' in the pattern of entities satisfying the predicate. This would allow reasoning over infinitary compositions by providing an intensional view of their interior structure. Surely, such a widget has already been invented. Can someone give me a reference? Best wishes, --greg -- L.G. Meredith Managing Partner Biosimilarity LLC 806 55th St NE Seattle, WA 98105 +1 206.650.3740 http://biosimilarity.blogspot.com From rrosebru@mta.ca Sun Sep 7 14:13:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Sep 2008 14:13:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcNiW-0007Ud-S5 for categories-list@mta.ca; Sun, 07 Sep 2008 14:06:28 -0300 Date: Fri, 05 Sep 2008 11:11:51 +0100 To: categories@mta.ca Subject: categories: New Association Computability in Europe formed MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: A.Beckmann@swansea.ac.uk (Arnold Beckmann) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 11 After four very successful conferences in Amsterdam in 2005, Swansea in 2006, Siena in 2007 and Athens in 2008, our community has officially formed the association Computability in Europe at the Annual General Meeting at this year's Computability in Europe conference in Athens. The object of the Association is to promote the development, particularly in Europe, of computability-related science, ranging over mathematics, computer science, and applications in various natural and engineering sciences such as physics and biology. This also includes the promotion of the study of philosophy and history of computing as it relates to questions of computability. A draft constitution of the Association can be found at http://www.amsta.leeds.ac.uk/~pmt6sbc/CiE.const.draft.pdf We invite every researcher interested in the object of the Association to become a member. The initial membership fee is set at zero, and lasts until 30 June 2010. To apply for membership of the Association, please complete and submit the form at http://www.cs.swan.ac.uk/acie/ Any enquiries concerning association CiE membership should be sent to the Membership Secretary, Arnold Beckmann, at a.beckmann@swansea.ac.uk. If you are not interested in becoming a member of this Association, we apologise for any inconvenience caused. With best regards, Association Computability in Europe *********************************************************************** Dr Arnold Beckmann | Swansea University | Computer Science | | a.beckmann@swansea.ac.uk | http://www.cs.swan.ac.uk/~csarnold/ From rrosebru@mta.ca Sun Sep 7 14:14:43 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Sep 2008 14:14:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcNk4-0007Zb-Kg for categories-list@mta.ca; Sun, 07 Sep 2008 14:08:04 -0300 Date: Fri, 5 Sep 2008 11:49:43 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Amusing fact MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 12 There is nothing deep in the following, but it is amusing and slightly surprising. In the well-known diagram below (the dots indicate spaces, which email doesn't handle well): ........0........0.......0.......... ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.......v.......... 0 ----> A' ----> A ----> A'' ----> 0 ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.... ..v.......... ........ ... f .. .. g .. .......... 0 ----> B' ----> B ----> B'' ----> 0 ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.......v.......... 0 ----> C' ----> C ----> C'' ----> 0 ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.......v.......... ........0........0.......0.......... it is widely known that if the three columns, middle row and one of the other two rows is exact, so is the remaining row. What if the upper and lower rows are exact (along with the three columns)? It might not be a complex, that is it might happen that gf \neq 0. Less widely known is that if ker(g) \inc im(f), then it is also exact. That is actually if and only if. That is, if either of K = ker(f) and I = im(f) contains the other, they are equal. Well, I got to wondering about that and eventually conjectured and proved that it is always the case that I/(I\cap K) is isomorphic to K/(I\cap K). This makes it transparent that if either contains the other, they have to be equal. There is just one point remaining. I did this by chasing elements around in Ab (so it is true in any abelian category). Does anyone see a clever proof using the snake lemma? I don't. Michael From rrosebru@mta.ca Sun Sep 7 14:15:51 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Sep 2008 14:15:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcNl3-0007dW-1P for categories-list@mta.ca; Sun, 07 Sep 2008 14:09:05 -0300 Date: Sat, 6 Sep 2008 12:48:14 +0200 (CEST) From: Johannes Huebschmann To: categories@mta.ca Subject: categories: Categories and functors, query MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 13 Dear All I somewhat recall that, a while ago, we discussed the origins of the notions of category and functor. S. Mac Lane had once pointed out to me these origins but from my recollections we did not entirely reproduce them. In his paper Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131 Saunders Mac Lane clearly pointed out the origins: "Category" from Kant (which I had known all the time) "Functor" from Carnap's book "Logical Syntax of Language" (which I had forgotten). Also I have a question, not directly related to the above issue: I have seen, on some web page, a copy of the referee's report about the Eilenberg-Mac Lane paper where Eilenberg-Mac Lane spaces are introduced. I cannot find this web page (or the report) any more. Can anyone provide me with a hint where I can possibly find it? Many thanks in advance Johannes HUEBSCHMANN Johannes Professeur de Mathematiques USTL, UFR de Mathematiques UMR 8524 Laboratoire Paul Painleve F-59 655 Villeneuve d'Ascq Cedex France http://math.univ-lille1.fr/~huebschm TEL. (33) 3 20 43 41 97 (33) 3 20 43 42 33 (secretariat) (33) 3 20 43 48 50 (secretariat) Fax (33) 3 20 43 43 02 e-mail Johannes.Huebschmann@math.univ-lille1.fr From rrosebru@mta.ca Sun Sep 7 20:00:55 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Sep 2008 20:00:55 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcTAD-0002JN-4V for categories-list@mta.ca; Sun, 07 Sep 2008 19:55:25 -0300 From: "R Brown" To: Subject: categories: Re: Categories and functors, query Date: Sun, 7 Sep 2008 22:33:48 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; 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: 15 There is another curiosity about the axioms for a category, namely the infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft told me that these axioms had influenced E-M. These axioms were well used in the algebra group at Chicago. However when I asked Sammy about this in 1985 he firmly said `no, and was why the notion of groupoid did not appear as an example in the E-M paper'! Perhaps it was a case of forgetting the influence? Ronnie ----- Original Message ----- From: "Johannes Huebschmann" To: Sent: Saturday, September 06, 2008 11:48 AM Subject: categories: Categories and functors, query > Dear All > > I somewhat recall that, a while ago, we discussed the origins of > the notions of category and functor. S. Mac Lane had once pointed > out to me these origins but from my recollections we did not > entirely reproduce them. > > In his paper > > Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131 > > Saunders Mac Lane clearly pointed out the origins: > > "Category" from Kant (which I had known all the time) > > "Functor" from Carnap's book "Logical Syntax of Language" (which I > had forgotten). > > > Also I have a question, not directly related to the above issue: > > I have seen, on some web page, a copy of > the referee's report about the Eilenberg-Mac Lane paper > where Eilenberg-Mac Lane spaces are introduced. > I cannot find this web page (or the report) > any more. Can anyone provide me with > a hint where I can possibly find it? > > Many thanks in advance > > Johannes > > > > HUEBSCHMANN Johannes > Professeur de Mathematiques > USTL, UFR de Mathematiques > UMR 8524 Laboratoire Paul Painleve > F-59 655 Villeneuve d'Ascq Cedex France > http://math.univ-lille1.fr/~huebschm > > TEL. (33) 3 20 43 41 97 > (33) 3 20 43 42 33 (secretariat) > (33) 3 20 43 48 50 (secretariat) > Fax (33) 3 20 43 43 02 > > e-mail Johannes.Huebschmann@math.univ-lille1.fr > > > -------------------------------------------------------------------------------- No virus found in this incoming message. Checked by AVG - http://www.avg.com Version: 8.0.169 / Virus Database: 270.6.17/1657 - Release Date: 06/09/2008 20:07 From rrosebru@mta.ca Mon Sep 8 08:02:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 08:02:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KceRD-0006t5-Hr for categories-list@mta.ca; Mon, 08 Sep 2008 07:57:43 -0300 From: Dana Scott To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v926) Subject: categories: Re: Categories and functors, query Date: Sun, 7 Sep 2008 18:25:39 -0700 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 16 On Sep 7, 2008, at 2:33 PM, R Brown wrote: > There is another curiosity about the axioms for a category, namely the > infuence of the known axioms for a groupoid (Brandt, 1926). Bill > Cockcroft > told me that these axioms had influenced E-M. These axioms were well > used in > the algebra group at Chicago. However when I asked Sammy about this > in 1985 > he firmly said `no, and was why the notion of groupoid did not > appear as an > example in the E-M paper'! > > Perhaps it was a case of forgetting the influence? I certainly heard Saunders mention Brandt groupoids as examples. (Not very good examples, since all maps are invertible.) But, as everyone knows, it is not the definition of a category that is the key part, but seeing that functors and natural transformations are interesting. From rrosebru@mta.ca Mon Sep 8 20:52:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 20:52:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcqRi-00021M-Rj for categories-list@mta.ca; Mon, 08 Sep 2008 20:47:02 -0300 Date: Mon, 8 Sep 2008 08:50:41 -0400 (EDT) From: Michael Barr To: categories@mta.ca Subject: categories: Re: Categories and functors, query 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: 18 Interesting speculation, but how can we verify or refute it? What I can add is that when I sat in on Sammy's category theory course (called homological algebra, but I am not sure Ext or Tor were ever mentioned), I do not recall that he so much as mentioned groupoids. I once mentioned to Charles Ehresmann that he appeared to view categories as a generalization of groupoids while Eilenberg and Mac Lane thought of them as a generalization of posets. Charles agreed. This reminds me of a speculation I have often had (although Saunders denied and he knew Birkhoff pretty well). In the 30s and 40s, the word "homomorphism" was regularly used but always meant surjective. By the late 40s and 50s people were talking about "homomorphism into" meaning not necessarily surjective. So groups had lattices of subgroups and lattices of quotient groups and Birkhoff invented lattice theory at least partly in the hope that the structure of those two lattices would tell you a lot about the structure of the group. I don't think this actually happened to any great extent. But I have wondered whether Birkhoff might instead have invented categories had our more general notion of homomorphism been rampant. As I said Saunders didn't think so, but it still sounds attractive to me. One of the things that astonishes me about "General theory of natural equivalences" is that they clearly knew about natural transformations in general but chose to talk only about equivalences. I once asked Sammy about that and he more or less said something like one generalization at a time. But they must have realized that the Hurevic map is a superior example. Still, Steenrod must have gotten the point immediately. Michael On Sun, 7 Sep 2008, R Brown wrote: > There is another curiosity about the axioms for a category, namely the > infuence of the known axioms for a groupoid (Brandt, 1926). Bill Cockcroft > told me that these axioms had influenced E-M. These axioms were well used in > the algebra group at Chicago. However when I asked Sammy about this in 1985 > he firmly said `no, and was why the notion of groupoid did not appear as an > example in the E-M paper'! > > Perhaps it was a case of forgetting the influence? > > Ronnie > > > > > ----- Original Message ----- > From: "Johannes Huebschmann" > To: > Sent: Saturday, September 06, 2008 11:48 AM > Subject: categories: Categories and functors, query > > >> Dear All >> >> I somewhat recall that, a while ago, we discussed the origins of >> the notions of category and functor. S. Mac Lane had once pointed >> out to me these origins but from my recollections we did not >> entirely reproduce them. >> >> In his paper >> >> Samuel Eilenberg and Categories, JPAA 168 (2002), 127-131 >> >> Saunders Mac Lane clearly pointed out the origins: >> >> "Category" from Kant (which I had known all the time) >> >> "Functor" from Carnap's book "Logical Syntax of Language" (which I >> had forgotten). >> >> >> Also I have a question, not directly related to the above issue: >> >> I have seen, on some web page, a copy of >> the referee's report about the Eilenberg-Mac Lane paper >> where Eilenberg-Mac Lane spaces are introduced. >> I cannot find this web page (or the report) >> any more. Can anyone provide me with >> a hint where I can possibly find it? >> >> Many thanks in advance >> >> Johannes >> >> >> >> HUEBSCHMANN Johannes >> Professeur de Mathematiques >> USTL, UFR de Mathematiques >> UMR 8524 Laboratoire Paul Painleve >> F-59 655 Villeneuve d'Ascq Cedex France >> http://math.univ-lille1.fr/~huebschm >> >> TEL. (33) 3 20 43 41 97 >> (33) 3 20 43 42 33 (secretariat) >> (33) 3 20 43 48 50 (secretariat) >> Fax (33) 3 20 43 43 02 >> >> e-mail Johannes.Huebschmann@math.univ-lille1.fr >> >> >> > > > -------------------------------------------------------------------------------- > > > > No virus found in this incoming message. > Checked by AVG - http://www.avg.com > Version: 8.0.169 / Virus Database: 270.6.17/1657 - Release Date: 06/09/2008 > 20:07 > > > From rrosebru@mta.ca Mon Sep 8 20:53:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 20:53:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcqT9-00025Z-6Y for categories-list@mta.ca; Mon, 08 Sep 2008 20:48:31 -0300 Date: Mon, 08 Sep 2008 12:00:58 -0400 From: Walter Tholen MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Categories and functors, query 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: 19 There is another aspect to the E-M achievement that I stressed in my CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the extent to which 20th-century mathematics was entrenched in set theory, it was a tremendous psychological step to put structure on "classes" and to dare regarding these (perceived) monsters as objects that one could study just as one would study individual groups or topological spaces. In my experience, skepticism towards category theory is often rooted in the fear of the "illegitimately large" size, till today. By comparison, Brandt groupoids lived in the cozy and familiar small world, and their definition was arrived at without having to leave the universe. With the definition of category (and functor and natural transformation) Eilenberg and Moore had to do a lot more than just repeating at the monoid level what Brandt did at the group level! In my view their big psychological step here is comparable to Cantor's daring to think that there could be different levels of infinity. Cheers, Walter. Dana Scott wrote: > > On Sep 7, 2008, at 2:33 PM, R Brown wrote: > >> There is another curiosity about the axioms for a category, namely the >> infuence of the known axioms for a groupoid (Brandt, 1926). Bill >> Cockcroft >> told me that these axioms had influenced E-M. These axioms were well >> used in >> the algebra group at Chicago. However when I asked Sammy about this >> in 1985 >> he firmly said `no, and was why the notion of groupoid did not >> appear as an >> example in the E-M paper'! >> >> Perhaps it was a case of forgetting the influence? > > > I certainly heard Saunders mention Brandt groupoids as examples. > (Not very good examples, since all maps are invertible.) But, as > everyone knows, it is not the definition of a category that > is the key part, but seeing that functors and natural > transformations are interesting. > > From rrosebru@mta.ca Mon Sep 8 21:15:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 21:15:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kcqns-0003LV-7Q for categories-list@mta.ca; Mon, 08 Sep 2008 21:09:56 -0300 MIME-Version: 1.0 To: "categories" Content-Type: text/plain; charset="utf-8" X-Atmail-Account: wlawvere@buffalo.edu Date: Mon, 8 Sep 2008 15:04:00 -0400 Subject: categories: Re: KT Chen's smooth CCC From: wlawvere@buffalo.edu Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 21 [Note from moderator: with apologies to the poster, this is being resent since some of you will have received a version with some corrupted characters.] There are no > objections to continuing to develop Chen's theory of "differentiable > spaces" Indeed on 8/17, 8/26, and 8/27 I urged the continuation of the development of Chen's theory (for example the smooth space of piecewise smooth paths), making use of recent experience of the range of possible categories. It is possible that >sheaves on a concrete site can never serve as a framework > for differential geometry with infinitesimals. But a proof would require a definition of what is meant by infinitesimals, as well as the constraint on the framework that the maps 1->R and R->R are the standard ones. Otherwise nonstandard analysis might fit. The nilpotents or germs capture the Heraclitian nature of motion in a way that abstract sets do not directly. The misplaced concreteness, according to which >spaces are [single] sets equipped with extra structure is only a "second aspect of the default paradigm. The first aspect, successfully overcome by the named pioneers, is the one generalizing the default category of topological spaces (or locales). Here "default" refers to the habitual response to the frequently occurring need to specify a background category of cohesion in which to interpret our algebra. The generalization from Sierpinski-valued functions (open sets) to real-valued, has also been proposed, but that sort of attempt never succeeded in yielding a simple theory of map spaces. In contrast to this "function-algebra X/R as primary" paradigm, the semi-dual "figure-geometry S/X as primary" has led to good map spaces (including internal function algebras) for many authors (Sebastiao e Silva, Fox, Hurewicz 60 years ago and several more recent). I believe that attempting to force nearly-perfect duality has in general not led to good results, but of course one studies the extent to which a monad (presumed identity on models S) approximates the identity on general spaces. For example, Froelicher=E2=80=99s duality condition applies not only to the line R but to the function space R^R, a non-trivial fact about the smooth case, derived by LSZ from a study of distributions of compact support (so citing it is not just name-dropping). Bill On Tue 09/02/08 6:00 PM , John Baez baez@math.ucr.edu sent: > Bill Lawvere wrote: >=20 > >By urging the study of the good geometrical ideas > and constructions of>Chen and Froelicher, as well as of Bott, Brown, > Hurewicz, Mostow, Spanier,>Steenrod, I am of course not advocating the > preferential resurrection of>the particular categories they tentatively > devised to contain the>constructions. >=20 > I chose Chen's framework when Urs Schreiber and I were doing some work > in mathematical physics and we needed a "convenient category" of > smoothspaces. I decided to choose one that was easy to explain to people > brainwashed by the "default paradigm", in which spaces are sets > equippedwith extra structure. Later I realized I needed to write a paper > establishing some properties of Chen's framework. By doing that I > guessI'm guilty of reinforcing the default paradigm, and for that I > apologize. > If I understand correctly, one can actually separate the objections > to continuing to develop Chen's theory of "differentiable > spaces"into two layers. >=20 > Let me remind everyone of Chen's 1977 definition. He didn't state > it this way, but it's equivalent: >=20 > There's a category S whose objects are convex subsets C of R^n > (n =3D 0,1,2,...) and whose maps are smooth maps between these. > This category admits a Grothendieck pretopology where a cover > is an open cover in the usual sense. >=20 > A differentiable spaces is then a sheaf X on S. We think of > X as a smooth space, and X(C) as the set of smooth maps from C to X. >=20 > But the way Chen sets it up, differentiable spaces are not all > the sheaves on S: just the "concrete" ones. >=20 > These are defined using the terminal object 1 in S. Any convex set > C has an underlying set of points hom(1,C). Any sheaf X on S has an > underlying set of points X(1). Thanks to these, any element of X(C) > has an underlying function from hom(1,C) to X(1). We say X is > "concrete"if for all C, the map sending elements of X(C) to their underly= ing > functions is 1-1. >=20 > The supposed advantage of concrete sheaves is that the underlying > set functor X |-> X(1) is faithful on these. So, we can think of > them as sets with extra structure. >=20 > But this advantage is largely illusory. The concreteness condition > is not very important in practice, and the concrete sheaves form not > a topos, but only a quasitopos. >=20 > That's one layer of objections. Of course, *these* objections > can be answered by working with the topos of *all* sheaves on S. > This topos contains some useful non-concrete objects: for example, > an object F such that F^X is the 1-forms on X. >=20 > But now comes a second layer of objections. This topos of sheaves > still lacks other key features of synthetic differential geometry. > Most importantly, it lacks the "infinitesimal arrow" object D > suchthat X^D is the tangent bundle of X. >=20 > The problem is that all the objects of S are ordinary > "non-infinitesimal"spaces. There should only be one smooth map from any = such space to D. > So as a sheaf on S, D would be indistinguishable from the 1-point > space. > So I guess the real problem is that the site S is concrete: that is, > the functor assigning to any convex set C its set of points hom(1,C) > is faithful. I could be jumping to conclusions, but it seems to me > that that sheaves on a concrete site can never serve as a framework > for differential geometry with infinitesimals. >=20 > Best, > jb >=20 >=20 >=20 >=20 >=20 >=20 >=20 >=20 From rrosebru@mta.ca Mon Sep 8 22:47:47 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Sep 2008 22:47:47 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcsEy-0000He-Gp for categories-list@mta.ca; Mon, 08 Sep 2008 22:42:00 -0300 Date: Mon, 8 Sep 2008 20:55:48 -0400 From: tholen@mathstat.yorku.ca To: categories@mta.ca Subject: categories: Re: Categories and functors, query MIME-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 22 Quoting Walter Tholen : > definition was arrived at without having to leave the universe. With the > definition of category (and functor and natural transformation) > Eilenberg and Moore had to do a lot more than just repeating at the > monoid level what Brandt did at the group level! In my view their big OOPS -- "Moore" should read "Mac Lane", of course. (Sorry, Saunders!) W. From rrosebru@mta.ca Tue Sep 9 09:02:02 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Sep 2008 09:02:02 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kd1q5-0007Wz-O7 for categories-list@mta.ca; Tue, 09 Sep 2008 08:56:57 -0300 Date: Tue, 9 Sep 2008 14:53:03 +0400 From: "Nikita Danilov" Subject: categories: Re: Categories and functors, query 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 Michael Barr writes: > > This reminds me of a speculation I have often had (although Saunders > denied and he knew Birkhoff pretty well). In the 30s and 40s, the word > "homomorphism" was regularly used but always meant surjective. By the > late 40s and 50s people were talking about "homomorphism into" meaning not > necessarily surjective. So groups had lattices of subgroups and lattices > of quotient groups and Birkhoff invented lattice theory at least partly in > the hope that the structure of those two lattices would tell you a lot > about the structure of the group. I don't think this actually happened to > any great extent. But I have wondered whether Birkhoff might instead have Noether's `set theoretic foundations of group theory', where group axioms are based on a notion of coset decomposition rather than multiplication, seems to be much earlier (20s) attempt to the same: http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty2.pdf > > Michael Nikita. From rrosebru@mta.ca Tue Sep 9 19:18:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Sep 2008 19:18:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KdBR0-0004nb-Sw for categories-list@mta.ca; Tue, 09 Sep 2008 19:11:42 -0300 Date: Tue, 09 Sep 2008 18:05:23 -0400 From: jim stasheff MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Categories and functors, query 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: 24 Michael But they must have realized that the Hurevic map is a superior example. Still, Steenrod must have gotten the point immediately. You lost me there. jim From rrosebru@mta.ca Wed Sep 10 12:08:20 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Sep 2008 12:08:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KdRCq-0000Do-5i for categories-list@mta.ca; Wed, 10 Sep 2008 12:02:08 -0300 Date: Tue, 09 Sep 2008 18:22:10 -0400 From: jim stasheff MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Categories and functors, query 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: 25 Walter, I beg to differ only with In my experience, skepticism towards category theory is often rooted in the fear of the "illegitimately large" size, till today. In my experience, disdain for cat theory is due to papers with a very high density of unfamiliar names reminiscent of the minutia of PST and the (in) famous comment (by some one) about something like: hereditary hemi-demi-semigroups with chain condition jim Tholen wrote: > There is another aspect to the E-M achievement that I stressed in my > CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the > extent to which 20th-century mathematics was entrenched in set theory, > it was a tremendous psychological step to put structure on "classes" and > to dare regarding these (perceived) monsters as objects that one could > study just as one would study individual groups or topological spaces. > In my experience, skepticism towards category theory is often rooted in > the fear of the "illegitimately large" size, till today. By comparison, > Brandt groupoids lived in the cozy and familiar small world, and their > definition was arrived at without having to leave the universe. With the > definition of category (and functor and natural transformation) > Eilenberg and Moore had to do a lot more than just repeating at the > monoid level what Brandt did at the group level! In my view their big > psychological step here is comparable to Cantor's daring to think that > there could be different levels of infinity. > > Cheers, > Walter. From rrosebru@mta.ca Wed Sep 10 13:33:51 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Sep 2008 13:33:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KdSVr-0000rv-8W for categories-list@mta.ca; Wed, 10 Sep 2008 13:25:51 -0300 Date: Tue, 9 Sep 2008 20:17:14 -0700 From: "Alex Hoffnung" To: categories@mta.ca Subject: categories: Equivalence of pseudo-limits 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: 26 Hi all, Given an indexing 2-category J, a pair of parallel functors F,G : J ----> CAT, and a natural equivalence f : F ==> G, the pseudo-limits of F and G should be equivalent. I am trying to find out what paper, if any, I can cite for this theorem. Or maybe this is just the type of thing that nobody has bothered to write down. Any help would be appreciated. best, Alex Hoffnung From rrosebru@mta.ca Wed Sep 10 19:57:47 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Sep 2008 19:57:47 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KdYYB-00059K-Sp for categories-list@mta.ca; Wed, 10 Sep 2008 19:52:39 -0300 Content-class: urn:content-classes:message MIME-Version: 1.0 Subject: categories: The disdain for categories Date: Wed, 10 Sep 2008 13:35:22 -0400 From: Andre Joyal To: 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: 27 Jim Stasheff wrote: >In my experience, disdain for cat theory is due to papers with a very >high density of unfamiliar names >reminiscent of the minutia of PST and the (in) famous comment (by some >one) about something like: >hereditary hemi-demi-semigroups with chain condition The chosen example is not too convincing,=20 since the notions involved are not typically categorical. Complicated sentences like this can be found in every fields. They are often the mark of a poor paper. My guess is that the disdain for categories has a mixed origin. Like logic, category theory has a taste for generalities. But most mathematicians are specialised and they find it hard to believe that important progresses can be made in their fields from the outside, as the result of general insights. But we all know that the division of mathematics into fields=20 is justified more by sociology than by science.=20 Category theory is a powerful tool for crossing=20 the boundaries between the fields. The unity of mathematics is growing stronger every day. andre -------- Message d'origine-------- De: cat-dist@mta.ca de la part de jim stasheff Date: mar. 09/09/2008 18:22 =C0: categories@mta.ca Objet : categories: Re: Categories and functors, query =20 Walter, I beg to differ only with In my experience, skepticism towards category theory is often rooted in the fear of the "illegitimately large" size, till today. In my experience, disdain for cat theory is due to papers with a very high density of unfamiliar names reminiscent of the minutia of PST and the (in) famous comment (by some one) about something like: hereditary hemi-demi-semigroups with chain condition jim Tholen wrote: > There is another aspect to the E-M achievement that I stressed in my > CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given = the > extent to which 20th-century mathematics was entrenched in set theory, > it was a tremendous psychological step to put structure on "classes" = and > to dare regarding these (perceived) monsters as objects that one could > study just as one would study individual groups or topological spaces. > In my experience, skepticism towards category theory is often rooted = in > the fear of the "illegitimately large" size, till today. By = comparison, > Brandt groupoids lived in the cozy and familiar small world, and their > definition was arrived at without having to leave the universe. With = the > definition of category (and functor and natural transformation) > Eilenberg and Moore had to do a lot more than just repeating at the > monoid level what Brandt did at the group level! In my view their big > psychological step here is comparable to Cantor's daring to think that > there could be different levels of infinity. > > Cheers, > Walter. From rrosebru@mta.ca Thu Sep 11 14:19:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:19:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KdpiQ-0000zs-MM for categories-list@mta.ca; Thu, 11 Sep 2008 14:12:22 -0300 Date: Wed, 10 Sep 2008 18:12:10 -0500 From: "Charles Wells" To: catbb Subject: categories: New version of Graph Based Logic and Sketches 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: 28 I have posted a new version of Graph Based Logic and Sketches, by Atish Bagchi and Charles Wells, here: http://www.cwru.edu/artsci/math/wells/pub/pdf/gbls.pdf This will eventually appear on ArXiv. -- Charles Wells professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://www.gyregimble.blogspot.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm personal website: http://www.abstractmath.org/Personal/index.html From rrosebru@mta.ca Thu Sep 11 14:20:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:20:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KdpjZ-00017R-Sp for categories-list@mta.ca; Thu, 11 Sep 2008 14:13:33 -0300 Date: Wed, 10 Sep 2008 17:20:45 -0700 From: Toby Bartels To: categories@mta.ca Subject: categories: Re: Categories and functors, query 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: 29 Dana Scott wrote in part: >But, as >everyone knows, it is not the definition of a category that >is the key part, but seeing that functors and natural >transformations are interesting. Indeed, the notion of natural isomorphism (or canonical isomorphism) should be available already to groupoid theorists before 1945. To what extent did they know about functors and natural isomorphisms, and to what extent did Saunders & Mac Lane have to tell them? Or, pace Walter's remarks, did they know about the ~small~ ones but not have the guts to apply them to large classes of strucures? It's been said before that the real insight of category theory --as something more general than groupoids, monoids, and posets-- is the notion of adjoint functors (including limits, etc). I'm inclined to agree, so I'm interested in why and whether groupoid theorists thought of (and applied) that which they ~did~ have. --Toby From rrosebru@mta.ca Thu Sep 11 14:32:31 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:32:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kdpuy-0002S0-5Q for categories-list@mta.ca; Thu, 11 Sep 2008 14:25:20 -0300 Date: Wed, 10 Sep 2008 21:25:24 -0400 From: jim stasheff MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: The disdain for categories 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: 30 Andre Joyal wrote: > Jim Stasheff wrote: > > =20 >> In my experience, disdain for cat theory is due to papers with a very >> high density of unfamiliar names >> reminiscent of the minutia of PST and the (in) famous comment (by some >> one) about something like: >> hereditary hemi-demi-semigroups with chain condition >> =20 > > The chosen example is not too convincing,=20 > since the notions involved are not typically categorical. > Complicated sentences like this can be found in every fields. > They are often the mark of a poor paper. > Category theory is a powerful tool for crossing=20 > the boundaries between the fields. > The unity of mathematics is growing stronger every day. > > andre > > =20 Indeed, the quote I was misremembering was NOT in category theory how's that for crossing the boundaries between the fields. ;-) in fact, it turns out that the correct usage is hemi-demi-semi-quaver - in music! jim > -------- Message d'origine-------- > De: cat-dist@mta.ca de la part de jim stasheff > Date: mar. 09/09/2008 18:22 > =C0: categories@mta.ca > Objet : categories: Re: Categories and functors, query > =20 > Walter, > > I beg to differ only with > > In my experience, skepticism towards category theory is often rooted in > the fear of the "illegitimately large" size, till today. > > In my experience, disdain for cat theory is due to papers with a very > high density of unfamiliar names > reminiscent of the minutia of PST and the (in) famous comment (by some > one) about something like: > hereditary hemi-demi-semigroups with chain condition > > jim > Tholen wrote: > =20 >> There is another aspect to the E-M achievement that I stressed in my >> CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given t= he >> extent to which 20th-century mathematics was entrenched in set theory, >> it was a tremendous psychological step to put structure on "classes" a= nd >> to dare regarding these (perceived) monsters as objects that one could >> study just as one would study individual groups or topological spaces. >> In my experience, skepticism towards category theory is often rooted i= n >> the fear of the "illegitimately large" size, till today. By comparison= , >> Brandt groupoids lived in the cozy and familiar small world, and their >> definition was arrived at without having to leave the universe. With t= he >> definition of category (and functor and natural transformation) >> Eilenberg and Moore had to do a lot more than just repeating at the >> monoid level what Brandt did at the group level! In my view their big >> psychological step here is comparable to Cantor's daring to think that >> there could be different levels of infinity. >> >> Cheers, >> Walter. >> =20 > > > > > =20 From rrosebru@mta.ca Thu Sep 11 14:33:08 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:33:08 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kdpw0-0002YJ-Gp for categories-list@mta.ca; Thu, 11 Sep 2008 14:26:24 -0300 Date: Thu, 11 Sep 2008 12:27:33 +1000 From: "Dominic Verity" To: categories@mta.ca Subject: categories: Re: Equivalence of pseudo-limits 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: 31 Hi Alex, This property is certainly known to hold for a much larger class of 2-categorical limits - the flexible limits, which class includes the classes of pseudo, lax and op-lax limits. I believe you will find a proof of this result in the Bird, Kelly, Power and Street paper entitled "Flexible Limits for 2-Categories". Failing that the Power and Robinson paper "A characterisation of PIE limits" probably contains this result is some form. You can also find explications of the pseudo and lax results in earlier works by Street, although obvious candidates for the best one to consult elude me at the moment. The most elementary proof of the flexible limit result starts by observing that all flexible limits can be constructed using products, splittings of idempotents and a couple of less familiar, exclusively 2-categorical, limits called inserters and equifiers. It is then straight forward to verify the result you mention for each of these particular limits and then to infer that it must therefore hold for all flexible limits. In the early 1990's Robert Pare introduced a class of limits called the persistent limits. These were defined for 2-categories, but made use of his double categorical approach to 2-limits. Persistent limits are precisely those limits which have the stability property you seek, but with regard to a slightly more general class of double categorical diagram transformations whose 1-cellular components are all equivalences. In my thesis (1992), I prove that the class of flexible limits introduced by Bird, Kelly, Power and Street is identical to Pare's class of persistent limits - thus closing the circle and demonstrating that the flexible limits are in a natural sense the largest class of 2-limits which have this property. Regards Dominic Verity 2008/9/10 Alex Hoffnung > Hi all, > > Given an indexing 2-category J, a pair of parallel functors > F,G : J ----> CAT, and a natural equivalence f : F ==> G, > the pseudo-limits of F and G should be equivalent. > > I am trying to find out what paper, if any, I can cite for this theorem. > Or > maybe this is just the type of thing that nobody has bothered to write > down. > Any help would be appreciated. > > best, > Alex Hoffnung > From rrosebru@mta.ca Thu Sep 11 14:34:43 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:34:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kdpww-0002ed-J6 for categories-list@mta.ca; Thu, 11 Sep 2008 14:27:22 -0300 To: categories@mta.ca From: "Edward A. Hirsch" Subject: categories: CSR-2009: First Call for Papers Date: Thu, 11 Sep 2008 10:51:27 +0400 (MSD) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 32 *************** We apologize for multiple copies ****************************** First Call for Papers 4th INTERNATIONAL COMPUTER SCIENCE SYMPOSIUM IN RUSSIA (CSR 2009) August 18-23, 2009, Novosibirsk, Russia http://math.nsc.ru/conference/csr2009/index.shtml ******************************************************************************* CSR 2009 intends to reflect the broad scope of international cooperation in computer science. It is the 4th conference in a series of regular events started with CSR 2006 in St.Petersburg (see LNCS 3967), CSR 2007 in Ekaterinburg (see LNCS 4649), and CSR 2008 in Moscow (see LNCS 5010). As usual, CSR 2009 consists of two tracks: Theory Track and Applications and Technology Track. IMPORTANT DATES: Deadline for submissions: November 26, 2008 Notification of acceptance: February 3, 2009 Conference dates: August 18-23, 2009 TOPICS Theory Track topics include * algorithms and data structures * complexity and cryptography * formal languages and automata * computational models and concepts * proof theory and applications of logic to computer science. Application Track topics include * abstract interpretation * model checking * automated reasoning * deductive methods * constraint solving * functional and declarative languages * type systems * software engineering * development methodologies for design, development, testing, analysis, and verification of correct and reliable systems. OPENING LECTURE: Andrei Voronkov (University of Manchester). PROGRAM COMMITTEES Program committee of Theory Track is: Farid Ablayev, Kazan State University Sergei N. Artemov, City University of New York Lev Beklemishev, Steklov Inst., Moscow Veronique Bruyere, Universite de Mons-Hainaut Cristian Calude, University of Auckland Christian Glasser, Universitaet Wuerzburg Dima Grigoriev, Institut de Recherche Mathematique de Rennes Dietrich Kuske, Universitaet Leipzig Larisa Maksimova, IM, Novosibirsk Andrei Mantsivoda, Irkutsk State University Yuri Matiyasevich, Steklov Institute, St.Petersburg Elvira Mayordomo, Universidad de Zaragoza Pierre McKenzie, Universite de Montreal Andrey S. Morozov, IM, Novosibirsk (co-chair) Jean-Eric Pin, LIAFA, Paris Kai Salomaa, Queen's University, Kingston, Canada Victor Selivanov, Novosibirsk Pedagogical University Ludwig Staiger, Universitaet Halle-Wittenberg Klaus W. Wagner, Universitaet Wuerzburg (co-chair) Program committee of Applications and Technology Track is: Thomas Ball, Microsoft Research Josh Berdine, Microsoft Research Bart Demoen, K.U. Leuven Franjo Ivancic, NEC Laboratories America Martin Leucker, TU Munich Rupak Majumdar, University of California, Los Angeles Greg Morrisett, Harvard University Arnd Poetzsch-Heffter, University of Kaiserslautern Andreas Rossberg, MPI-SWS Andrey Rybalchenko, MPI-SWS (chair) Alexander Serebrenik, TU Eindhoven Henny Sipma, Stanford University Natasha Sharygina, University of Lugano Helmut Veith, TU Darmstadt Eran Yahav, IBM Research Andreas Zeller, Universitaet des Saarlandes ORGANIZERS: Sobolev Institute of Mathematics SB RAS. Conference chair: Anna Frid SUBMISSIONS: Authors are invited to submit an extended abstract or a full paper of at most 10 pages in the LNCS format (the instructions on it can be found here: http://www.springer.com/computer/lncs?SGWID=0-164-7-72376-0). Proofs and other material omitted due to space constraints are to be put into a clearly marked appendix to be read at discretion of the referees. Papers must present original (and not previously published) research. Simultaneous submissions to journals or to other conferences with published proceedings are not allowed. The proceedings of the symposium will be published in Springer's LNCS series. Submissions should be uploaded at EasyChair Conference system: http://www.easychair.org/conferences/?conf=csr09 . FURTHER INFORMATION AND CONTACTS: Web: http://math.nsc.ru/conference/csr2009/index.shtml Email: csr2009@math.nsc.ru From rrosebru@mta.ca Thu Sep 11 14:38:31 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:38:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kdq1M-00037y-PB for categories-list@mta.ca; Thu, 11 Sep 2008 14:31:56 -0300 From: "R Brown" To: Subject: categories: Re: Categories and functors, query Date: Thu, 11 Sep 2008 10:05:10 +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 Dear All, I agree with Andre that part of the matter is sociological. It is also quite fundamental, and is about the proper aims of mathematics. The need is for discussion, rather than total agreement. Miles Reid's infamous comment was "The study of category theory for its own sake (surely one of the most sterile of intellectual pursuits) also dates from this time; Grothendieck can't necessarily be blamed for this, [!!!] since his own use of categories was very successful in solving problems. " (My riposte in a paper was to suggest a game: `I can think of a more intellectually sterile pursuit than you can!') This suggests the view that solving problems, presumably already formulated ones, is the key part of mathematics. (Miles did tell me he expected his student to use topoi or whatever!) A 1974 report on graduate mathematicians in employment suggested they were good at solving problems but not so good at formulating them. Grothendieck in one letter to me wrote on his aim for `understanding'. (see my article on `Promoting Mathematics' on my Popularisation web page) I believe many students come into mathematics because they like finding out why things are true, they want to understand. Loday told me he thought one of the strengths of French mathematics was to try to realise this aim. By contrast, I once asked Frank Adams why he wrote that a certain nonabelian cohomology was trivial and he said `you just do a calculation' - Frank was a determined problem solver! So people have asked: "Where are the big theorems, the big problems, in category theory?" Are they there? Does it matter if they are not there? Atiyah in his article on `20th century mathematics' (Bull LMS, 2001) talks about the unity of mathematics, but the word `category' does not occur in his article. (Neither does groupoid.) He states a dichotomy between geometry (good)and algebra (bad) but fails to recognise the combination given by, say, Grothendieck's work, and also by higher categorical structures. Indeed, underlying structures and processes may be of various types, all very useful to know. I am *very* impressed by Henry Whitehead's finding so many of these. A word often omitted in mathematics teaching is `analogy'. Yet this is what abstraction is about, and why it is so powerful. Category theory allows for powerful analogies. I am always puzzled, even horrified, by mathematicians who use the word `nonsense' to describe the work of others (as is all too common), yet often themselves cannot well define professionalism in the subject. Indeed they often cannot believe the direction others may take is chosen for good professional reasons! They sometimes say `not mainstream'. Yet history shows `the mainstream' shifts its course radically over the years. The lack is of a consistent and well maintained mathematical criticism, recognising historical trends and not just the `great man (or woman)', or famous problem, approach. I believe we need to have prepared an answer to: What has category theory done for mathematics? And indeed for evaluation of any subject areas. But a good case is that category theory leads, or can lead, and has led, to clarity, to understanding and development of the rich variety of structures there are and to be found. However this does not rate for million $ prizes (as it should, of course!). When I see all the current fuss (rightly) about the LHC in Geneva, I do wonder: who is going to speak up for mathematics, to attract students into the subject, by getting over a message as to its value and achievements? and also getting this message over to students studying the subject! (see `Promoting Mathematics' and Tim and my article on `the methodology of mathematics') Ronnie www.bangor.ac.uk/r.brown/publar.html ----- Original Message ----- From: "jim stasheff" To: Sent: Tuesday, September 09, 2008 11:22 PM Subject: categories: Re: Categories and functors, query > Walter, > > I beg to differ only with > > In my experience, skepticism towards category theory is often rooted in > the fear of the "illegitimately large" size, till today. > > In my experience, disdain for cat theory is due to papers with a very > high density of unfamiliar names > reminiscent of the minutia of PST and the (in) famous comment (by some > one) about something like: > hereditary hemi-demi-semigroups with chain condition > > jim > Tholen wrote: >> There is another aspect to the E-M achievement that I stressed in my >> CT06 talk for the Eilenberg - Mac Lane Session at White Point. Given the >> extent to which 20th-century mathematics was entrenched in set theory, >> it was a tremendous psychological step to put structure on "classes" and >> to dare regarding these (perceived) monsters as objects that one could >> study just as one would study individual groups or topological spaces. >> In my experience, skepticism towards category theory is often rooted in >> the fear of the "illegitimately large" size, till today. By comparison, >> Brandt groupoids lived in the cozy and familiar small world, and their >> definition was arrived at without having to leave the universe. With the >> definition of category (and functor and natural transformation) >> Eilenberg and Moore had to do a lot more than just repeating at the >> monoid level what Brandt did at the group level! In my view their big >> psychological step here is comparable to Cantor's daring to think that >> there could be different levels of infinity. >> >> Cheers, >> Walter. > From rrosebru@mta.ca Fri Sep 12 11:42:09 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 11:42:09 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ke9kd-0005Ie-IM for categories-list@mta.ca; Fri, 12 Sep 2008 11:35:59 -0300 Date: Thu, 11 Sep 2008 18:54:11 +0100 (BST) From: Richard Garner To: categories@mta.ca Subject: categories: Re: Equivalence of pseudo-limits 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: 34 In the case singled out by Alex a rather direct proof can also be given. Let Psd denote the 2-category of 2-functors, pseudonatural transformations and modifications P -> Cat. The pseudolimit of a 2-functor F: P -> Cat may be identified with the hom-category Psd(1,F); and accordingly we have a pseudolimit 2-functor lim = Psd(1,-): Psd -> Cat, which sends pseudonatural equivalences F =~ G to equivalences of categories lim(F) =~ lim(G). The corresponding result for pseudolimits in other 2-categories now follows by the Cat-enriched Yoneda lemma. Richard --On 11 September 2008 12:27 Dominic Verity wrote: > Hi Alex, > > This property is certainly known to hold for a much larger class of > 2-categorical limits - the flexible limits, which class includes the classes > of pseudo, lax and op-lax limits. I believe you will find a proof of this > result in the Bird, Kelly, Power and Street paper entitled "Flexible Limits > for 2-Categories". Failing that the Power and Robinson paper "A > characterisation of PIE limits" probably contains this result is some form. > > You can also find explications of the pseudo and lax results in earlier > works by Street, although obvious candidates for the best one to consult > elude me at the moment. > > The most elementary proof of the flexible limit result starts by observing > that all flexible limits can be constructed using products, splittings of > idempotents and a couple of less familiar, exclusively 2-categorical, limits > called inserters and equifiers. It is then straight forward to verify the > result you mention for each of these particular limits and then to infer > that it must therefore hold for all flexible limits. > > In the early 1990's Robert Pare introduced a class of limits called the > persistent limits. These were defined for 2-categories, but made use of his > double categorical approach to 2-limits. Persistent limits are precisely > those limits which have the stability property you seek, but with regard to > a slightly more general class of double categorical diagram transformations > whose 1-cellular components are all equivalences. > > In my thesis (1992), I prove that the class of flexible limits introduced by > Bird, Kelly, Power and Street is identical to Pare's class of persistent > limits - thus closing the circle and demonstrating that the flexible limits > are in a natural sense the largest class of 2-limits which have this > property. > > Regards > > Dominic Verity > > 2008/9/10 Alex Hoffnung > >> Hi all, >> >> Given an indexing 2-category J, a pair of parallel functors >> F,G : J ----> CAT, and a natural equivalence f : F ==> G, >> the pseudo-limits of F and G should be equivalent. >> >> I am trying to find out what paper, if any, I can cite for this theorem. >> Or >> maybe this is just the type of thing that nobody has bothered to write >> down. >> Any help would be appreciated. >> >> best, >> Alex Hoffnung >> > > From rrosebru@mta.ca Fri Sep 12 11:43:01 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 11:43:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ke9lq-0005PT-Ja for categories-list@mta.ca; Fri, 12 Sep 2008 11:37:14 -0300 Date: Thu, 11 Sep 2008 17:12:37 -0400 From: Walter Tholen MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Bourbaki and Categories 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: 35 Here is some up-lifting press about categories that I saw in an article by Karl Heinrich Hofmann entitled "Bourbaki in T"ubingen und in den USA, Erinnerungen an die franz"osische Revolution in der Mathematik", which may translate as "Bourbaki in Tubingen and in the USA, reminiscenses of the French revolution in mathematics", and which appeared in the "Mitteilungen der DMV" (the German equivalent of the AMS Notices, which is distributed to all members), vol 16.2 (2008), pp128-136. While the author has a lot of praise for Bourbaki's work, he lists also a number of "defects of the Bourbaki concept", and the following appears quite prominently in his article (my translation, okayed by the author): "Since Bourbaki is considered as the exponent of the theory of mathematical structures, it is truly surprising that the theory of categories (S. Eilenberg and S. Mac Lane, 1946) was almost demonstrably ignored as the mother of all structure theories. This was hardly sustainable in commutative algebra anymore, and the discord between Grothendieck and Bourbaki may well have been rooted in this rejection. This dismissive position is even more surprising since Eilenberg as one of the few non-French people belonged to the early Bourbaki group, and since the French founder of category theory, Charles Ehresmann, was at times closely connected with Bourbaki. In my view this failure of Bourbaki is grave." From rrosebru@mta.ca Fri Sep 12 11:44:39 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 11:44:39 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ke9nW-0005bE-Q3 for categories-list@mta.ca; Fri, 12 Sep 2008 11:38:58 -0300 Date: Thu, 11 Sep 2008 19:01:57 -0400 From: edubuc To: categories@mta.ca Subject: categories: categories and disdain 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: 36 this is about the recent thread "categories and functors" and "disdain for categories" Ten points: 1) It seems clear that E-M arrived to categories and functors by abstraction from the usual large categories of sets, groups, boolean algebras, modules, etc etc 2) It seems (less clearly) that Ehresman arrived to categories and functors by generalization from groupoids and morphisms of groupoids. 3) I agree with "It's been said before that the real insight of category theory --as something more general than groupoids, monoids, and posets-- is the notion of adjoint functors (including limits, etc)." Concerning this, I think that real breakthrough made by categories is the simple fact that they furnish the appropriate abstract structure to define the Bourbaki's concept of universal property. The fact that the singleton set is characterized by being a terminal object opens the way to characterize thousands of objects and constructions by being the terminal object in the appropriate category. Yoneda's lemma is the milestone. Everything is due to it. 4) I think the small-large business played no role at all in the rise of the concept of categories, neither in the rise of the disdain to them by many mathematicians. 5) "working" mathematicians were never afraid about paradoxes. In consequence, I think that phrases as "dare regarding these (perceived) monsters ...", "fear of the "illegitimately large" size", etc etc, are misleading and out of place. 6) Cantor did not "dare to think that there could be different levels of infinity", he discovered that they were different levels of infinity, and proceed to study this phenomena. This was not a bold action, he was just fascinated by the existence of different levels of infinity. He was not afraid of paradoxes either, he was very well aware of Russell paradox, but for him it was just another theorem. 7) It is often repeated that axiomatic set theory arise in order to eliminate paradoxes. False, axiomatic set theory arise in an attempt to understand Von Neumann accumulation process: Which were the axioms satisfied by the output of that process ? Answer: axiomatic set theory. 8) "In my experience, disdain for cat theory is due to papers with a very high density of unfamiliar names", I agree with this in the sense that this fact contributed to the rise of the disdain, but not as the single reason. I agree also with "Complicated sentences like this can be found in every fields. They are often the mark of a poor paper". It follows there must be other reasons (besides the abundance of poor papers in category theory, a fact that I found true) to explain the disdain. 9) A profound reason could be an instinctive opposition to real change in many people. The instinctive reaction against progress that may disrupt their own comfortable position. 10) The so self proclaimed "problem solvers" who disdain abstract theories often do not resolve any real problem. The just do "concrete nonsense". People who really solve true problems usually have a great respect for abstract theories. Of course, they are also many who just do "abstract nonsense" instead of contribute to the meaningful development of theories. eduardo dubuc From rrosebru@mta.ca Fri Sep 12 11:45:20 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 11:45:20 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ke9oX-0005gn-I0 for categories-list@mta.ca; Fri, 12 Sep 2008 11:40:01 -0300 Date: Fri, 12 Sep 2008 02:56:31 -0700 (PDT) From: Jeff Egger Subject: categories: Another terminological question... To: categories@mta.ca 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: 37 Dear all, In ``basic concepts of enriched category theory'', Kelly writes: > Since the cone-type limits have no special position of > dominancein the general case, we go so far as to call > weighted limits simply ``limits'', where confusion > seems unlikely. My question is this: why does he not apply the same principle to the concept of powers? Instead, he introduces the word ``cotensor'', apparently in order to reserve the word ``power'' for that special case which could sensibly be called ``discrete power''. [This leads to the unfortunate scenario that a ``cotensor'' is a sort of limit, while dually a ``tensor'' is a sort of colimit.] Is there perhaps some genuinely mathematical objection to calling cotensors powers (and tensors copowers) which I may have overlooked? Cheers, Jeff. P.S. I specify ``genuinely mathematical'' because I know that some people are opposed to any change of terminology for any reason whatsoever. Obviously, I disagree; in particular, I don't see that minor terminological schisms such as monad/triple (even compact/rigid/autonomous) are in any way detrimental to the subject. I also disagree with the notion (symptomatic of the curiously feudal mentality which seems to permeate the mathematical community) that prestigious mathematicians have more right to set terminology than the rest of us. I see no correlation between mathematical talent and good terminology; nor do I understand that a great mathematician can be ``dishonoured'' by anything less than strict adherence to their terminology---or notation, for that matter. __________________________________________________________________ Looking for the perfect gift? Give the gift of Flickr! http://www.flickr.com/gift/ From rrosebru@mta.ca Fri Sep 12 15:02:21 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 15:02:21 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KeCt5-00055p-3h for categories-list@mta.ca; Fri, 12 Sep 2008 14:56:55 -0300 Date: Fri, 12 Sep 2008 17:57:00 +0200 From: "zoran skoda" Subject: categories: Re: Bourbaki and Categories 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: RO X-Status: X-Keywords: X-UID: 38 > > sustainable in commutative algebra anymore, and the discord between > Grothendieck and Bourbaki may well have been rooted in this rejection. I can not recall where, but I read more than once more detailed descriptions on what Bourbaki did not accept from Grothendieck. The conservativeness of Bourbaki who did not accept the usage of category theory (not only "neglect") and non-acceptance of a very general approach of Grothendieck to the notion of "manifold" he envisioned for the future Bourbaki works were some of the main points of departure. The remark that as a proponent of "structures" Bourbaki had to include categories is anyway a bit lacking an argument. First of all, because of the size problems one can not take big categories on equal footing with, say groups, and considering only small categories would be strange and lacking most interesting examples. On the other hand, Grothendieck judged the lack stemming in conservativeness rather than in consistency of the structure-oriented style. Indeed, according to Dieudonne, Bourbaki felt comfortable only in including to the books already (meta)stable, "dead" mathematics and not the structures in the unstable "living" phase of development. This was the intended scope and self-conscious (according to Dieudonne) limitation of the work. One can accept this and still cry for an exception for so economic tool as the category theory (if taken in conservative and very basic sense), especially in the vision of the wish for generality, Bourbaki followed otherwise. Zoran Skoda From rrosebru@mta.ca Fri Sep 12 21:14:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 21:14:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KeIhF-0000S6-60 for categories-list@mta.ca; Fri, 12 Sep 2008 21:09:05 -0300 From: Colin McLarty To: categories@mta.ca Date: Fri, 12 Sep 2008 14:46:11 -0400 MIME-Version: 1.0 Subject: categories: Re: Bourbaki and Categories Content-Type: text/plain; charset=iso-8859-1 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: 39 From=3A zoran skoda =3Czskoda=40gmail=2Ecom=3E Date=3A Friday=2C September 12=2C 2008 2=3A06 pm wrote=2C among other things =3E main points of departure=2E The remark that as a proponent of = =3E =22structures=22 Bourbaki =3E had to include categories is anyway a bit lacking an argument=2E = =3E First of all=2C because =3E of the size problems one can not take big categories on equal = =3E footing with=2C say groups=2C =3E and considering only small categories would be strange and lacking = =3E most interesting examples=2E The claim is not that Bourbaki should have studied categories as structures=2E It is that Bourbaki was doomed to fail in trying to use their structure theory=2E Leo Corry shows in his book =22Modern Algebra = and the Rise of Mathematical Structures=22 (Birkh=E4user 1996) that they did = fail=2E = And they should have seen this coming=2C because their theory had been = =22superseded by that of category and functor=2C which includes it under = a more general and convenient form=22 (Dieudonn=E9 =22The Work of Nicholas Bourbaki=22 1= 970)=2E best=2C Colin From rrosebru@mta.ca Fri Sep 12 21:15:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 21:15:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KeIiK-0000Vl-TC for categories-list@mta.ca; Fri, 12 Sep 2008 21:10:12 -0300 Date: Fri, 12 Sep 2008 16:34:55 -0400 (EDT) From: Robert Seely To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 40 For those who like compliments: the triples website has had (for a while now) a link to a lecture by Voevodsky given at the American Academy of Arts and Sciences in October 2002, in which he describes "categories [as] one of the most important ideas of 20th century mathematics". The video of the talk may be found at http://claymath.msri.org/voevodsky2002.mov (the compliment isn't the only reason for watching!). And there are a few other categorical links on our site at http://www.math.mcgill.ca/triples/ Suggestions are always welcome. -= rags =- -- From rrosebru@mta.ca Fri Sep 12 21:15:45 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 21:15:45 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KeIj1-0000YX-Ee for categories-list@mta.ca; Fri, 12 Sep 2008 21:10:55 -0300 Date: Fri, 12 Sep 2008 23:05:56 +0200 (CEST) Subject: categories: Bourbaki and categories (references) From: pierre.ageron@math.unicaen.fr To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 41 On the Bourbaki's attitude towards category theory, there are various references (check online Liliane Beaulieu's bibliography on Bourbaki), bu= t the fundamental study is undoubtedly that of Ralf Kr=F6mer. A very very g= ood paper, limited only by the non-availability of Bourbaki's archives for th= e sixties : KR=D6MER, Ralf La =AB machine de Grothendieck =BB se fonde-t-elle seulement sur des voca= bles m=E9tamath=E9matiques? Bourbaki et les cat=E9gories au cours des ann=E9es cinquante, Revue d'histoire des math=E9matiques 12-1 (2006), pages 119-16= 2 Thank you to Paul for remembering my talk at Amiens. It was certainly related to the subject and also relied on Bourbaki's archives, but dealt with an earlier "pre-categories" period. Only a 2 pages abstract is so far available (I might write down a longer version some day in sha'a l-lah) : AGERON, Pierre Autour d=92Ehresmann : Bourbaki, Cavaill=E8s, Lautman, Cahiers de topolog= ie et de g=E9om=E9trie diff=E9rentielle cat=E9goriques XLVI-3, pages 165-166 (available online via NUMDAM) From rrosebru@mta.ca Sat Sep 13 19:13:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 13 Sep 2008 19:13:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KedGs-0005gP-VR for categories-list@mta.ca; Sat, 13 Sep 2008 19:07:15 -0300 From: Colin McLarty To: categories@mta.ca Date: Fri, 12 Sep 2008 21:25:50 -0400 MIME-Version: 1.0 Content-Language: en Subject: categories: Re: Bourbaki and Categories Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 42 From: Robert Seely Date: Friday, September 12, 2008 8:17 pm noted > For those who like compliments: the triples website has had (for a > while now) a link to a lecture by Voevodsky given at the American > Academy of Arts and Sciences in October 2002, in which he describes > "categories [as] one of the most important ideas of 20th century > mathematics". The video of the talk may be found at > > http://claymath.msri.org/voevodsky2002.mov > > (the compliment isn't the only reason for watching!). It is a terrific lecture. The line "I think that at the heart of 20th century mathematics lies one particular notion and that is the notion of a category" occurs at minute 16. best, Colin From rrosebru@mta.ca Sat Sep 13 19:13:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 13 Sep 2008 19:13:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KedHZ-0005iJ-Oh for categories-list@mta.ca; Sat, 13 Sep 2008 19:07:57 -0300 From: "George Janelidze" To: Subject: categories: Re: Bourbaki and Categories Date: Sat, 13 Sep 2008 16:31:19 +0200 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: RO X-Status: X-Keywords: X-UID: 43 Dear Colleagues, I think the first things to say about "Bourbaki and Categories" are: (a) It is very obvious that the invention of category theory was by far the greatest discovery of 20th century mathematics. (b) Bourbaki Tractate is another great event, of a very different kind of course, which will be a treasure for the Historians of next centuries. It shows how the members of a very leading group of a leading mathematical country were thinking in the middle of the same century (well, up to their internal disagreements; after all, Eilenberg and Grothendieck were also there at some point...). (c) Accordingly, Bourbaki Tractate is the best evidence showing how hard it was to understand (even and especially for such brilliant mathematicians!) that there is something even better that Cantor paradise. (d) Defining structures, Bourbaki makes very clear that morphisms are important (and some form of universal properties are important). But morphisms are NOT defined in general: it is simply a class of maps between structures of a given type closed under composition and having isomorphisms (which ARE defined) as its invertible members. And... every interested student will ask: if so, why not defining a category? Let me also add what is less important but still comes to my mind: (e) Bourbaki approach to structures has a hidden very primitive form of what was later discovered by topos theorists: in order to define a structure they need a 'scales of sets', which is build using finite products and power sets (no unions and no colimits of any kind!). (f) According to Walter Tholen's message, Karl Heinrich Hofmann says: "...it is truly surprising that the theory of categories (S. Eilenberg and S. Mac Lane, 1946) was almost demonstrably ignored as the mother of all structure theories. This was hardly sustainable in commutative algebra anymore...". Very true (except 1946), but it is much-much-much worse in homological algebra, where the absence of categories and functors (having a section called "Functoriality" though) in Bourbaki's presentation is most amazing. (g) A few days ago Tom Leinster has explained to us that "disinformation is *deliberate* false information, false information *intended* to mislead". Fine, but sometimes false information is created by ignorance so badly, that it sounds right to call it disinformation (Don't you agree, Tom?). And... look at http://en.wikipedia.org/wiki/Bourbaki : There is a section called "Criticism of the Bourbaki perspective", which, among other things, says: "The following is a list of some of the criticisms commonly made of the Bourbaki approach:^[13]..." (where [13] is a book of Pierre Cartier; I have not seen that book, and so I am not making any conclusions about it). The list has seven items with no category theory in it! George Janelidze From rrosebru@mta.ca Sat Sep 13 19:16:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 13 Sep 2008 19:16:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KedKh-0005rt-1I for categories-list@mta.ca; Sat, 13 Sep 2008 19:11:11 -0300 MIME-Version: 1.0 Subject: categories: Re: Bourbaki and Categories Date: Sat, 13 Sep 2008 13:17:23 -0400 From: Andre Joyal To: 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: 44 Dear Colin, Zoran, Robert, Eduardo and All, I find the present discussion on Bourbaki and category theory very = important. =20 I recall asking the question to Samuel Eilenberg 25 years ago and more = recently to Pierre Cartier. =20 If my recollection is right, Bourbaki had essentially two options: = rewrite the whole treaty using categories,=20 or just introduce them in the book on homological algebra,=20 The second option won, essentially because of the enormity of the task = of rewriting everything.=20 Other factors may have contributed on a smaller scale, like some = unresolved foundational questions.=20 In any cases, it was the beginning of end for Bourbaki. Bourbaki was a great humanistic and scientific enterprise. Advanced mathematics was made available to a large number of students, possibly over the head of their bad teachers.=20 It defended the unity and rationality of science in an age of growing irrationalism (it was conceived in the mid thirties). I have personally learned a lot of mathematics by reading Bourbaki. =20 Everything was proved, and the proofs were logically very clear. It was a like a continuation of Euclid Elements two thousand years = later! But after a while, I stopped reading it. I had realised that something important was missing: the motivation.=20 The historical notes were very sketchy and not integrated to the text. I remember my feeling of frustration in reading the books of functional = analysis, because the applications to partial differential equations were not = described. Everything was presented in a deductive order, from top to down. We all know that learning is very much an inductive process, from the particular to the general. This is true also of mathematical = research.=20 Bourbaki is dead but I hope that the humanistic philosophy behind the = enterprise is not. =20 Unfortunately, we presently live in an era of growing irrationalism. Science still needs to be defended against religion. Civilisation maybe at a turning point with the problem of climate = change.=20 Millions of people need and want to learn science and mathematics.=20 Should we not try to give Bourbaki a second life?=20 It will have to be different this time. Possibly with a new name. Obviously, internet is the medium of choice. What do you think? Andre -------- Message d'origine-------- De: cat-dist@mta.ca de la part de Colin McLarty Date: ven. 12/09/2008 14:46 =C0: categories@mta.ca Objet : categories: Re: Bourbaki and Categories =20 From: zoran skoda Date: Friday, September 12, 2008 2:06 pm wrote, among other things > main points of departure. The remark that as a proponent of=20 > "structures" Bourbaki > had to include categories is anyway a bit lacking an argument.=20 > First of all, because > of the size problems one can not take big categories on equal=20 > footing with, say groups, > and considering only small categories would be strange and lacking=20 > most interesting examples. The claim is not that Bourbaki should have studied categories as structures. It is that Bourbaki was doomed to fail in trying to use their structure theory. Leo Corry shows in his book "Modern Algebra and the Rise of Mathematical Structures" (Birkh=E4user 1996) that they did = fail. =20 And they should have seen this coming, because their theory had been=20 "superseded by that of category and functor, which includes it under a more general and convenient form" (Dieudonn=E9 "The Work of Nicholas Bourbaki" 1970). best, Colin From rrosebru@mta.ca Sun Sep 14 13:58:59 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 14 Sep 2008 13:58:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Keuqh-0006zb-Ns for categories-list@mta.ca; Sun, 14 Sep 2008 13:53:23 -0300 From: "R Brown" To: Subject: categories: Re: Bourbaki and Categories Date: Sun, 14 Sep 2008 11:24:18 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; 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 Dear All, The importance of Bourbaki should be stessed, as it was started when, so = we=20 are told, texts were very bad. There are many beautiful things in the boo= ks:=20 I developed part of an undergraduate course from the account of the=20 classification of closed subgroups of R^n. This relates to old questions = on=20 orbits of the planets, and also gives some nice exercises and even exam=20 questions of a calculation type. It is good to present students with a=20 classification theorem. The difficulties for Bourbaki seem to arise from the presentation (a) as = a=20 final and definitive view in toto, and (b) without enough context, as And= re=20 points out. On (a), there is the old childish joke: what happens if you put worms in = a=20 straight line from Marble Arch to Picadilly Circus? One of them would be=20 bound to wriggle and spoil it all! So some mathematical worms have not on= ly=20 wriggled but grown large and marched off in a different direction. On (b), there is the old debating society tag: text without context is merely pretext. See more questions in Tim and my article on `Mathematics in Context'. What is wrong is to present, or take, the whole account as totally=20 authoritative, and will last indefinitely. What Bourbaki also shows is the value for at least the writers of taking = a=20 viewpoint and following it through as far as it will go: if it seems in t= he=20 end to go too far, or to be inadequate, then that is valuable information= =20 for them and others. See my Dirac quote in `Out of Line'. Ronnie ----- Original Message -----=20 From: "Andre Joyal" To: Sent: Saturday, September 13, 2008 6:17 PM Subject: categories: Re: Bourbaki and Categories Dear Colin, Zoran, Robert, Eduardo and All, I find the present discussion on Bourbaki and category theory very=20 important. I recall asking the question to Samuel Eilenberg 25 years ago and more=20 recently to Pierre Cartier. If my recollection is right, Bourbaki had essentially two options: rewrit= e=20 the whole treaty using categories, or just introduce them in the book on homological algebra, The second option won, essentially because of the enormity of the task of= =20 rewriting everything. Other factors may have contributed on a smaller scale, like some unresolv= ed=20 foundational questions. In any cases, it was the beginning of end for Bourbaki. Bourbaki was a great humanistic and scientific enterprise. Advanced mathematics was made available to a large number of students, possibly over the head of their bad teachers. It defended the unity and rationality of science in an age of growing irrationalism (it was conceived in the mid thirties). I have personally learned a lot of mathematics by reading Bourbaki. Everything was proved, and the proofs were logically very clear. It was a like a continuation of Euclid Elements two thousand years later= ! But after a while, I stopped reading it. I had realised that something important was missing: the motivation. The historical notes were very sketchy and not integrated to the text. I remember my feeling of frustration in reading the books of functional=20 analysis, because the applications to partial differential equations were not=20 described. Everything was presented in a deductive order, from top to down. We all know that learning is very much an inductive process, from the particular to the general. This is true also of mathematical research= . Bourbaki is dead but I hope that the humanistic philosophy behind the=20 enterprise is not. Unfortunately, we presently live in an era of growing irrationalism. Science still needs to be defended against religion. Civilisation maybe at a turning point with the problem of climate change. Millions of people need and want to learn science and mathematics. Should we not try to give Bourbaki a second life? It will have to be different this time. Possibly with a new name. Obviously, internet is the medium of choice. What do you think? Andre -------- Message d'origine-------- De: cat-dist@mta.ca de la part de Colin McLarty Date: ven. 12/09/2008 14:46 =C0: categories@mta.ca Objet : categories: Re: Bourbaki and Categories From: zoran skoda Date: Friday, September 12, 2008 2:06 pm wrote, among other things > main points of departure. The remark that as a proponent of > "structures" Bourbaki > had to include categories is anyway a bit lacking an argument. > First of all, because > of the size problems one can not take big categories on equal > footing with, say groups, > and considering only small categories would be strange and lacking > most interesting examples. The claim is not that Bourbaki should have studied categories as structures. It is that Bourbaki was doomed to fail in trying to use their structure theory. Leo Corry shows in his book "Modern Algebra and the Rise of Mathematical Structures" (Birkh=E4user 1996) that they did fa= il. And they should have seen this coming, because their theory had been "superseded by that of category and functor, which includes it under a more general and convenient form" (Dieudonn=E9 "The Work of Nicholas Bourbaki" 1970). best, Colin From rrosebru@mta.ca Sun Sep 14 13:59:43 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 14 Sep 2008 13:59:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Keurs-00074B-Jq for categories-list@mta.ca; Sun, 14 Sep 2008 13:54:36 -0300 From: "George Janelidze" To: Categories Subject: categories: Non-cartesian categorical algebra Date: Sun, 14 Sep 2008 15:39:11 +0200 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: 46 Dear Colleagues, I would like to make a remark concerning my CT2008 talk. First let me recall: A lot of mathematics (e.g. of Galois theory) can be done in the context of adjoint functors between abstract categories with finite limits - and since one gets all finite limits our of finite products and equalizers, one can try a further generalization with monoidal structure plus equalizers. The point was that this seemingly primitive old idea actually works very seriously and should be taken as the idea of developing non-cartesian categorical algebra. And "non-cartesian" is the right idea of "non-commutative" and "quantum", although what Ross Street means by "quantum" is more involved and also important. In particular non-cartesian internal categories are to be taken seriously. At the end of my talk Jeff Egger told us that he knows someone studied such generalized internal categories, and later sent me an email with the name: Marcelo Aguiar; and gave the home page address http://www.math.tamu.edu/~maguiar/ , and... I realized that it is the third time I am informed about this work! Recently (winter 2007) I spend two very nice months in Warsaw invited by Piotr Hajac, and discussing mathematics with him, Tomasz Brzezinski, Tomasz Maszczyk, and a few others - and, among other interesting things, Tomasz Brzezinski showed me Marcelo Aguiar's website, including PhD, where those generalized internal categories were studied. I also recall now an email message from Steven Chase (from 2002) where he mentions "...the notion of a category internal to a monoidal category which was developed by my former doctoral student, Marcelo Aguiar, in his thesis, "Internal Categories and Quantum Groups" (available on line...". In fact the whole story begins, in some sense, with the book [S. U. Chase and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics 97, Springer 1969], which does not use monoidal categories yet, but very clearly shows that the commutative case is much easier (for Galois theory) because it makes tensor product (of algebras) (co)cartesian. There are many other important further contributions by other authors of different generations. Knowing them personally, I can name Bodo Pareigis, Stefaan Caenepeel, Peter Schauenburg, and the aforementioned Polish mathematicians (although Tomasz Brzezinski is in UK now), but I am not ready to give any reasonably complete list. There are also things-to-be-corrected happening: for instance by far not enough comparisons have been made with the Australian work on abstract monoidal categories, and some authors use words like "coring"... George Janelidze From rrosebru@mta.ca Mon Sep 15 08:34:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 08:34:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfCGg-0001Zw-9J for categories-list@mta.ca; Mon, 15 Sep 2008 08:29:22 -0300 Date: Mon, 15 Sep 2008 06:55:55 +0200 From: Andre.Rodin@ens.fr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 47 zoran skoda wrote: >The remark that as a proponent of "structures" >Bourbaki had to include categories is anyway a bit lacking an argument. I think that as a 'proponent of "structures"' Bourbaki had NOT include categories - and not only because of the size problem. A more fundamental reason seems me to be this. Structures are things determined up to isomor= phism; in the structuralist mathematics the notion of isomorphism is basic and t= he notion of general morphism is derived (as in Bourbaki). In CT this is th= e other way round: the notion of general morphism is basic while isos are d= efined through a specific property (of reversibility). This is why the inclusion of CT would require a revision of fundamentals = of Bourbaki's structuralist thinking. Although CT for obvious historical rea= sons is closely related to structuralist mathematics it is not, in my understa= nding, a part of structuralist mathematics - at least not if one takes CT *serio= usly*, i.e. as foundations. best, andrei From rrosebru@mta.ca Mon Sep 15 08:34:24 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 08:34:24 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfCG2-0001X6-55 for categories-list@mta.ca; Mon, 15 Sep 2008 08:28:42 -0300 Date: Sun, 14 Sep 2008 13:53:12 -0600 (MDT) Subject: categories: Re: Bourbaki and Categories From: mjhealy@ece.unm.edu 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: 48 Andre Joyal's message was inspiring. I think a new Bourbaki type of effort (this time with motivation and category theory) is called for. I would like a text on categorical algebra oriented toward those who have studied algebra and know enough category theory to use adjunctions, monad= s and the like. The same goes for categorical logic and model theory. I read Andre's closing remarks as a commentary on the emerging crisis in overcoming misconceptions about and outright hostility toward science and mathemtics. I have a current project, using my own meager funds and time= , to advance science teaching using a book available through the National Academies Press, "Science, Evolution, and Creationism". From the title, = I think you can see my motivation. In a similar (although less aprocryphal) vein, when I worked with compute= r scientists and applied mathematicians in industry ( and also when I've submitted papers to certain neural network journals) I encountered misconceptions about and outright hostility toward category theory. For example, in the dynamic systems community there seems to be a widespread myth that "category theory was tried and failed". I have followed this u= p to some extent and haven't found any basis for it. I am often told that the best way to counter skepticism is with a working application. Having tried that, and tried again, I've come to the conclusion that Yes, you need applications, but applications cannot by themselves counter a refusal to give a theory credit for being consistent with the data. You need a good, clear presentation of the theory couched in a language oriented toward the intended audience. As with biology teaching that shows clearly the importance of the theory of evolution, maybe mathematic= s teaching that incorporates category theory needs to begin in 6th Grade (i= n schools in the USA) if not sooner. Maybe a new Bourbaki project could have an extension into this level of instruction. Best Regards, Mike Please excuse my deviating from mathematics From rrosebru@mta.ca Mon Sep 15 08:37:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 08:37:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfCJP-0001pm-V4 for categories-list@mta.ca; Mon, 15 Sep 2008 08:32:12 -0300 Date: Mon, 15 Sep 2008 09:58:33 +0200 From: Andree Ehresmann To: categories@mta.ca Subject: categories: Re: Bourbaki and 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: O X-Status: X-Keywords: X-UID: 49 Dear all, I would add some information on Bourbaki/categories/France. Charles Ehresmann has been an active member of Bourbaki from 1936 up =20 to the end of the war, when he began to no more regularly participate =20 and wanted to resign (it was not accepted but replaced by an age limit =20 for active participation). What Andre says: >Bourbaki had essentially two options: rewrite the whole treaty using > categories, or just introduce them in the book on homological algebra, >The second option won, essentially because of the enormity of the task > of rewriting everything. is more easily understood if we take into account that communication =20 between France and the USA were entirely broken during the war, so =20 that mathematical ideas could not circulate and categories were only =20 heard of after the war, at a time where the more general parts of the =20 treatise were written or at least prepared (the successive versions =20 process was very slow). Charles said to me that he did not recall to have read Eilenberg =20 & Mac Lane's paper before the fifties, or at least not seen its =20 interest. Naturally he had sooner made a large use of groupoids in is =20 foundation of differential geometry, and he had even defined the =20 general "composition of jets" and given its properties, but without =20 linking it to the notion of a category. He exposed it in a course in =20 Rio de Janeiro in the early fifties, and one of his students =20 (Constantino de Barros who later came to Paris to prepare a thesis =20 with him) suggested that there was a connection with categories. =20 Charles' first large use of categories is in his seminal paper =20 "Gattungen von lokalen Strukturen" (1957, reprinted in "Charles =20 Ehresmann: Oeuvres completes et commentees" Part I). It is around this date that the word "category" began to circulate =20 in France. In 1957, Choquet (with whom I prepared my thesis) =20 suggested that I learnt more on the notion of category which he did =20 not know but seemed to have many applications (it was the reason for =20 which I first went to see Charles!). It should be noted that Choquet =20 was less conservative than many French mathematicians. In 1959, he =20 defended the development of probabilities by inviting Loomis to give a =20 course (I remember Henri Cartan saying then to Paul-Andre Meyer that =20 he should not study this domain for it would be bad for his career!). =20 And later on, he defended Logic which was very badly considered. A final remark: the "disdain" for categories (not to be confused =20 with 'ignorance') came only later on, since Charles was given the =20 "Prix Petit d'Ormoy" by the French Academy in 1965, essentially for =20 his recent work on categories... Andree C. Ehresmann From rrosebru@mta.ca Mon Sep 15 19:20:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:20:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMJQ-0003Qk-Ek for categories-list@mta.ca; Mon, 15 Sep 2008 19:12:52 -0300 Date: Mon, 15 Sep 2008 07:59:53 -0400 (EDT) From: Michael Barr To: Andre.Rodin@ens.fr cc: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 50 I don't know about this. I took several courses in the late 1950s that seem to have been influenced by the structuralist ideas (certainly categories weren't mentioned; I never heard the word until Dave Harrison arrived in 1959) and each of them started by defining an appropriate notion of "admissible map". I do not recall any special point being made of isomorphism and I think in general it was used for what we now call a bimorphism (1-1 and onto) even in cases, such as topological groups, when they were not isomorphisms. To be sure Bourbaki was not mentioned either, but this structuralist influence seemed strong. Michael On Mon, 15 Sep 2008, Andre.Rodin@ens.fr wrote: > > zoran skoda wrote: > > >> The remark that as a proponent of "structures" >> Bourbaki had to include categories is anyway a bit lacking an argument. > > > > I think that as a 'proponent of "structures"' Bourbaki had NOT include > categories - and not only because of the size problem. A more fundamental > reason seems me to be this. Structures are things determined up to isomorphism; > in the structuralist mathematics the notion of isomorphism is basic and the > notion of general morphism is derived (as in Bourbaki). In CT this is the > other way round: the notion of general morphism is basic while isos are defined > through a specific property (of reversibility). > This is why the inclusion of CT would require a revision of fundamentals of > Bourbaki's structuralist thinking. Although CT for obvious historical reasons > is closely related to structuralist mathematics it is not, in my understanding, > a part of structuralist mathematics - at least not if one takes CT *seriously*, > i.e. as foundations. > > best, > andrei > > > From rrosebru@mta.ca Mon Sep 15 19:22:04 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:22:04 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMKr-0003Wj-9Y for categories-list@mta.ca; Mon, 15 Sep 2008 19:14:21 -0300 From: Joost Vercruysse To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v926) Subject: categories: Re: Non-cartesian categorical algebra Date: Mon, 15 Sep 2008 14:57:55 +0200 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 51 On 14-sep-08, at 15:39, George Janelidze wrote: Dear George and all, > There are also things-to-be-corrected happening: > for instance by far not enough comparisons have been made with the > Australian work on abstract monoidal categories, and some authors > use words > like "coring"... I hope the following information can be of help here: Indeed, Marcello Aguilar gave a definition of `internal categories'. Although the abstract definition of a `coring' looks formally the same as the one of an internal category (or, if you wish, an internal cocategory), corings provide examples of these internal cocategories, but they (usually) refer to a much more concrete situation: a coring is a co-monoid in the monoidal category of bimodules over a given (possibly non-commutative) ring, this dualizes usual ring extensions. The theory of corings is in fact quite young, and grew from a pure algebraic theory to something more and more categorical in the last few years (this might cause some confusion, `internal corings', which can be defined in certain monoidal categories (the regular ones from aguilar) or bicategories, are indeed the same objects as internal cocategories, there is no need for two names for the same thing at this level of generality). Therefore, I find the above remark ``not enough comparision have been made ...'' indeed correct: I believe that people from corings can learn from more from the pure category theory side, and hopefully the other way around as well. Best wishes, Joost. From rrosebru@mta.ca Mon Sep 15 19:25:06 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:25:06 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMNb-0003h0-4W for categories-list@mta.ca; Mon, 15 Sep 2008 19:17:11 -0300 To: categories@mta.ca Subject: categories: Categories in Algebra, Geometry and Logic - Brussels 10, 11 October 2008 From: Rudger Kieboom Date: Mon, 15 Sep 2008 18:03:27 +0200 Content-Transfer-Encoding: quoted-printable 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: 52 Dear All, This is the second announcement (and final announcement on this categories - mailing list) of the meeting in Brussels on Friday 10 and Saturday 11 October 2008 in honour of Francis Borceux and Dominique Bourn on the occasion of their 60th birthdays. =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D Categories in Algebra, Geometry and Logic =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D Friday 10 and Saturday 11 October 2008, at the Royal Flemish Academy of Belgium for Sciences and the Arts Paleis der Academie=EBn / Palais des Academies Hertogsstraat 1 Rue Ducale 1000 Brussel / Bruxelles Belgium For all information, and downloadable poster, see the conference web page: http://www.math.ua.ac.be/bbdays/ Further (final) announcements (on the conference dinner, and other practical information) will be sent by e-mail to the registered participants. Abstracts of the talks will be made available on the web page as soon as possible. Keynote Speakers (55 min. talks): ---------------------------------------------- J. Ad=E1mek (Braunschweig) J. B=E9nabou (Paris) M. M. Clementino (Coimbra) A. Ehresmann (Amiens) G. Janelidze (Cape Town) P. T. Johnstone (Cambridge) F. W. Lawvere (Buffalo) J. Penon (Paris) J. Rosick=FD (Brno) W. Tholen (Toronto) Shorter Communications (25 min. talks): ------------------------------------------------------ S. Caenepeel (Brussel) Z. Janelidze (Cape Town) S. Mantovani (Milano) D. Rodelo (Faro) I. Stubbe (Antwerpen) T. Van der Linden (Coimbra, Brussel) From rrosebru@mta.ca Mon Sep 15 19:26:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:26:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMQA-0003ri-Vw for categories-list@mta.ca; Mon, 15 Sep 2008 19:19:51 -0300 Date: Mon, 15 Sep 2008 20:26:56 +0100 (BST) From: Dusko Pavlovic To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 53 i think that we should try to heed andre joyal's call for action. he calls for a new collaborative effort a la bourbaki, this time based on categories from the outset. it is true that very ambitious efforts usually fail, and this would be an extremely ambitious one. moreover, taking action sounds like something people used to do in 20th century, and not in these times of fox news and smooth crowd control. but there are two points that make me think that andre's call is different: 1) he is pointing to the reasons for action, that are slowly but surely catching up with every scientist, no matter how much we try to ignore them. 2) he is suggesting a medium (web, internet) that may make a difference between... well between being able to make a difference and not being able to make a difference. ad (2), i would like to add that the web tools facilitate in a substantial way not only dissemination, but also collaboration. there are methods to support more efficient knowledge aggregation from a broader base than ever before. developing a suitable collaboration process may be hard (at least as hard as developing a suitable voting procedure), but it may be worth while. eg, the wikipedia process can be criticized from many angles; but wikipedia has the amazing property that it is an *evolutionary* knowledge repository, which can easily correct any observed shortcomings, and recover from any misinterpretations, almost like science itself. at the moment, the wikipedia process is probably not optimal for presenting subtle or many faceted concepts, and the discussions of everyone with everyone else are not the most productive way. that is perhaps why most of us (with some very honorable exceptions!) have been staying away from it. but an improved process, combining the integrity, and perhaps the structure of the categories@mta community with the available wiki-methods may bring categorical methods into a dynamic environment, perhaps more natural for them than books and papers. just my 2c, -- dusko PS like an unwanted pop song, the name Nicolas Bourwiki just emerged in my head! can someone please propose a worse one, or i am stuck. oh, i already have a worse one... On Sep 13, 2008, at 10:17 AM, Andre Joyal wrote: > Bourbaki is dead but I hope that the humanistic philosophy behind the > enterprise is not. Unfortunately, we presently live in an era of > growing irrationalism. > Science still needs to be defended against religion. > Civilisation maybe at a turning point with the problem of climate > change. > Millions of people need and want to learn science and mathematics. > > Should we not try to give Bourbaki a second life? > It will have to be different this time. > Possibly with a new name. > Obviously, internet is the medium of choice. > What do you think? > > Andre From rrosebru@mta.ca Mon Sep 15 19:26:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:26:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMPb-0003q1-93 for categories-list@mta.ca; Mon, 15 Sep 2008 19:19:15 -0300 Mime-Version: 1.0 (Apple Message framework v753) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed To: Content-Transfer-Encoding: quoted-printable From: David Spivak Subject: categories: Re: Bourbaki and Categories Date: Mon, 15 Sep 2008 11:51:24 -0700 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 54 I agree with Andre. Encapsulating a group of mathematicians inside a =20= single named entity fosters a kind of collaborative spirit in which =20 good ideas are not kept for personal use later but are shared amongst =20= the community. When ideas are shared in real time, good mathematics =20 can be produced faster. Anyone who wants to join the collective can =20 do so, and the collective produces highly useful material. Of course =20= such an enterprise is orthogonal to name-recognition, and maybe to =20 getting tenure! But there is certainly something good about it, as =20 there is about wikipedia and the open source movement. I also agree that the internet could be used in a better way to =20 transfer knowledge of mathematics. Math papers are written linearly, =20= in the bottom-up (Euclid/Bourbaki) style, to some extent. Whereas =20 words on paper are in this sense one-dimensional, computers offer =20 many more dimensions for knowledge transfer. Even more interesting to me would be a kind of zoom-feature on =20 proofs. Proofs are in the eye of the beholder: for example it has =20 been debated as to whether Perelman's 70 pages was a full proof of =20 geometrization. Given a proof with a statement which one does not =20 understand, a mathematician may find himself reproving something that =20= was obvious to (or wrongly assumed to be obvious by) another =20 mathematician. The community could benefit if a mathematician who =20 proves such a statement then uploaded the proof, even in rough form, =20 to some kind of math wiki. If it were well-organized, this math wiki =20= could revolutionize how mathematics is done. In fact, choosing the =20 "right way" to organize such a site may itself be a problem which =20 could produce interesting mathematics. Whatever the case may be, I am all for the idea of a new Bourbaki-=20 style enterprise in some form or another. I think it may first =20 require interested parties to get together at some physical location. David On Sep 13, 2008, at 10:17 AM, Andre Joyal wrote: > Dear Colin, Zoran, Robert, Eduardo and All, > > I find the present discussion on Bourbaki and category theory very =20 > important. > I recall asking the question to Samuel Eilenberg 25 years ago and =20 > more recently to Pierre Cartier. > If my recollection is right, Bourbaki had essentially two options: =20 > rewrite the whole treaty using categories, > or just introduce them in the book on homological algebra, > The second option won, essentially because of the enormity of the =20 > task of rewriting everything. > Other factors may have contributed on a smaller scale, like some =20 > unresolved foundational questions. > In any cases, it was the beginning of end for Bourbaki. > > Bourbaki was a great humanistic and scientific enterprise. > Advanced mathematics was made available to a large number > of students, possibly over the head of their bad teachers. > It defended the unity and rationality of science in an age > of growing irrationalism (it was conceived in the mid thirties). > > I have personally learned a lot of mathematics by reading Bourbaki. > Everything was proved, and the proofs were logically very clear. > It was a like a continuation of Euclid Elements two thousand years =20= > later! > But after a while, I stopped reading it. > I had realised that something important was missing: the motivation. > The historical notes were very sketchy and not integrated to the text. > I remember my feeling of frustration in reading the books of =20 > functional analysis, > because the applications to partial differential equations were not =20= > described. > Everything was presented in a deductive order, from top to down. > We all know that learning is very much an inductive process, from > the particular to the general. This is true also of mathematical =20 > research. > > Bourbaki is dead but I hope that the humanistic philosophy behind =20 > the enterprise is not. > Unfortunately, we presently live in an era of growing irrationalism. > Science still needs to be defended against religion. > Civilisation maybe at a turning point with the problem of climate =20 > change. > Millions of people need and want to learn science and mathematics. > > Should we not try to give Bourbaki a second life? > It will have to be different this time. > Possibly with a new name. > Obviously, internet is the medium of choice. > What do you think? > > Andre > > From rrosebru@mta.ca Mon Sep 15 19:28:59 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:28:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMRd-0003yV-04 for categories-list@mta.ca; Mon, 15 Sep 2008 19:21:21 -0300 Date: Tue, 16 Sep 2008 06:58:46 +1000 Subject: categories: Re: Another terminological question... From: Steve Lack To: 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: 55 Dear Jeff, I had a chat about this with a couple of other long-time users of the terms tensor and cotensor (Ross Street and Dominic Verity). We all think that, given the current overburdening of the word tensor, this would be a sensible change. Regards, Steve Lack. On 12/09/08 7:56 PM, "Jeff Egger" wrote: > Dear all, > > In ``basic concepts of enriched category theory'', > Kelly writes: > >> Since the cone-type limits have no special position of >> dominancein the general case, we go so far as to call >> weighted limits simply ``limits'', where confusion >> seems unlikely. > > My question is this: why does he not apply the same > principle to the concept of powers? Instead, he > introduces the word ``cotensor'', apparently in order > to reserve the word ``power'' for that special case > which could sensibly be called ``discrete power''. > [This leads to the unfortunate scenario that a > ``cotensor'' is a sort of limit, while dually a > ``tensor'' is a sort of colimit.] Is there perhaps > some genuinely mathematical objection to calling > cotensors powers (and tensors copowers) which I may > have overlooked? > > Cheers, > Jeff. > > P.S. I specify ``genuinely mathematical'' because I > know that some people are opposed to any change of > terminology for any reason whatsoever. Obviously, > I disagree; in particular, I don't see that minor > terminological schisms such as monad/triple (even > compact/rigid/autonomous) are in any way detrimental > to the subject. > > I also disagree with the notion (symptomatic of the > curiously feudal mentality which seems to permeate the > mathematical community) that prestigious mathematicians > have more right to set terminology than the rest of us. > I see no correlation between mathematical talent and > good terminology; nor do I understand that a great > mathematician can be ``dishonoured'' by anything less > than strict adherence to their terminology---or notation, > for that matter. > > From rrosebru@mta.ca Tue Sep 16 21:12:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:12:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkYK-0003hh-P1 for categories-list@mta.ca; Tue, 16 Sep 2008 21:05:52 -0300 From: "George Janelidze" To: Subject: categories: Re: Bourbaki and Categories Date: Tue, 16 Sep 2008 02:03:12 +0200 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: 56 Dear Andree, Could you please explain this better?: The only Bourbaki member I new personally was Sammy Eilenberg. As many of us, I knew him very well and I would say that he was more skeptical about the Bourbaki Tractate then one can conclude from Andre's message. Having in mind not just this but the content of Bourbaki's "Homological algebra" and what we see today from the followers of that Bourbaki group, I protest against Andre's "two options" and I insist that Bourbaki group simply did not see the importance of category theory (in spite of being brilliant mathematicians, as I said in my previous message). I hope Andre will forgive me and even agree with me. However, there were three great category-theorists in that group (plus there is this mysterious story about Chevalley's book of category theory lost in the train), and "did not see" cannot be said about them of course. On the other hand I have never heard of any joint work of Charles Ehresmann with any of the two others, Eilenberg and Grothendieck (and nothing jointly from them). I think apart from the time issues you describe, the relationship between Bourbaki Tractate and category theory should have been determined by their separate or joint influence and therefore also by their communication with each other (if any). Is this true, and could you please give details? Respectfully, and with best regards- George From rrosebru@mta.ca Tue Sep 16 21:13:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:13:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkZX-0003mE-LM for categories-list@mta.ca; Tue, 16 Sep 2008 21:07:07 -0300 Date: Tue, 16 Sep 2008 08:52:36 +0200 From: Andrej Bauer MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 57 Dear friends, the usual kind of wiki is not suitable for collaborative science, but recently there has been news of a special wiki for scientists which has good support for references, keeps track of who said what, and has a rating system. You can read more about it in Nature Genetics here http://www.nature.com/ng/journal/v40/n9/full/ng.f.217.html and see it working here: http://www.wikigenes.org/ They even have movies for those who are too lazy to click: http://www.wikigenes.org/app/info/movie.html It looks however that they are not offering the software that runs the whole thing. The next Bourbaki, if there is going to be one, should not only advance one particular kind of knowledge, but also show everyone that linearly written text is not the only option. My opinion is that we have not yet found the right way to do "hive-science", but when we do, it will be a revolution. (A good start would be to get out of the hold that the evil publishers have on us.) Best regards, Andrej From rrosebru@mta.ca Tue Sep 16 21:16:23 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:16:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkcR-000403-I0 for categories-list@mta.ca; Tue, 16 Sep 2008 21:10:07 -0300 Date: Tue, 16 Sep 2008 01:57:50 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 58 Bourbaki redone as Bourwiki (thanks, Dusko!) with the benefit of category theoretic insights will hopefully clarify some segments of mathematics. What troubles me in this discussion however is its assumed scope of "some." I get the sense that there are people who want it to be mandated as "all." Perhaps it should be. Just now I looked through an issue of American Mathematical Monthly that came to hand to get a sense of the likely alignment of Bourwiki with what the mathematical community generally regards as the scope of its subject. Actually I do this periodically, and I don't see much change between the issue I picked up just now and any of the other issues I've looked at in the past with just this question in mind. If the subject Bourwiki is proposing to serve is mathematics, then perhaps it is time that the American Mathematical Monthly, along with the Putnam Mathematical Competition, the International Mathematics Olympiad, and the Journal of the AMS, abandon their pretense of being about mathematics and come up with a suitable name for their subject. Not only do categories, functors, natural transformations, adjunctions, and monads go unused in these 20th century icons of mathematics, they go unacknowledged. Clearly they have not gotten with the modern mathematical program and fall somewhere between a throwback to a golden age and a backwater of mathematics. When they die off like the dinosaurs they are, real mathematics will be able to advance unfettered into the 21st century and beyond. Judging from the talks at BLAST in Denver last month (B = Boolean algebras, L = lattices, A = (universal) algebra, S = set theory, T = topology), at least the algebraic community is moving very slightly in this direction. Things will hopefully improve yet further when algebraic geometry gets over its snit with equational model theory. Meanwhile if you need a witness for seven degrees of separation, look no further than AMM and CT. (I confess to being an unreconstructed graph theorist and algebraist myself. I may have to preemptively volunteer myself for re-education before it becomes involuntary.) Vaughan Pratt From rrosebru@mta.ca Tue Sep 16 21:18:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:18:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkeM-0004BB-Ir for categories-list@mta.ca; Tue, 16 Sep 2008 21:12:06 -0300 Date: Tue, 16 Sep 2008 12:27:07 +0200 From: Andre.Rodin@ens.fr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 59 When one defines, say, a group =E0 la Borbaki, i.e. structurally, it usua= lly goes without saying that the defined structure is defined up to isomorphism. T= he notion of isomorphism plays in this case the role similar to that of equa= lity in the (naive) arithmetic. In most structural constexts the distinction b= etween the "same" structure and isomorphic structures is mathematically trivial = just like the distinction between the "same" number and equal numbers. It may = be not specially discussed in this case exactly because it is very basic. The no= tion of admissible map, say, that of group homomorphism, on the contrary, requ= ires a definition, which may be non-trivial. The idea to do mathematics up to isomorphism is not Bourbaki's invention;= it goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In th= is sense the modern axiomatic method is structuralist. In his often-quoted l= etter to Frege Hilbert explicitely says that a theory is "merely a framework" w= hile domains of their objects are multiple and transform into each other by "univocal and reversible one-one transformations". Those who trace the hi= story of mathematical structuralism back to Hilbert are quite right, in my view= . I have in mind two issues related to CT, which suggest that CT goes in a *different* direction - in spite of the fact that MacLane and many other workers in CT had (and still have) structuralist motivations. The first i= s Functorial Semantics, which brings a *category* of models, not just one m= odel up to isomorphism. From the structuralist viewpoint the presence of non-isomorphic models (i.e. non-categoricity) is a shortcoming of a given theory. From the perspective of Functorial Semantics it is a "natural" fe= ature of mathematical theories to be dealt with rather than to be remedied. The second thing I have in mind is Sketch theory. I cannot see that Hilbe= rt's basic structuralist intuition applies in this case. In my understanding t= hings work in Sketch theory more like in Euclid. Think about circle and straigh= t line as a sketch of the theory of the first four books of Euclid's "Elements".= I would particularly appreciate, Michael, your comment on this point since = I learnt a lot of Sketch theory from your works. I have also a comment about the idea to rewrite Bourbaki's "Elements" fro= m a new categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for h= is work just like did Hilbert writing his "Gundlagen". In my view, this is t= his long-term Euclidean tradition of "working foundations", which is worth to= be saved and further developed, in particular in a categorical setting. I'm = less sure that Bourbaki's example should be followed in a more specific sense. Bourbaki tries to cover too much - and doesn't try to distinguish between= what belongs to foundations and what doesn't. As a result the work is too long= and not particularly usefull for (early) beginners. I realise that today's mathematics unlike mathematics of Euclid's time is vast, so it is more difficult to present its basics in a concentrated form. But consider Hilb= ert's "Grundlagen". It covers very little - actually near to nothing - of geome= try of its time. But at the same time it provided a very powerful model of how t= o do mathematics in a new way, which greatly influenced mathematics education = and mathematical research in 20th century. In my view, Euclid's "Elements" an= d Hilbert's "Grundlagen" are better examples to be followed. best, andrei le 15/09/08 12:59, Michael Barr =E0 barr@math.mcgill.ca a =E9crit : > I don't know about this. I took several courses in the late 1950s that > seem to have been influenced by the structuralist ideas (certainly > categories weren't mentioned; I never heard the word until Dave Harriso= n > arrived in 1959) and each of them started by defining an appropriate > notion of "admissible map". I do not recall any special point being ma= de > of isomorphism and I think in general it was used for what we now call = a > bimorphism (1-1 and onto) even in cases, such as topological groups, wh= en > they were not isomorphisms. > > To be sure Bourbaki was not mentioned either, but this structuralist > influence seemed strong. > > Michael > From rrosebru@mta.ca Tue Sep 16 21:20:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:20:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkhB-0004N6-Eg for categories-list@mta.ca; Tue, 16 Sep 2008 21:15:01 -0300 From: Tom Hirschowitz To: categories@mta.ca Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable Mime-Version: 1.0 (Apple Message framework v926) Subject: categories: Full professor position Date: Tue, 16 Sep 2008 13:40:52 +0200 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 60 Dear all, Here is a bit of additional information about the full professor position at Universit=E9 de Savoie I advertised in July (full message below). The schedule for 'qualification' is available here = http://www.education.gouv.fr/personnel/enseignant_superieur/enseignant_che= rcheur/calendrier_qualification.htm=20 . In summary: - registration is now open, and will close on October 14th (5pm, =20 Paris time), - applicants should defend their habilitation before December 10th. Hoping to see some categorists show up, Tom Original message: ------------------------ A full professor position will be available at Universit=E9 de Savoie in Chamb=E9ry (France), from September 1, 2009. The new professor will be a member of the LIMD group (Logique, Informatique et Math=E9matiques Discr=E8tes) and is expected to do his research in proof theory in relation to computer science, preferably in one or more of the following areas: lambda-calculus, type theory, realizability, denotational semantics (games, categories, ...), linear logic, concurrency and mobility (process algebras, bisimilarity, semantics, ...), mathematics of programming languages (design, typing, compilation, ...), ... The LIMD group is a UMR (mixed CNRS-University research unit), part of the mathematics laboratory (LAMA) at Universit=E9 de Savoie. The LAMA (*) has presently 27 permanent researchers in total; the LIMD group has 3 full Professors, 4 "Maitre de conferences" and 2 CNRS researchers. The position is made available by the retirement of one of the professors. The teaching assignments are: mathematics and/or computer science for students from L1 to M2 (in the new european terminology). To get this position, it is necessary to - speak french fluently, and - have been accepted on the so-called "liste de qualification". If you are not yet qualified, note that the deadline for applying is usually around mid-october. For further details, please don't hesitate to contact me. Tom Hirschowitz (*) http://www.lama.univ-savoie.fr/ From rrosebru@mta.ca Tue Sep 16 21:20:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:20:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfkgZ-0004KF-IW for categories-list@mta.ca; Tue, 16 Sep 2008 21:14:23 -0300 Date: Tue, 16 Sep 2008 07:24:43 -0400 (EDT) From: Michael Barr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories MIME-Version: 1.0 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: 61 I don't know what to say about the suggestion that a circle and a line=20 make a sketch of which Euclidean plane geometry is a model. I would thin= k=20 you would need a point too, since intersections are crucial. Maybe=20 complex projective geometry since then two lines intersect in one point=20 (unless they coincide), a line and a circle in two (unless they are=20 tangent or equal) and every pair of circles in four (ditto). Maybe the=20 exceptions could be handled in some sketch. At any rate, it wold e=20 interesting to try to sketch this in detail. At any rate, I never though= t=20 about this before. Michael On Tue, 16 Sep 2008, Andre.Rodin@ens.fr wrote: > > When one defines, say, a group =E0 la Borbaki, i.e. structurally, it us= ually goes > without saying that the defined structure is defined up to isomorphism.= The > notion of isomorphism plays in this case the role similar to that of eq= uality > in the (naive) arithmetic. In most structural constexts the distinction= between > the "same" structure and isomorphic structures is mathematically trivia= l just > like the distinction between the "same" number and equal numbers. It ma= y be not > specially discussed in this case exactly because it is very basic. The = notion > of admissible map, say, that of group homomorphism, on the contrary, re= quires a > definition, which may be non-trivial. > The idea to do mathematics up to isomorphism is not Bourbaki's inventio= n; it > goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In = this > sense the modern axiomatic method is structuralist. In his often-quoted= letter > to Frege Hilbert explicitely says that a theory is "merely a framework"= while > domains of their objects are multiple and transform into each other by > "univocal and reversible one-one transformations". Those who trace the = history > of mathematical structuralism back to Hilbert are quite right, in my vi= ew. > I have in mind two issues related to CT, which suggest that CT goes in = a > *different* direction - in spite of the fact that MacLane and many othe= r > workers in CT had (and still have) structuralist motivations. The first= is > Functorial Semantics, which brings a *category* of models, not just one= model > up to isomorphism. From the structuralist viewpoint the presence of > non-isomorphic models (i.e. non-categoricity) is a shortcoming of a giv= en > theory. From the perspective of Functorial Semantics it is a "natural" = feature > of mathematical theories to be dealt with rather than to be remedied. > The second thing I have in mind is Sketch theory. I cannot see that Hil= bert's > basic structuralist intuition applies in this case. In my understanding= things > work in Sketch theory more like in Euclid. Think about circle and strai= ght line > as a sketch of the theory of the first four books of Euclid's "Elements= ". I > would particularly appreciate, Michael, your comment on this point sinc= e I > learnt a lot of Sketch theory from your works. > I have also a comment about the idea to rewrite Bourbaki's "Elements" f= rom a new > categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for= his > work just like did Hilbert writing his "Gundlagen". In my view, this is= this > long-term Euclidean tradition of "working foundations", which is worth = to be > saved and further developed, in particular in a categorical setting. I'= m less > sure that Bourbaki's example should be followed in a more specific sens= e. > Bourbaki tries to cover too much - and doesn't try to distinguish betwe= en what > belongs to foundations and what doesn't. As a result the work is too lo= ng and > not particularly usefull for (early) beginners. I realise that today's > mathematics unlike mathematics of Euclid's time is vast, so it is more > difficult to present its basics in a concentrated form. But consider Hi= lbert's > "Grundlagen". It covers very little - actually near to nothing - of geo= metry of > its time. But at the same time it provided a very powerful model of how= to do > mathematics in a new way, which greatly influenced mathematics educatio= n and > mathematical research in 20th century. In my view, Euclid's "Elements" = and > Hilbert's "Grundlagen" are better examples to be followed. > > best, > andrei > > > le 15/09/08 12:59, Michael Barr =E0 barr@math.mcgill.ca a =E9crit : > >> I don't know about this. I took several courses in the late 1950s tha= t >> seem to have been influenced by the structuralist ideas (certainly >> categories weren't mentioned; I never heard the word until Dave Harris= on >> arrived in 1959) and each of them started by defining an appropriate >> notion of "admissible map". I do not recall any special point being m= ade >> of isomorphism and I think in general it was used for what we now call= a >> bimorphism (1-1 and onto) even in cases, such as topological groups, w= hen >> they were not isomorphisms. >> >> To be sure Bourbaki was not mentioned either, but this structuralist >> influence seemed strong. >> >> Michael >> > From rrosebru@mta.ca Tue Sep 16 21:22:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:22:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kfkir-0004Vb-Nh for categories-list@mta.ca; Tue, 16 Sep 2008 21:16:45 -0300 Date: Tue, 16 Sep 2008 15:09:56 +0200 From: Andre.Rodin@ens.fr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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 Of course, you are right about a point, I missed it! I must confess I did= n't think about this example in precise terms. My claim is that sketch theory doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly precisely fits the ancient Euclidean pattern either but there is a sugges= tive analogy, which concerns the idea that certain basic objects like point, l= ine and circle *generate* the rest. A further claim is this: a specific reason *why* sketch theory doesn't fi= t the structuralist pattern is that in sketch theory (like in CT in general) isomorphisms don't have the same distinguished status. andrei >I don't know what to say about the suggestion that a circle and a line >make a sketch of which Euclidean plane geometry is a model. I would thi= nk >you would need a point too, since intersections are crucial. Maybe >complex projective geometry since then two lines intersect in one point >(unless they coincide), a line and a circle in two (unless they are >tangent or equal) and every pair of circles in four (ditto). Maybe the >exceptions could be handled in some sketch. At any rate, it wold e >interesting to try to sketch this in detail. At any rate, I never thoug= ht >about this before. >Michael From rrosebru@mta.ca Tue Sep 16 21:23:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:23:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kfkjs-0004aV-K7 for categories-list@mta.ca; Tue, 16 Sep 2008 21:17:48 -0300 Date: Tue, 16 Sep 2008 10:20:25 -0400 From: jim stasheff MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 63 David Spivak wrote: > I agree with Andre. Encapsulating a group of mathematicians inside a > single named entity fosters a kind of collaborative spirit in which > good ideas are not kept for personal use later but are shared amongst > the community. When ideas are shared in real time, good mathematics > can be produced faster. Anyone who wants to join the collective can > do so, and the collective produces highly useful material. Of course > such an enterprise is orthogonal to name-recognition, and maybe to > getting tenure! A partial solution: those who already have sufficient name recognition should proclaim the value contributed by those without it and especially in support of tenure for them. jim From rrosebru@mta.ca Tue Sep 16 21:25:01 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:25:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kfkl4-0004gc-9r for categories-list@mta.ca; Tue, 16 Sep 2008 21:19:02 -0300 Date: Tue, 16 Sep 2008 10:47:27 -0400 (EDT) From: Michael Barr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 64 Still, it might be interesting and even instructive to try to build a sketch whose objects are (interpreted as) sets of points sets of lines, sets of circles and intersection is an operation. I guess incidence would have to be a relation. It might work. Michael From rrosebru@mta.ca Tue Sep 16 21:25:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Sep 2008 21:25:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kfklv-0004k3-El for categories-list@mta.ca; Tue, 16 Sep 2008 21:19:55 -0300 Date: Tue, 16 Sep 2008 17:32:11 +0200 From: Andre.Rodin@ens.fr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 65 Certainly! I shall try. Thank you for encouraging! This issue seems me al= so interesting from a different viewpoint. Even if the New Maths was a pedag= ogical failure it is still remarkable that the traditional school mathematics ca= n be wholly spelled out in the Bourbaki-like terms. I wonder if CT allows for anything of this sort. andrei >Still, it might b interesting and even instructive to try to build a >sketch whose objects are (interpreted as) sets of points sets of lines, >sets of circles and intersection is an operation. I guess incidence wou= ld >have to a relation. It might work. >Michael From rrosebru@mta.ca Thu Sep 18 10:33:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:33:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJY0-0005Vz-Tw for categories-list@mta.ca; Thu, 18 Sep 2008 10:27:52 -0300 Date: Wed, 17 Sep 2008 11:30:53 +1000 Subject: categories: Re: Bourbaki and Categories From: Steve Lack To: , 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: 66 Dear Andrei, Sketches are not mathematical objects in their own right, in the same sense that groups or spaces are. They are presentations (for theories), and have status similar to other sorts of presentations (for groups, rings, etc.) Of course that is in no way meant to suggest that they are not important and worthy of study. Regards, Steve Lack. On 16/09/08 11:09 PM, "Andre.Rodin@ens.fr" wrote: > > Of course, you are right about a point, I missed it! I must confess I didn't > think about this example in precise terms. My claim is that sketch theory > doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly > precisely fits the ancient Euclidean pattern either but there is a suggestive > analogy, which concerns the idea that certain basic objects like point, line > and circle *generate* the rest. > A further claim is this: a specific reason *why* sketch theory doesn't fit the > structuralist pattern is that in sketch theory (like in CT in general) > isomorphisms don't have the same distinguished status. > > andrei > > > > > >> I don't know what to say about the suggestion that a circle and a line >> make a sketch of which Euclidean plane geometry is a model. I would think >> you would need a point too, since intersections are crucial. Maybe >> complex projective geometry since then two lines intersect in one point >> (unless they coincide), a line and a circle in two (unless they are >> tangent or equal) and every pair of circles in four (ditto). Maybe the >> exceptions could be handled in some sketch. At any rate, it wold e >> interesting to try to sketch this in detail. At any rate, I never thought >> about this before. > >> Michael > > > From rrosebru@mta.ca Thu Sep 18 10:34:25 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:34:25 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJYa-0005Zd-PL for categories-list@mta.ca; Thu, 18 Sep 2008 10:28:28 -0300 Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Ross Street Subject: categories: Re: Non-cartesian categorical algebra Date: Wed, 17 Sep 2008 12:41:06 +1000 To: Joost Vercruysse , Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 67 There is an embedding theorem on which we have put Cayley's name: if M is a monoid in a closed category then the structural coretraction M --> [M,M] into the endohom is a nice monoid map. A bicategorical version of this gives a nice module (distributor) A --|--> A^{op} #A for any (pro)monoidal V-category A. This leads to a monoidal embedding of any such A into the category of A-bimodules. (E.g. see Section 4 of Pastro-St: http://www.tac.mta.ca/tac/volumes/21/4/21-04.pdf however Brian Day also knew about these things.) So the abstract case is not so much more abstract. I think Peter Johnstone says somewhere that one view of the Abelian Category Embedding Theorem is not so much that it means we should use module-proofs to work in abelian categories but rather, when working in categories of modules, we might as well work in an abelian category. I think the same applies here for monoidal categories. The coring people I have spoken to seem quite comfortable with this development. Luckily we all have our own sources of motivation. Ross On 15/09/2008, at 10:57 PM, Joost Vercruysse wrote: > cocategory), corings provide examples of these internal cocategories, > but they (usually) refer to a much more concrete situation: a coring > is a co-monoid in the monoidal category of bimodules over a given > (possibly non-commutative) ring, this dualizes usual ring extensions. From rrosebru@mta.ca Thu Sep 18 10:34:48 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:34:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJZF-0005e2-Vt for categories-list@mta.ca; Thu, 18 Sep 2008 10:29:10 -0300 Date: Wed, 17 Sep 2008 06:36:49 +0200 From: Andre.Rodin@ens.fr To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 68 Dear Steve, > Sketches are not mathematical objects in their own right, in the same s= ense > that groups or spaces are. Of course, they are not. >They are presentations (for theories), and have > status similar to other sorts of presentations (for groups, rings, etc.= ) I think about a sketch as an alternative to a string of formulae, which represents (axioms of) a theory. I didn't try to compare sketch theory wi= th group theory. I tried to compare sketch theory with the general setting,= in which group theory is developed a la Bourbaki (along with many other theories). A classical account of this general setting (which differs at = certain points with Bourbaki's version) is Tarski's model theory. The notion of presentation in my understanding implies that what a given presentation is a presentation *of* is somehow given in advance. I try to= think of a sketch as a means to build a theory, not to present a ready-made the= ory. Perhaps *representation* is a better word for it than *presentation*. best, andrei From rrosebru@mta.ca Thu Sep 18 10:36:15 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:36:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJaU-0005o5-8O for categories-list@mta.ca; Thu, 18 Sep 2008 10:30:26 -0300 From: "R Brown" To: "Vaughan Pratt" , Subject: categories: Re: Bourbaki and Categories Date: Wed, 17 Sep 2008 10:17:54 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; 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: 69 There are already some pretty good categorical entries on wiki; I have modified some of the entries on groups, actions, equivalence relations, to include references to groupoids, which has resulted in hits. But we should also consider planetmath.org (entries are contributed under the terms of the GNU Free Documentation License (FDL)) which allows for group work and is not so open as wiki to general modification. It needs a group of you wonderful energetic people to engage with reviewing what is on wiki and planetmath and making sure they express what is fealt to be desirable! In the old days, a graduate book would have an appendix on say set theory, and maybe basic algebra, as needed for the rest of the text. It would be very useful to have basic category theory (in terms of `the basic facts of life') on the web available to all, with nice accounts of say `left adjoints preserve colimits', etc. , with many convincing examples, and maybe history, to which a text could refer. Something initially less ambitious like this might actually get done. Being electronic, it would be seen as a `current', rather than `final account', and so would better reflect the way mathematics develops, in which a slight shift of emphasis, or notation (like --> for a function), can have profound consequences. There is perhaps a case for a separate collected electronic account, with hyperref, and also a printed version, since a book is a useful portable random access device. Print on Demand allows this to be produced quite cheaply, with a 35% royalty on retail sales, and available on amazon. Ronnie ----- Original Message ----- From: "Vaughan Pratt" To: Sent: Tuesday, September 16, 2008 9:57 AM Subject: categories: Re: Bourbaki and Categories > Bourbaki redone as Bourwiki (thanks, Dusko!) with the benefit of > category theoretic insights will hopefully clarify some segments of > mathematics. > > What troubles me in this discussion however is its assumed scope of > "some." I get the sense that there are people who want it to be > mandated as "all." > > Perhaps it should be. > > Just now I looked through an issue of American Mathematical Monthly that > came to hand to get a sense of the likely alignment of Bourwiki with > what the mathematical community generally regards as the scope of its > subject. Actually I do this periodically, and I don't see much change > between the issue I picked up just now and any of the other issues I've > looked at in the past with just this question in mind. > > If the subject Bourwiki is proposing to serve is mathematics, then > perhaps it is time that the American Mathematical Monthly, along with > the Putnam Mathematical Competition, the International Mathematics > Olympiad, and the Journal of the AMS, abandon their pretense of being > about mathematics and come up with a suitable name for their subject. > Not only do categories, functors, natural transformations, adjunctions, > and monads go unused in these 20th century icons of mathematics, they go > unacknowledged. Clearly they have not gotten with the modern > mathematical program and fall somewhere between a throwback to a golden > age and a backwater of mathematics. When they die off like the > dinosaurs they are, real mathematics will be able to advance unfettered > into the 21st century and beyond. > > Judging from the talks at BLAST in Denver last month (B = Boolean > algebras, L = lattices, A = (universal) algebra, S = set theory, T = > topology), at least the algebraic community is moving very slightly in > this direction. Things will hopefully improve yet further when > algebraic geometry gets over its snit with equational model theory. > > Meanwhile if you need a witness for seven degrees of separation, look no > further than AMM and CT. > > (I confess to being an unreconstructed graph theorist and algebraist > myself. I may have to preemptively volunteer myself for re-education > before it becomes involuntary.) > > Vaughan Pratt > > From rrosebru@mta.ca Thu Sep 18 10:41:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:41:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJfh-0006Sb-Hi for categories-list@mta.ca; Thu, 18 Sep 2008 10:35:49 -0300 MIME-Version: 1.0 Subject: categories: Re: Bourbaki and Categories Date: Wed, 17 Sep 2008 13:13:21 -0400 From: Andre Joyal To: "George Janelidze" , 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: 70 Dear George, I thank you for your message. You wrote: >I insist that Bourbaki group simply did >not see the importance of category theory. It is difficult to know. The Boubaki group had shielded itself in secrecy, like a free mason cell.=20 You are surely aware of the interview of Pierre Cartier=20 in the Mathematical Intelligencer No1 1998.=20 Everyone interested in the history of Bourbaki should read it. http://ega-math.narod.ru/Bbaki/Cartier.htm Let me stress a few passages:=20 >The fourth generation was more or less a group of students of = Grothendieck. But at that time Grothendieck had already left Bourbaki.=20 >He belonged to Bourbaki for about ten years but he left in anger. The = personalities were very strong at the time.=20 >I remember there were clashes very often. There was also, as usual, a = fight of generations, like in any family.=20 >I think a small group like that repeated more or less the psychological = features of a family.=20 >So we had clashes between generations, clashes between brothers, and so = on.=20 >But they did not distract Bourbaki from his main goal, even though they = were quite brutal occasionally.=20 >At least the goal was clear. There were a few people who could not take = the burden of this psychological style,=20 >for instance Grothendieck left and also Lang dropped out. > It is amazing that category theory was more or less the brainchild of = Bourbaki. The two founders were Eilenberg and MacLane.=20 > MacLane was never a member of Bourbaki, but Eilenberg was, and MacLane = was close in spirit. The first textbook on homo-logical=20 >algebra was Cartan-Eilenberg, which was published when both were very = active in Bourbaki. Let us also mention Grothendieck, who=20 >developed categories to a very large extent. I have been using = categories in a conscious or unconscious way in much of my work,=20 >and so had most of the Bourbaki members. But because the way of = thinking was too dogmatic, or at least the presentation in the=20 >books was too dogmatic, Bourbaki could not accommodate a change of = emphasis, once the publication process was started. >In accordance with Hilbert's views, set theory was thought by Bourbaki = to provide that badly needed general framework. If you need=20 > some logical foundations, categories are a more flexible tool than set = theory. The point is that categories offer both a general=20 > philosophical foundation=97that is the encyclopedic, or taxonomic = part=97and a very efficient mathematical tool, to be used in=20 >mathematical situations. That set theory and structures are, by = contrast, more rigid can be seen by reading the final chapter in=20 >Bourbaki set theory, with a monstrous endeavor to formulate categories = without categories. The interview ends with the following passage. The bold faces are from me. >When I began in mathematics the main task of a mathematician was to = bring order and make a synthesis of existing material, to create=20 >what Thomas Kuhn called normal science. Mathematics, in the forties and = fifties, was undergoing what Kuhn calls a solidification period.=20 >In a given science there are times when you have to take all the = existing material and create a unified terminology, unified standards,=20 >and train people in a unified style. The purpose of mathematics, in the = fifties and sixties, was that, to create a new era of normal science.=20 >Now we are again at the beginning of a new revolution. Mathematics is = undergoing major changes. We don't know exactly where it will go. > It is not yet time to make a synthesis of all these things=97MAYBE IN = TWENTY OR THIRTY CENTURY IT WILL BE TIME FOR A NEW BOURBAKI. >I consider myself very fortunate to have had two lives, a life of = normal science and a life of scientific revolution. Is it the time for a new Bourbaki? Best regards,=20 Andr=E9 -------- Message d'origine-------- De: cat-dist@mta.ca de la part de George Janelidze Date: lun. 15/09/2008 20:03 =C0: categories@mta.ca Objet : categories: Re: Bourbaki and Categories =20 Dear Andree, Could you please explain this better?: The only Bourbaki member I new personally was Sammy Eilenberg. As many = of us, I knew him very well and I would say that he was more skeptical = about the Bourbaki Tractate then one can conclude from Andre's message. Having = in mind not just this but the content of Bourbaki's "Homological algebra" = and what we see today from the followers of that Bourbaki group, I protest against Andre's "two options" and I insist that Bourbaki group simply = did not see the importance of category theory (in spite of being brilliant mathematicians, as I said in my previous message). I hope Andre will = forgive me and even agree with me. However, there were three great category-theorists in that group (plus = there is this mysterious story about Chevalley's book of category theory lost = in the train), and "did not see" cannot be said about them of course. On = the other hand I have never heard of any joint work of Charles Ehresmann = with any of the two others, Eilenberg and Grothendieck (and nothing jointly = from them). I think apart from the time issues you describe, the relationship between Bourbaki Tractate and category theory should have been = determined by their separate or joint influence and therefore also by their = communication with each other (if any). Is this true, and could you please give details? Respectfully, and with best regards- George From rrosebru@mta.ca Thu Sep 18 10:43:36 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:43:36 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJhb-0006kv-FO for categories-list@mta.ca; Thu, 18 Sep 2008 10:37:47 -0300 From: Topos8@aol.com Date: Thu, 18 Sep 2008 08:50:33 EDT Subject: categories: Princeton Companion To Mathematics - slight revision 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: 71 I was just paging through a copy of "The Princeton Companion to Mathematics. Tim Gowers and all who contributed to this volume did a superb job in producing such a comprehensive and in-depth look at the culture and content of mathematics. There is, I think, one lacuna in this otherwise wonderful book. Eugenia Cheng wrote a very fine two page explanation of the concept of a category for this volume. But in the 350 pages devoted to the various branches of mathematics, 350 pages divided into 26 topics such as Algebraic Geometry, Algebraic Topology, Differential Topology, Harmonic Analysis etc, there is no article on Category Theory. This should be contrasted with the 11 page article on Set Theory, and the 12 page article on Logic and Model Theory. What does this say about the general perception of the proper role for category theory in mathematics? Carl Futia From rrosebru@mta.ca Fri Sep 19 13:25:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:25:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kgifd-0003RE-Fq for categories-list@mta.ca; Fri, 19 Sep 2008 13:17:25 -0300 Date: Thu, 18 Sep 2008 10:31:24 -0400 (EDT) From: Michael Barr To: Steve Lack , categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 72 Of course sketches are mathematical objects in their own right. Of course, the functor that assigns to each sketch the corresponding theory is not full or faithful. But the definition is precise, the notion of model is also precise, so I have no idea what, if any, content there is in the claim. Incidentally, you might with equal justice claim that triples are not mathematical objects since two distinct triples can have isomorphic categories of Eilenberg-Moore algebras. In fact there are triples (or theories) on Set that have infinitary operations, yet whose category of models is isomorphic to Set. Michael On Wed, 17 Sep 2008, Steve Lack wrote: > Dear Andrei, > > Sketches are not mathematical objects in their own right, in the same sense > that groups or spaces are. They are presentations (for theories), and have > status similar to other sorts of presentations (for groups, rings, etc.) > > Of course that is in no way meant to suggest that they are not important and > worthy of study. > > Regards, > > Steve Lack. > > > On 16/09/08 11:09 PM, "Andre.Rodin@ens.fr" wrote: > >> >> Of course, you are right about a point, I missed it! I must confess I didn't >> think about this example in precise terms. My claim is that sketch theory >> doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly >> precisely fits the ancient Euclidean pattern either but there is a suggestive >> analogy, which concerns the idea that certain basic objects like point, line >> and circle *generate* the rest. >> A further claim is this: a specific reason *why* sketch theory doesn't fit the >> structuralist pattern is that in sketch theory (like in CT in general) >> isomorphisms don't have the same distinguished status. >> >> andrei >> >> >> >> >> >>> I don't know what to say about the suggestion that a circle and a line >>> make a sketch of which Euclidean plane geometry is a model. I would think >>> you would need a point too, since intersections are crucial. Maybe >>> complex projective geometry since then two lines intersect in one point >>> (unless they coincide), a line and a circle in two (unless they are >>> tangent or equal) and every pair of circles in four (ditto). Maybe the >>> exceptions could be handled in some sketch. At any rate, it wold e >>> interesting to try to sketch this in detail. At any rate, I never thought >>> about this before. >> >>> Michael >> >> >> > > > From rrosebru@mta.ca Fri Sep 19 13:26:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:26:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kgigd-0003Xt-I5 for categories-list@mta.ca; Fri, 19 Sep 2008 13:18:27 -0300 Date: Thu, 18 Sep 2008 10:36:27 -0400 (EDT) From: Michael Barr To: R Brown , categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 73 The three books I have authored or co-authored all contain a chapter of about 40 pages that is an introduction to category theory. They are largely identical (the one for Acyclic Models has a an added section on categories of fractions, needed for that book). The one in TTT is already freely available and, with Charles's permission, I would happily post that in whatever place you would like. Michael On Wed, 17 Sep 2008, R Brown wrote: > There are already some pretty good categorical entries on wiki; I have > modified some of the entries on groups, actions, equivalence relations, to > include references to groupoids, which has resulted in hits. But we should > also consider planetmath.org (entries are contributed under the terms of the > GNU Free Documentation License (FDL)) which allows for group work and is not > so open as wiki to general modification. It needs a group of you wonderful > energetic people to engage with reviewing what is on wiki and planetmath and > making sure they express what is fealt to be desirable! > ... From rrosebru@mta.ca Fri Sep 19 13:26:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:26:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgihO-0003dK-1J for categories-list@mta.ca; Fri, 19 Sep 2008 13:19:14 -0300 Date: Thu, 18 Sep 2008 18:18:46 +0200 (CEST) From: Peter Schuster To: Categories Subject: categories: Constructive Topology - Workshop G. Sambin 60 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: 74 Advances in Constructive Topology and Logical Foundations Workshop in Honour of the 60th Birthday of Giovanni Sambin Padua, Italy, 8-11 October 2008 http://www.math.unipd.it/60thsambin/ From rrosebru@mta.ca Fri Sep 19 13:26:38 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:26:38 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kgigz-0003aW-Iy for categories-list@mta.ca; Fri, 19 Sep 2008 13:18:49 -0300 Date: Thu, 18 Sep 2008 11:42:33 -0400 From: edubuc MIME-Version: 1.0 To: Categories list Subject: categories: bourbaki_and_disdain 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: 75 Hello 1) I agree completely with Andre, or, more properly, with Samuel Eilemberg and Pierre Cartier (Andre just tel us what they said to him when he ask them the question) So the problem is not wether to agree with Andre or not, but wether to agree with Eilemberg-Cartier or not. They said to Andre: "Bourbaki had essentially two options: rewrite the whole treaty using categories, or just introduce them in the book on homological algebra, The second option won, essentially because of the enormity of the task of rewriting everything." 2) We can see that this makes sense too, the Bourbaki Tractate was already written, and Grothendieck's proposal was to entirely rewrite the thing !! 3) Also, Bourbaki Tractate, as it is, is a masterful book. It is just perfect, or as close to perfection as possible. It sets a way, philosophy and style to write mathematics. I myself (as Andre) have learned a lot by reading Bourbaki, and, more than that, I enjoyed the reading (and the task to understand it) as much as I enjoy any reading where I can see perfection in every sense (as in Borges for example). 4) So, it is a lost for mathematics and for category theory that category theory fit in Bourbaki's goal as a foundational theory (as set theory is) , and not just as one more part of mathematics (as set theory is not). What a marvelous book just on category theory Bourbaki could have written, can you imagine !! A Boubaki book on category theory !! We miss it . . . 5) Concerning "desdain", I recall to all of you that most of the people who show desdain for category theory, also show desdain for Bourbaki. Eduardo. From rrosebru@mta.ca Fri Sep 19 13:27:33 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:27:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgiiN-0003ie-Vx for categories-list@mta.ca; Fri, 19 Sep 2008 13:20:16 -0300 To: "Michael Barr" , Subject: categories: Re: Bourbaki and Categories Date: Thu, 18 Sep 2008 21:38:31 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: From: cat-dist@mta.ca Status: RO X-Status: X-Keywords: X-UID: 76 Michael, That seems a very good and generous idea! It could initially be posted on your site, except that it should also be indefinitely available. The pdf should have hyperref, and perhaps there should be a complete version and also a parcelled version, so that wiki and Planet Math could have links to specific parts. What do you think? Ronnie ----- Original Message ----- From: "Michael Barr" To: "R Brown" Cc: "Vaughan Pratt" ; Sent: Thursday, September 18, 2008 3:36 PM Subject: Re: categories: Re: Bourbaki and Categories > The three books I have authored or co-authored all contain a chapter of > about 40 pages that is an introduction to category theory. They are > largely identical (the one for Acyclic Models has a an added section on > categories of fractions, needed for that book). The one in TTT is already > freely available and, with Charles's permission, I would happily post that > in whatever place you would like. > > Michael > From rrosebru@mta.ca Fri Sep 19 13:28:10 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:28:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kgij4-0003pt-9X for categories-list@mta.ca; Fri, 19 Sep 2008 13:20:58 -0300 Date: Thu, 18 Sep 2008 23:52:47 +0200 From: Andree Ehresmann To: categories@mta.ca Subject: categories: Re: Bourbaki and 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: 77 George Janedlize writes > Could you please explain this better... > the Bourbaki group simply did not see the importance of category theory... > However, there were three great category-theorists in that group... > I have never heard of any joint work of Charles Ehresmann with any =20 > of the two others, Eilenberg and Grothendieck... ...the relation =20 > between Bourbaki Tractate and category theory should have been =20 > determined by their separate or joint influence and therefore also =20 > by their communication with each other (if any). I'll try to explain why there is no contradiction. 1. Charles only participated actively to the Bourbaki group from 1935 =20 to the mid forties, at a time he did not know category theory. In 1935 =20 he had written a first version for the volume "Theorie des ensembles" =20 where he introduced the notions of local structures and associated =20 pseudogroups of transformations (not so far from groupoids!), but this =20 version was not accepted and he did not like the final version =20 published much later. After the war, he only participated irregularly =20 because he felt that he was no more able to make himself heard, the =20 decisions being taken by "those who spoke the more loudly" (as he said =20 to me). 2. Around 1950 it was decided that active participation ended at 45 =20 (the age Charles had then), lessening the influence of those =20 (Eilenberg, Cartan, Chevalley and Dieudonne) who could have stressed =20 the importance of categories. I don't know exactly when Grothendieck =20 became a member, but it was much later, and I think he did not remain =20 for long. Later on, disdain for category theory had developed in =20 France... 3. As for the communication between Charles and the other =20 category-theorists, he had no contact with Grothendieck who was much =20 younger. He was friendly with Eilenberg but did not see him often. =20 Before the war he lived in Paris and regularly met Henri Cartan, =20 Dieudonne, and more specially, Chevalley (both had regular exchanges =20 with the philosophers Cavailles and Lautman). But their communication =20 almost ceased after the war when he developed all his activity in =20 Strasbourg (up to 1955) and was out of France for a long part of the =20 year. Anyway, before our joint work (from the mid sixties up to his =20 death), Charles worked essentially alone and published no joint work =20 at all, except 6 Notes on Topology or Geometry with some of his =20 students. When he began to specialize in category theory in the =20 sixties, it was not well understood by other French mathematicians, =20 and his influence dwindled up to a real opposition in the seventies. Andree From rrosebru@mta.ca Fri Sep 19 13:29:50 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 19 Sep 2008 13:29:50 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgijZ-0003tq-K4 for categories-list@mta.ca; Fri, 19 Sep 2008 13:21:29 -0300 Date: Fri, 19 Sep 2008 03:00:32 -0700 From: John Baez To: categories Subject: categories: Bourbaki and Categories Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 78 On Wed, Sep 17, 2008 at 01:13:21PM -0400, Andre Joyal wrote: > Is it the time for a new Bourbaki? Do we imagine this new Bourbaki as just systematizing and presenting what we know already, or struggling to create brand new mathematics? I can imagine a new Bourbaki who tries to explain all of mathematics in the language of categories. But I can also imagine a new Bourbaki who tries to explain all of mathematics in the language of infinity-categories. It may be a bit too late for the first one, and a bit too early for the second one. Perhaps we need both! Best, jb From rrosebru@mta.ca Sat Sep 20 10:13:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 10:13:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh2Bc-0006No-9C for categories-list@mta.ca; Sat, 20 Sep 2008 10:07:44 -0300 From: "R Brown" To: "edubuc" , "Categories list" Subject: categories: Re: bourbaki_and_disdain Date: Fri, 19 Sep 2008 21:22:08 +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: 79 In this discussion I would like to mention some things about Alexander = Grothendieck: In 1958 at Edinburgh ICM (me very callow!) I happened to hear Raoul Bott = say 2 things which expressed his own amazement about AG: One was that AG could play with concepts and make something real of it. = (Compare AG's comment to me much later: `The introduction of the cipher = 0 or the group concept was general nonsense too, and mathematics was = more or less stagnating for thousands of years because nobody was around = to take such childish steps ...',) I like the idea that `childish steps' = should still be possible in mathematics, and may be more fun than trying = the famous problem line. In fact Saul Ulam told me at a conference in = Sicily in 1964 that a young person may think that the most ambitious = thing to do is to try for solving a famous problem; but this might = distract them from developing the mathematics most appropriate to them. = I was not in danger of the former, but I thought it interesting that = someone so good as Ulam should make this comment, and think it worth = publicising, since it is relevant to aims. =20 The other comment of Bott was that AG was prepared to work very hard to = make things become tautologous. Should one say this is also (except in = the exercises!) the intention of Bourbaki ?=20 AG also mentioned to me that his initial direction at Paris was not = viewed favourably until it proved the generalisation of the Riemann-Roch = theorem!=20 Ronnie ----- Original Message -----=20 From: "edubuc" To: "Categories list" Sent: Thursday, September 18, 2008 4:42 PM Subject: categories: bourbaki_and_disdain > Hello >=20 > 1) I agree completely with Andre, or, more properly, with Samuel > Eilemberg and Pierre Cartier (Andre just tel us what they said to him > when he ask them the question) >=20 > So the problem is not wether to agree with Andre or not, but wether to > agree with Eilemberg-Cartier or not. They said to Andre: >=20 > "Bourbaki had essentially two options: rewrite the whole treaty using > categories, > or just introduce them in the book on homological algebra, > The second option won, essentially because of the enormity of the task > of rewriting everything." >=20 > 2) We can see that this makes sense too, the Bourbaki Tractate was > already written, and Grothendieck's proposal was to entirely rewrite = the > thing !! >=20 ... From rrosebru@mta.ca Sat Sep 20 10:14:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 10:14:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh2DR-0006SN-GV for categories-list@mta.ca; Sat, 20 Sep 2008 10:09:37 -0300 Date: Fri, 19 Sep 2008 18:21:14 -0400 From: "Zinovy Diskin" Subject: categories: Re: Bourbaki and Categories 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: RO X-Status: X-Keywords: X-UID: 80 Dear Michael, Still some content in Steve's claim could be imagined. A working mathematician (WM) works with Borubaki's structures like groups or vector spaces and leaves all worries about what his proofs actually mean for a working math logician. For such a WM, sketches (as any other syntactical machineries) are indeed technical minutiae rather than mathematical objects. It'd be perhaps a reasonable view unless a bunch of strong semantic results (Tarski, Mal'cev,Robinson) that our WM values so much, which are provided by bringing syntax onto the stage. Zinovy On Thu, Sep 18, 2008 at 10:31 AM, Michael Barr wrote: > Of course sketches are mathematical objects in their own right. Of > course, the functor that assigns to each sketch the corresponding theory > is not full or faithful. But the definition is precise, the notion of > model is also precise, so I have no idea what, if any, content there is in > the claim. Incidentally, you might with equal justice claim that triples > are not mathematical objects since two distinct triples can have > isomorphic categories of Eilenberg-Moore algebras. In fact there are > triples (or theories) on Set that have infinitary operations, yet whose > category of models is isomorphic to Set. > > Michael > From rrosebru@mta.ca Sat Sep 20 10:15:22 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 10:15:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh2EK-0006UJ-Fl for categories-list@mta.ca; Sat, 20 Sep 2008 10:10:32 -0300 Date: Sat, 20 Sep 2008 00:27:41 +0200 (CEST) Subject: categories: Re: Bourbaki and Categories From: Mark.Weber@pps.jussieu.fr To: "Michael Barr" , 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: 81 Dear Michael, Semantically, as Lawvere observed long ago, a monad gives rise not just t= o a category of algebras but also to a forgetful functor into the category on which the monad acts. For any category C the functor "semantics" : Mnd(C)^op --> CAT/C whose object map sends a monad on C to its associated forgetful functor i= s full and faithful. Thus a pair of monads on C giving rise to isomorphic forgetful functors must necessarily be isomorphic. So your observations about different monads giving rise to the same algebras, while correct, d= o not tell the whole story on the semantic side. The situation is of course different for sketches: they too give rise to forgetful functors (into Set), but this does not suffice to determine a given sketch up to isomorphism in the same way, and this justifies Steve Lack's perspective of "sketches as presentations of theories". Mark Weber > Previously, Michael Barr wrote: > > Of course sketches are mathematical objects in their own right. Of > course, the functor that assigns to each sketch the corresponding theor= y > is not full or faithful. But the definition is precise, the notion of > model is also precise, so I have no idea what, if any, content there is= in > the claim. Incidentally, you might with equal justice claim that tripl= es > are not mathematical objects since two distinct triples can have > isomorphic categories of Eilenberg-Moore algebras. In fact there are > triples (or theories) on Set that have infinitary operations, yet whose > category of models is isomorphic to Set. > > Michael > From rrosebru@mta.ca Sat Sep 20 11:11:35 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 11:11:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh357-0001MI-Eq for categories-list@mta.ca; Sat, 20 Sep 2008 11:05:05 -0300 Date: Fri, 19 Sep 2008 15:53:13 -0400 (EDT) Subject: categories: Re: Non-Cartesian Homological Algebra 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: 82 Here are some further remarks on the interesting concepts discussed in George's post and the replies to it. Incidentally, I prefer the term "non-commutative category theory" to "non-Cartesian ...", but that's a personal matter. If A is an algebra over a field k, then, in the context of the internal category theory introduced in Aguiar's 1997 thesis, an A-coring is simpl= y an internal category in the dual of the monoidal category Vec(k) of=20 k-spaces. On the other hand, the notion of internal category in Vec(k) itself includes (but is more general than) the small k-linear categories in Mitchell's 1972 paper, "Rings with Several Objects". I think it is useful to view these 2 concepts as special cases of the same general theory. Some of the constructions for corings in the book of Brzezinski and Wisbauer are special cases of categorical notions developed in Aguiar's thesis. Corings have been around for awhile, as Sweedler's original paper introducing them was published in 1975, and they appeared in at least a few other papers during the ensuing 20 years (e.g., Takeuchi's monograph on a Morita theory for monoidal categories of bimodules, which = I think was published in the Journal of the Math. Society of Japan).=20 Sweedler's version of Jacobson's theorem in his paper seems to be a sort of non-commutative analogue of the connection between equivalence relations and quotient spaces. Aguiar's framework is probably not general enough to cover some cases of interest. It may be that one should begin simply with a monad in an arbitrary bicategory, since at least a few of the basic concepts for internal categories can be developed in that setting. Taking the bicategory to be Vec(k) with a single object, the important notion of entwined structure, discussed in work of Caenepeel, Brzezinski, Pareigis, and others, then appears as a sort of distributive law, but relating a monad and a comonad rather than 2 monads. The theorem that an entwined structure is equivalent to a certain type of coring is then apparently an analogue of Beck's theorem. One would like the theory to include Takeuchi's notion of X-bialgebra (which generalizes concepts developed earlier by Sweedler and, independently, David Winter). However, to that end it appears useful to enrich the bicategory so that the 1-endomorphisms of an object (and 2-morphisms between them) constitute a 2-monoidal category in the sense of [M. Aguiar and S. Mahajan, Hopf Monoids in Species and Associated Hopf Algebras] (see Chapter 5, although most of that monograph, not yet completed, is on a quite different subject). There is a link to the monograph from Aguiar's website. He an= d I have had some discussions on these matters during the summer, but there is still much about this situation that we (or, at least, I) don't understand. Steve Chase From rrosebru@mta.ca Sat Sep 20 11:11:35 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 11:11:35 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh35t-0001Oh-Kf for categories-list@mta.ca; Sat, 20 Sep 2008 11:05:53 -0300 Date: Fri, 19 Sep 2008 22:16:31 -0400 From: jim stasheff MIME-Version: 1.0 To: categories Subject: categories: Re: Bourbaki and Categories 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: 83 John Baez wrote: > On Wed, Sep 17, 2008 at 01:13:21PM -0400, Andre Joyal wrote: > > >> Is it the time for a new Bourbaki? >> > > Do we imagine this new Bourbaki as just systematizing and > presenting what we know already, or struggling to create > brand new mathematics? > > I can imagine a new Bourbaki who tries to explain all of > mathematics in the language of categories. But I can > also imagine a new Bourbaki who tries to explain all of > mathematics in the language of infinity-categories. It > may be a bit too late for the first one, and a bit too > early for the second one. > > Perhaps we need both! > > Best, > jb > > > > > Is it necessary to have a global point of view to appreciate Bourbaki? I found them quite valuable locally - i.e.. just afew of the chapters by themselves earned my appreciaiton. jim From rrosebru@mta.ca Sat Sep 20 11:11:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 11:11:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh36d-0001Rp-Sm for categories-list@mta.ca; Sat, 20 Sep 2008 11:06:39 -0300 Date: Sat, 20 Sep 2008 06:59:05 +0200 (CEST) Subject: categories: Re: Bourbaki and Categories ... minor correction From: Mark.Weber@pps.jussieu.fr To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 84 In my last post I said that sketches give rise to forgetful functors into Set, which is obviously wrong -- the forgetful functor arising from a given sketch is into the functor category [A,Set], where A is the underlying category of the sketch, and this forgetful functor doesn't determine the sketch up to isomorphism. Mark From rrosebru@mta.ca Sun Sep 21 10:50:37 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Sep 2008 10:50:37 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhPFJ-0006mv-Oy for categories-list@mta.ca; Sun, 21 Sep 2008 10:45:05 -0300 Date: Sat, 20 Sep 2008 13:17:19 -0400 From: "Zinovy Diskin" Subject: categories: Re: Bourbaki and Categories 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: 85 Let me add my two cents :) On Tue, Sep 16, 2008 at 4:57 AM, Vaughan Pratt wrote: > If the subject Bourwiki is proposing to serve is mathematics, then > perhaps it is time that the American Mathematical Monthly, along with > the Putnam Mathematical Competition, the International Mathematics > Olympiad, and the Journal of the AMS, abandon their pretense of being > about mathematics and come up with a suitable name for their subject. We can think of two definitions of math. The first is based on the subject matter ("what"): math is the study of Bourbaki's structures. The other is based on "how:" math is the study of structures in a well-structured way. With this definition, a good computer programmer or, say, Henry Ford, who applied conveyor to assembling, are more mathematicians than some of the guys responsible for what Vaughan wrote about. If we imagine a two dimensional plane with the "what" axis being vertical and the "how" horizontal, then we get two mathematics and resp. two sorts of mathematicians: vertical and horizontal. (CT and CT-rists go, of course, along the harmonized diagonal :). Historically, Bourbaki put a bold point on the line of Euclid-Peano-Hilbert and indeed formed the vertical dimension of the modern mathematical space. However, while the very texts written by Bourbaki are mostly enjoyable, his epigones have created a special literary style, which is good for writing/producing but hardly for reading mathematical papers. Bourbaki should not be blamed for wide dissemination of this indigestible style but...Vladimir Arnold once said that "bourbakization" of modern mathematics should perhaps be called "oBourbachivanie" in Russian (which rhymes with the Russian "oDourachivanie", which refers to fooling with someone :). > Not only do categories, functors, natural transformations, adjunctions, > and monads go unused in these 20th century icons of mathematics, they go > unacknowledged. Clearly they have not gotten with the modern > mathematical program and fall somewhere between a throwback to a golden > age and a backwater of mathematics. When they die off like the > dinosaurs they are, real mathematics will be able to advance unfettered > into the 21st century and beyond. > Dinosaurs would not normally die off themselves. Some causes are needed, and here's one (somewhat speculative though). CT can change the very notion of what a formal definition is. In the modern style, the notion of ordered pair/tuple and its derivatives like formula and term are central. This quite simple syntax (as is often happens with simple syntax, think, for example, of a Java program) can hide complex structures so that a tuple-based formal definition is not actually formal and implicitly involves intuitive concepts. If CT will sometime indeed reshape the criteria of being a formal specification (of a Bourbaki's structure), then dinosaurs would be forced to acquire CT (or die off). Zinovy > Judging from the talks at BLAST in Denver last month (B = Boolean > algebras, L = lattices, A = (universal) algebra, S = set theory, T = > topology), at least the algebraic community is moving very slightly in > this direction. Things will hopefully improve yet further when > algebraic geometry gets over its snit with equational model theory. > > Meanwhile if you need a witness for seven degrees of separation, look no > further than AMM and CT. > > (I confess to being an unreconstructed graph theorist and algebraist > myself. I may have to preemptively volunteer myself for re-education > before it becomes involuntary.) > > Vaughan Pratt > > > From rrosebru@mta.ca Sun Sep 21 10:52:11 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Sep 2008 10:52:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhPGz-0006tf-0y for categories-list@mta.ca; Sun, 21 Sep 2008 10:46:49 -0300 Date: Sat, 20 Sep 2008 13:34:01 -0400 From: "Zinovy Diskin" Subject: categories: Re: are sketches math objects? To: "Michael Barr" , 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: 86 Okay, sketches are presentations of theories but Steve's claim was that they are not mathematical objects. Michael's and mine bewilderment is about why does the former imply the latter? (at least, why "of course" :) Zinovy On Fri, Sep 19, 2008 at 6:27 PM, wrote: > Dear Michael, > > Semantically, as Lawvere observed long ago, a monad gives rise not just to > a category of algebras but also to a forgetful functor into the category > on which the monad acts. For any category C the functor > > "semantics" : Mnd(C)^op --> CAT/C > > whose object map sends a monad on C to its associated forgetful functor is > full and faithful. Thus a pair of monads on C giving rise to isomorphic > forgetful functors must necessarily be isomorphic. So your observations > about different monads giving rise to the same algebras, while correct, do > not tell the whole story on the semantic side. > > The situation is of course different for sketches: they too give rise to > forgetful functors (into Set), but this does not suffice to determine a > given sketch up to isomorphism in the same way, and this justifies Steve > Lack's perspective of "sketches as presentations of theories". > > Mark Weber > From rrosebru@mta.ca Sun Sep 21 10:53:05 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Sep 2008 10:53:05 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhPIL-0006x0-U4 for categories-list@mta.ca; Sun, 21 Sep 2008 10:48:13 -0300 Date: Sat, 20 Sep 2008 14:13:56 -0500 From: "Charles Wells" To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories (fwd) 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: 87 I think it's a great idea and I have no objection. Wikibooks has a draft of a book about category theory at http://en.wikibooks.org/wiki/Category_theory and a stub for course notes at http://en.wikiversity.org/wiki/Introduction_to_Category_Theory It wouldn't be a good idea to simply replace what is already written there with the intro part of TTT but some of it could be put in to cover additional topics and provide more examples. I have done some work on Wikipedia and need to warn you that pretty soon what you wrote may be changed in many ways, in the small and in the large ; if you are anal about what you write this can be upsetting. In any case the original TTT will remain on the web as it is. On Fri, Sep 19, 2008 at 5:21 AM, Michael Barr wrote: > Any objection? > > Mike > From rrosebru@mta.ca Sun Sep 21 10:56:16 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Sep 2008 10:56:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhPLw-0007Bx-1A for categories-list@mta.ca; Sun, 21 Sep 2008 10:51:56 -0300 MIME-Version: 1.0 Subject: categories: Bourbaki and Categories Date: Sat, 20 Sep 2008 16:21:15 -0400 From: Andre Joyal To: 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: 88 On Fri, Sep 19, 2008, John Baez wrote: >Do we imagine this new Bourbaki as just systematizing and >presenting what we know already, or struggling to create >brand new mathematics? This is an important question. We need to have a clear view of the goal of such an enterprise.=20 >From my point of view, the goal should be "educational": to help students and researchers to learn mathematics and cross the boundary between fields.=20 Mathematics is vast, and every mathematician is a permanent student.=20 The traditional way to learn is to read the litterature and to discuss = with a master. I was told that Grothendieck had learned algebraic geometry by = discussing with Serre. But few peoples have this chance. Obviously, Internet is opening new avenues for learning.=20 Many peoples (and myself) have learned a lot by reading your bulletin=20 "This Week's Finds in Mathematical Physics".=20 You have a real talent to explain a subject by exposing the heuristic! A discussion forum like the "Categories list" is also very helpful. Wikipedia is a useful place to gather informations about a subject. But the Bourbaki Tractate was offering something more: a unified presentation of mathematics, including the proofs. The Bourbaki Tractate was the result of a sustained collaboration=20 of many generations of mathematicians from different fields.=20 Conflicts are inevitable and mathematics evolve quickly. A unified, final presentation seems impossible. On Thur, Sep 18, 2008, Ronnie Brown wrote:=20 > Something initially less ambitious like this >might actually get done. Being electronic, it would be seen as a = `current', >rather than `final account', and so would better reflect the way = mathematics >develops, in which a slight shift of emphasis, or notation (like --> = for a >function), can have profound consequences. A partially unified evolving presentation of mathematics seems possible. Today's litt=E9rature is already offering something like that! But I find it cahotic and complicated. But mathematics is naturally organised and simple! The complexity of the litterature is often artificial. Many proofs are complicated, simply because the author ignores the abstract argument that could simplify everything. Some statements are left unproved, and peremptory declared obvious when they are not. The reader who cant see the obvious thing is = terrorised. He may as well quit mathematics.=20 What can we do? Let me submit a few ideas for disccusion. Mathematics is naturally self-organised.=20 The proof of most theorems can be broken in small steps of the form=20 A_1,..,A_n --->B, where A_1,..,A_n is the list of hypothesis and B is the conclusion.=20 Each step may have a simple proof. A complete proof maybe obtained by working backward from the statement of the theorem to=20 the axioms ot to known theorems.=20 Everyone who knows a nice proof of a meaningful=20 implication A_1,..,A_n --->B should write a paper about it and put it in a special section of the arXiv (the NB = section ?). On Mon, Sep 19, 2008, Michael Spivack wrote: >Even more interesting to me would be a kind of zoom-feature on=20 >proofs. Proofs are in the eye of the beholder: for example it has=20 >been debated as to whether Perelman's 70 pages was a full proof of=20 >geometrization. Given a proof with a statement which one does not=20 >understand, a mathematician may find himself reproving something that=20 >was obvious to (or wrongly assumed to be obvious by) another=20 >mathematician. The community could benefit if a mathematician who=20 >proves such a statement then uploaded the proof, even in rough form,=20 >to some kind of math wiki. If it were well-organized, this math wiki=20 >could revolutionize how mathematics is done. In fact, choosing the=20 >"right way" to organize such a site may itself be a problem which=20 >could produce interesting mathematics. On Tues, Sep 16, 2008, Andrej Bauer wrote: >My opinion is that we have not yet found the right way to do >"hive-science", but when we do, it will be a revolution. (A good start >would be to get out of the hold that the evil publishers have on us.) A special database for mathematics should be created (but I dont know how). Papers in the NB section of the arXves could be selected, modified and organised with a system of references, to give a partially unified presentation of mathematics.=20 The same database could support different presentations realised by different competing teams. =20 Each team could work like a mathematical journal, with an editor in chief and an editorial board. What do you think? Andre From rrosebru@mta.ca Mon Sep 22 11:46:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 11:46:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Khmaj-00007u-6f for categories-list@mta.ca; Mon, 22 Sep 2008 11:40:45 -0300 Message-Id: Date: Mon, 22 Sep 2008 09:55:29 +1000 Subject: categories: Re: are sketches math objects? From: Steve Lack To: Zinovy Diskin , Mime-version: 1.0 Content-type: text/plain; charset="US-ASCII" Content-transfer-encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 115 On 21/09/08 3:34 AM, "Zinovy Diskin" wrote: > Okay, sketches are presentations of theories but Steve's claim was > that they are not mathematical objects. Michael's and mine > bewilderment is about why does the former imply the latter? (at > least, why "of course" :) > > Zinovy > What I actually said was this: "Sketches are not mathematical objects in their own right, in the same sense that groups or spaces are. They are presentations (for theories), and have status similar to other sorts of presentations (for groups, rings, etc.) Of course that is in no way meant to suggest that they are not important and worthy of study." So I did not say that "they are not mathematical objects", and I used the words "of course" only in clarifying that I was not suggesting that they were unimportant. What I was saying was that they have a different flavour to such mathematical objects as groups or spaces. I was saying this in response to the observation that sketches did not seem to fit into the Bourbaki notion of structure, and so in particular, that the notion of isomorphism of sketch was not as crucial as that of isomorphism of group. Michael Barr asked what the content of the statement might be. I certainly wasn't trying to make a precise mathematical statement, although Michael himself indicated one that could be made. I guess that my second sentence (that sketches are presentations) is the content. So the content, if you like, is "whatever status you give to group presentations, you should give the same to sketches". For my part, I think that presentations are extremely important technical tools, which need to be studied and understood; but which nonetheless are just that: technical tools for dealing with the real objects of study (the things they present). Steve Lack. From rrosebru@mta.ca Mon Sep 22 11:47:49 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 11:47:49 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhmcE-0000RW-6D for categories-list@mta.ca; Mon, 22 Sep 2008 11:42:18 -0300 Date: Sun, 21 Sep 2008 23:54:38 -0700 From: "Meredith Gregory" To: "Andre Joyal" , categories@mta.ca Subject: categories: Re: Bourbaki and Categories MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 116 All, i have been utterly delighted by this conversation. What i can't help but think about, however, is that with the internet we have a different sort of opportunity. Let me try to describe it. - What is missing in most mathematical presentations is a view into the often very human and very messy process of getting to the presentation. = What young mathematicians need -- in my view -- is a view of mathematicians d= oing mathematics. They need to see very top-down orientations rubbing elbows = with very bottoms-up orientations. They need to see highly inventive, unifyin= g viewpoints come up against skeptical viewpoints armed with vast arrays o= f counter-examples. They need to see people desperately trying to organize while others are desperately trying to de-construct. This is where the l= ife of mathematics is. This is how people bring mathematics to life. - With the internet we have the opportunity to record not just the final artifact, tractate or wiki, but the process. Ever since Andre Joyal mentioned a 2nd life for Bourbaki i can't stop thinking about a Bourbaki colloquium run in Second Life -- so that whatever the outcome of a given process is in terms of artifact, people = can go back and look at the process, itself. They can see how people argued = and counter-argued. There is getting to be a precendent for this, from Harvardto Intel , to run serious technical conversation in Second Life. Perhaps this idea is too far out, but i would urge those who seriously consider a second life for Bourbaki to remember to record the living part a= s well as the outcome. After all, looking over the last many emails to categories so much of it is an attempt to recover process -- how things got to be where they are. Best wishes, --greg On Sat, Sep 20, 2008 at 1:21 PM, Andre Joyal wrote: > On Fri, Sep 19, 2008, John Baez wrote: > > >Do we imagine this new Bourbaki as just systematizing and > >presenting what we know already, or struggling to create > >brand new mathematics? > > This is an important question. > We need to have a clear view of the goal of such an enterprise. > From my point of view, the goal should be "educational": > to help students and researchers to learn mathematics > and cross the boundary between fields. > Mathematics is vast, and every mathematician is a permanent student. > The traditional way to learn is to read the litterature and to discuss wi= th > a master. > I was told that Grothendieck had learned algebraic geometry by discussing > with Serre. > But few peoples have this chance. > Obviously, Internet is opening new avenues for learning. > Many peoples (and myself) have learned a lot by reading your bulletin > "This Week's Finds in Mathematical Physics". > You have a real talent to explain a subject by exposing the heuristic! > A discussion forum like the "Categories list" is also very helpful. > Wikipedia is a useful place to gather informations about a subject. > But the Bourbaki Tractate was offering something more: > a unified presentation of mathematics, including the proofs. > > The Bourbaki Tractate was the result of a sustained collaboration > of many generations of mathematicians from different fields. > Conflicts are inevitable and mathematics evolve quickly. > A unified, final presentation seems impossible. > From rrosebru@mta.ca Mon Sep 22 20:29:46 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 20:29:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Khuii-00044k-FR for categories-list@mta.ca; Mon, 22 Sep 2008 20:21:32 -0300 Date: Mon, 22 Sep 2008 08:12:48 -0600 From: ICLP 08 Message-Id: <200809221412.m8MECmHI006441@pippo.cs.nmsu.edu> Subject: categories: ICLP'08 CALL FOR PARTICIPATION To: undisclosed-recipients:; Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 117 CALL FOR PARTICIPATION ICLP'08 24th International Conference on Logic Programming Udine, Italy, December 9th-13th, 2008 http://iclp08.dimi.uniud.it We are pleased to announce the 24th International Conference on Logic Programming, to be held in Udine, Italy, in December 2008. The ICLP'08 program includes 37 regular presentations, 26 short presentations, 4 tutorials, 5 workshops (ASPOCP, WG17, ALPSWS, WLPE and CICLOPS) and the traditional Prolog programming contest. The tutorials include presentations given by: - Carla Piazza e Alberto Policriti (Systems Biology: Models and Logics) - Angelo Montanari (Temporal Logics) - Tom Schrjivers (Constraint Handling Rules) - Peter O'Hearn (Separation Logic) Moreover the program features the invited talk by Vitor Santos Costa (The Life of a Logic Programming System) and a second invited talk TBA. The conference celebrates the 20th anniversary of the stable-model semantics with a special session at ICLP 2008 dedicated to answer-set programming. The session will feature invited talks by Michael Gelfond, Vladimir Lifschitz, Nicola Leone and David Pearce, as well as by other major contributors to the field, presenting personal perspectives on the stable-model semantics, its impact and its future. Online registration for ICLP is now open at: http://iclp08.dimi.uniud.it/ICLPRegistrationForm/ICLP2008registration.html Deadline for early registration is October 15, 2008. ---------------------------------------------------- For further information: iclp08@cs.nmsu.edu http://iclp08.dimi.uniud.it From rrosebru@mta.ca Mon Sep 22 20:30:40 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 20:30:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Khujx-00048u-Uf for categories-list@mta.ca; Mon, 22 Sep 2008 20:22:50 -0300 Date: Mon, 22 Sep 2008 13:49:37 -0400 From: "Zinovy Diskin" To: "Steve Lack" , categories@mta.ca Subject: categories: Re: are sketches math objects? 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: 118 Yes, all this "discussion" is mainly misunderstanding, and I apologize if I've contributed to it. It seems it was triggered by this piece: On Wed, Sep 17, 2008 at 12:36 AM, wrote: > Dear Steve, > >> Sketches are not mathematical objects in their own right, in the same sense >> that groups or spaces are. > > Of course, they are not. > ... So, the issue is closed. Still there is some point to mention, and I again apologize if I'm peering into it too much. Our entire mis-discussion is, perhaps, a result of two different attitudes. CT favors and prefers to work in a presentation-free setting while engineering applications are all about presentations; and this mismatch may contribute to the disdain of CT from the practitioners' side. (Of course, this is not meant to anyhow diminish the elegance, value and usefulness even for practical problems such concepts as triple or classifying category :). Zinovy On Sun, Sep 21, 2008 at 7:55 PM, Steve Lack wrote: > On 21/09/08 3:34 AM, "Zinovy Diskin" wrote: > >> Okay, sketches are presentations of theories but Steve's claim was >> that they are not mathematical objects. Michael's and mine >> bewilderment is about why does the former imply the latter? (at >> least, why "of course" :) >> >> Zinovy >> > > What I actually said was this: > > "Sketches are not mathematical objects in their own right, in the same sense > that groups or spaces are. They are presentations (for theories), and have > status similar to other sorts of presentations (for groups, rings, etc.) > > Of course that is in no way meant to suggest that they are not important and > worthy of study." > > So I did not say that "they are not mathematical objects", and I used the > words "of course" only in clarifying that I was not suggesting that they > were unimportant. What I was saying was that they have a different flavour > to such mathematical objects as groups or spaces. I was saying this in > response to the observation that sketches did not seem to fit into the > Bourbaki notion of structure, and so in particular, that the notion of > isomorphism of sketch was not as crucial as that of isomorphism of group. > > Michael Barr asked what the content of the statement might be. I certainly > wasn't trying to make a precise mathematical statement, although Michael > himself indicated one that could be made. I guess that my second sentence > (that sketches are presentations) is the content. So the content, if you > like, is "whatever status you give to group presentations, you should give > the same to sketches". For my part, I think that presentations are extremely > important technical tools, which need to be studied and understood; but > which nonetheless are just that: technical tools for dealing with the real > objects of study (the things they present). > > Steve Lack. > > From rrosebru@mta.ca Mon Sep 22 20:30:40 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 20:30:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Khul9-0004DX-Ky for categories-list@mta.ca; Mon, 22 Sep 2008 20:24:03 -0300 MIME-Version: 1.0 Subject: categories: Apology Date: Mon, 22 Sep 2008 14:29:28 -0400 From: Andre Joyal To: 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: 119 Dear All, My apology to David Spivack. In my last message on Bourbaki and Categories,=20 I wrote Michael Spivack instead of DAVID Spivack. Andre From rrosebru@mta.ca Mon Sep 22 20:32:37 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 20:32:37 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Khun2-0004Jv-OE for categories-list@mta.ca; Mon, 22 Sep 2008 20:26:00 -0300 Date: Mon, 22 Sep 2008 14:10:57 -0700 From: John Baez To: categories Subject: categories: Sketches 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: 120 Zinovy Diskin wrote: >Okay, sketches are presentations of theories but Steve's claim was >that they are not mathematical objects. He didn't say that. He wisely said something much more cautious. Best, jb From rrosebru@mta.ca Mon Sep 22 20:32:37 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 20:32:37 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhumW-0004Hr-3C for categories-list@mta.ca; Mon, 22 Sep 2008 20:25:28 -0300 Date: Mon, 22 Sep 2008 13:54:51 -0700 From: John Baez To: categories Subject: categories: Bourbaki and Categories 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: 121 On Fri, Sep 19, 2008 at 10:16:31PM -0400, jim stasheff wrote: > John Baez wrote: > >I can imagine a new Bourbaki who tries to explain all of > >mathematics in the language of categories. But I can > >also imagine a new Bourbaki who tries to explain all of > >mathematics in the language of infinity-categories. > Is it necessary to have a global point of view to appreciate Bourbaki? I > found them quite valuable locally - i.e.. just a few of the chapters by > themselves earned my appreciation. It's easy to appreciate their books locally - but I think they sought a global systematic viewpoint while writing them. It's possible that a "neo-Bourbaki" should take a less systematic approach. Mathematics may be too much in a state of foundational flux for a systematic approach to be successful right now. Maybe the best we can hope for is something a bit more like Wikipedia, where different people contribute different portions of text, and they don't cohere in a polished whole. But presumably anyone calling for a new Bourbaki wants something different from Wikipedia. There's "Scholarpedia": http://www.scholarpedia.org/ but it doesn't seem to be doing anything with math yet, and if it ever does, I bet it'll take a "midde-of-the-road" approach instead of pushing a specific intellectual agenda. I would like to see lots of people try lots of different things. Best, jb From rrosebru@mta.ca Mon Sep 22 20:36:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Sep 2008 20:36:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Khuq5-0004X8-Ic for categories-list@mta.ca; Mon, 22 Sep 2008 20:29:09 -0300 Date: Mon, 22 Sep 2008 17:09:24 -0400 From: Jacques Carette MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 122 I hereby volunteer my time and the technical resources I have at my disposal to 'host' a "New Bourbaki based on CT" project. Why me? 1. I am an interested _user_ of category theory, but not an 'inventor' in the field. Neutrality is frequently an asset in a "keeper of the infrastructure". 2. I need the results of such a project! My core work (for 11 years in industry and now 6 in academia) is in "mechanized mathematics", or computer-based tools that help automate the mathematics process, explicitly including both computation (i.e. what Maple does) and proof (i.e. what Coq does). (The curious can see http://imps.mcmaster.ca/mathscheme/publications.html for some of our work and my personal publications page for more). One cannot simultaneously build a library of mathematics and know category theory without seeing its tendrils everywhere. 3. I have a real stake in seeing "mathematics on computers": not only is it my day-to-day research, I am also the Chair of the Electronic Services Committee for the Canadian Mathematical Society (CMS). I am actively searching for *good* applications of "electronic services" where we can get involved. I cannot promise the resources of the CMS for this, but I can certainly promise to bring this up as an agenda item for our December meeting in Ottawa. However, I would be extremely proud if I could claim, years from now, that "Bourwiki" was a ``truly international project which benefited from a strong involvement of the Canadian Mathematical Society''. Jacques Carette PS: Below here are some minor comments/thoughts on the various topics in this email thread -- the only important parts are above. 1. I am fairly fond of the (awfully titled) book "Post-modern Algebra" http://www.amazon.com/Post-Modern-Algebra-Applied-Mathematics-Wiley-Interscience/dp/0471127388 by J. Smith and A. Romanwska, as a middle-ground between full-fledged use of categories and classical algebra in an introductory text. The organization of the text is definitely non-classical and tries to organize concepts according to mathematical criteria rather than historical timelines. 2. I definitely believe that a proper encoding of modern mathematics into a 'library' should be done top-done by specializing abstract concepts. The "Little Theories" method from theorem proving is essentially the recognition that a large body of mathematics is a huge diagram in an appropriate category of theories. These issues are all too frequently treated as meta-mathematical. The constructive theory of "representations of theories" (via sketches or otherwise) can play a very important role in software construction. 3. Heuristics and examples are crucial for human understanding of mathematics. Thus, while a "mechanized mathematics system" may be built from highly abstract pieces, it needs to present itself to its users in as concrete a manner as possible. This dichotomy between system-building and usability is further explored in a paper "High-Level Theories" (available officially at http://www.springerlink.com/content/b1122523vtm88w73/, unofficially at http://imps.mcmaster.ca/doc/hlt.pdf ) 4. I cannot over-emphasize my agreement with Joyal's statement that "But mathematics is naturally organised and simple!" 5. To a certain degree, some libraries (like those of the large theorem provers) attempt "partially unified presentations of mathematics"; some even have pseudo-databases. These are unfortunately quite classical in their structure, with the exception of the work of the Kestrel Institute (in computer science) which is very categorical. The algebra library of the programming language 'Aldor' (www.aldor.org) was definitely influenced by categories. Category theory is actively influencing the development of the language 'Haskell'. 6. What I propose to help with are the technical and online aspects of Andre Joyal's proposal: > A special database for mathematics should be created > (but I dont know how). > Papers in the NB section of the arXves could be selected, > modified and organised with a system of references, > to give a partially unified presentation of mathematics. > The same database could support different > presentations realised by different competing teams. > Each team could work like a mathematical journal, > with an editor in chief and an editorial board. > using means like the ones which Andrej Bauer suggested. From rrosebru@mta.ca Tue Sep 23 08:59:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2008 08:59:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ki6SC-00040U-8J for categories-list@mta.ca; Tue, 23 Sep 2008 08:53:16 -0300 Date: Mon, 22 Sep 2008 19:43:27 -0500 From: "Charles Wells" To: catbb Subject: categories: Group presentations _are_ sketches 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: 123 A group presentation consists of a bunch of symbols and some equations between strings of some of the symbols and some of their formal inverses. The symbols are the arrows of the graph, and the diagrams exhibit the equations. The type of theory (instead of FP theory, FL theory, coherent theory or whatever) is that of categories in which all arrows have inverses (groupoids) to which if you insist you may add any two arrows compose (groups). A model of a sketch in a category takes each arrow to an isomorphism in such a way that the diagrams commute. The presentation generates a group, and that group is the theory. The category of actions of the theory on category is clearly equivalent to the category of models of the presentation as sketch. This brings up a point. An FP theory (for example) is typically thought of as having models in an FP category, but in fact it can have models in any category. The necessary finite products must exist, but others can be missing. Nevertheless, most of the examples that have actually been considered are in categories that do have all finite products (in other words are within the doctrine). In the case of a group presentation, it is _normal_ to consider models in categories in which most arrows don't have isomorphisms (they are not within the doctrine). This is only a psychological difference, but it is interesting. -- Charles Wells professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://www.gyregimble.blogspot.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm personal website: http://www.abstractmath.org/Personal/index.html From rrosebru@mta.ca Tue Sep 23 08:59:30 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2008 08:59:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ki6Sb-00042F-5Y for categories-list@mta.ca; Tue, 23 Sep 2008 08:53:41 -0300 Date: Tue, 23 Sep 2008 01:43:46 -0400 (EDT) From: Robert Seely To: Categories List Subject: categories: Octoberfest in Montreal (Oct 4-5) 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: 124 A brief update: if you intend to attend and haven't emailed me, please do so asap so we have a good idea of how many folks to expect. Even more importantly, if you wish to give a talk, let me know asap as well. The list of speakers currently stands as follows: M Makkai TBA V Harnik TBA M Barr Duality of $Z$-groups M Warren Types and groupoids (abstract) P Hofstra From poset to quantifier (abstract) R Cockett TBA B Redmond Safe recursion revisited G Lukacs TBA R Lucyshyn-Wright TBA C Nourani Functorial Parallel Worlds Model Computations (abstract) N Yanofsky On The Algorithmic Informational Content of Categories S Niefield Par-valued lax functors and exponentiability (abstract) D Pronk Translation Groupoids and Orbifold Homotopy Theory G Seal TBA (If any speakers have given me a title, and it's not listed above, or if you've indicated a wish to speak and you're not on the list above, please email me again - you have my apology for losing track of your previous email! - and ditto for abstracts: I seem to only have the 4 mentioned above.) Full details about the meeting may be found on the webpage http://www.math.mcgill.ca/triples/octoberfest08.html -= rags =- -- From rrosebru@mta.ca Tue Sep 23 09:02:27 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2008 09:02:27 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ki6Uu-00049T-3D for categories-list@mta.ca; Tue, 23 Sep 2008 08:56:04 -0300 MIME-Version: 1.0 Subject: categories: iBourbaki Date: Mon, 22 Sep 2008 21:45:04 -0400 From: Andre Joyal To: "Jacques Carette" , 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: 125 Dear Jacques Carrette, Your wrote: >I hereby volunteer my time and the technical resources I have at my >disposal to 'host' a "New Bourbaki based on CT" project. Your offer to help is enormously appreciated. The realisation of the project depends on establishing a close=20 collaboration between mathematicians and computer scientists. I suggest to call it=20 iBourbaki =20 after an idea of Valeria de Paiva. She was very helpful in answering some of my questions. Let me try to describe the problem as I see it. What is needed is an online system which can be consulted by everybody but only modified in a restricted way. The user should be able to navigate in the system by clicking on words or by doing a search. He should be able to suggest a modification or a correction. The typography is very important, it should be of the highest standard: = Tex. It should be possible to draw diagrams and planar figures. The presentation of mathematics supported by the system should be the work of various teams, collaborating or competing.=20 The system should be able to support different presentations. Let me stress that the iBourbaki project is not about theorem proving or = verification.=20 The goal is more modest: to create an interface for an efficient=20 organisation and dissimination of mathematical knowledge (for students = and researchers).=20 The system will have to be very user-friendly, almost invisible to the = user. In addition to presenting mathematics in a logical fashion, it should give informations about the applications, the history, the motivations = and the heuristics. The project will have to be very practical. I am aware of the enormous difficulties of realising iBourbaki. Equally enormous will be the benefits to the community if it succeed. Andre -------- Message d'origine-------- De: cat-dist@mta.ca de la part de Jacques Carette Date: lun. 22/09/2008 17:09 =C0: categories@mta.ca Objet : categories: Re: Bourbaki and Categories =20 I hereby volunteer my time and the technical resources I have at my disposal to 'host' a "New Bourbaki based on CT" project. Why me? 1. I am an interested _user_ of category theory, but not an 'inventor' in the field. Neutrality is frequently an asset in a "keeper of the infrastructure". 2. I need the results of such a project! My core work (for 11 years in industry and now 6 in academia) is in "mechanized mathematics", or computer-based tools that help automate the mathematics process, explicitly including both computation (i.e. what Maple does) and proof (i.e. what Coq does). (The curious can see http://imps.mcmaster.ca/mathscheme/publications.html for some of our work and my personal publications page for more). One cannot simultaneously build a library of mathematics and know category theory without seeing its tendrils everywhere. 3. I have a real stake in seeing "mathematics on computers": not only is it my day-to-day research, I am also the Chair of the Electronic Services Committee for the Canadian Mathematical Society (CMS). I am actively searching for *good* applications of "electronic services" where we can get involved. I cannot promise the resources of the CMS for this, but I can certainly promise to bring this up as an agenda item for our December meeting in Ottawa. However, I would be extremely proud if I could claim, years from now, that "Bourwiki" was a ``truly international project which benefited from a strong involvement of the Canadian Mathematical = Society''. Jacques Carette PS: Below here are some minor comments/thoughts on the various topics in this email thread -- the only important parts are above. 1. I am fairly fond of the (awfully titled) book "Post-modern Algebra" http://www.amazon.com/Post-Modern-Algebra-Applied-Mathematics-Wiley-Inter= science/dp/0471127388 by J. Smith and A. Romanwska, as a middle-ground between full-fledged use of categories and classical algebra in an introductory text. The organization of the text is definitely non-classical and tries to organize concepts according to mathematical criteria rather than historical timelines. 2. I definitely believe that a proper encoding of modern mathematics into a 'library' should be done top-done by specializing abstract concepts. The "Little Theories" method from theorem proving is essentially the recognition that a large body of mathematics is a huge diagram in an appropriate category of theories. These issues are all too frequently treated as meta-mathematical. The constructive theory of "representations of theories" (via sketches or otherwise) can play a very important role in software construction. 3. Heuristics and examples are crucial for human understanding of mathematics. Thus, while a "mechanized mathematics system" may be built from highly abstract pieces, it needs to present itself to its users in as concrete a manner as possible. This dichotomy between system-building and usability is further explored in a paper "High-Level Theories" (available officially at http://www.springerlink.com/content/b1122523vtm88w73/, unofficially at http://imps.mcmaster.ca/doc/hlt.pdf ) 4. I cannot over-emphasize my agreement with Joyal's statement that "But mathematics is naturally organised and simple!" 5. To a certain degree, some libraries (like those of the large theorem provers) attempt "partially unified presentations of mathematics"; some even have pseudo-databases. These are unfortunately quite classical in their structure, with the exception of the work of the Kestrel Institute (in computer science) which is very categorical. The algebra library of the programming language 'Aldor' (www.aldor.org) was definitely influenced by categories. Category theory is actively influencing the development of the language 'Haskell'. 6. What I propose to help with are the technical and online aspects of Andre Joyal's proposal: > A special database for mathematics should be created > (but I dont know how). > Papers in the NB section of the arXves could be selected, > modified and organised with a system of references, > to give a partially unified presentation of mathematics. > The same database could support different > presentations realised by different competing teams. > Each team could work like a mathematical journal, > with an editor in chief and an editorial board. > using means like the ones which Andrej Bauer suggested. ------_=_NextPart_001_01C91D1E.004E3E13 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable iBourbaki

Dear Jacques Carrette,

Your wrote:
>I hereby volunteer my time and the technical resources I have at = my
>disposal to 'host' a "New Bourbaki based on CT" = project.

Your offer to help is enormously appreciated.
The realisation of the project depends on establishing a close
collaboration between mathematicians and computer scientists.
I suggest to call it

iBourbaki 

after an idea of Valeria de Paiva.
She was very helpful in answering some of my questions.
Let me try to describe the problem as I see it.

What is needed is an online system which can be consulted by = everybody
but only modified in a restricted way.
The user should be able to navigate in the system
by clicking on words or by doing a search.
He should be able to suggest a modification or a correction.
The typography is very important, it should be of the highest standard: = Tex.
It should be possible to draw diagrams and planar figures.
The presentation of mathematics supported by the system
should be the work of various teams, collaborating or competing.
The system should be able to support different presentations.

Let me stress that the iBourbaki project is not about theorem proving or = verification.
The goal is more modest: to create an interface for an efficient
organisation and dissimination of mathematical knowledge (for students = and researchers).
The system will have to be very user-friendly, almost invisible to the = user.
In addition to presenting mathematics in a logical fashion, it = should
give informations about the applications, the history, the motivations = and
the heuristics. The project will have to be very practical.

I am aware of the enormous difficulties of realising iBourbaki.
Equally enormous will be the benefits to the community if it = succeed.

Andr=E9


-------- Message d'origine--------
De: cat-dist@mta.ca de la part de Jacques Carette
Date: lun. 22/09/2008 17:09
=C0: categories@mta.ca
Objet : categories: Re: Bourbaki and Categories

I hereby volunteer my time and the technical resources I have at my
disposal to 'host' a "New Bourbaki based on CT" project.

Why me?
1. I am an interested _user_ of category theory, but not an = 'inventor'
in the field.  Neutrality is frequently an asset in a "keeper = of the
infrastructure".
2. I need the results of such a project!  My core work (for 11 = years in
industry and now 6 in academia) is in "mechanized = mathematics", or
computer-based tools that help automate the mathematics process,
explicitly including both computation (i.e. what Maple does) and = proof
(i.e. what Coq does).  (The curious can see
http://imps= .mcmaster.ca/mathscheme/publications.html for some of our
work and my personal publications page for more).  One cannot
simultaneously build a library of mathematics and know category = theory
without seeing its tendrils everywhere.
3. I have a real stake in seeing "mathematics on computers": = not only is
it my day-to-day research, I am also the Chair of the Electronic
Services Committee for the Canadian Mathematical Society (CMS).  I = am
actively searching for *good* applications of "electronic = services"
where we can get involved.

I cannot promise the resources of the CMS for this, but I can = certainly
promise to bring this up as an agenda item for our December meeting = in
Ottawa.  However, I would be extremely proud if I could claim, = years
from now, that "Bourwiki" was a ``truly international project = which
benefited from a strong involvement of the Canadian Mathematical = Society''.

Jacques Carette

PS: Below here are some minor comments/thoughts on the various topics = in
this email thread -- the only important parts are above.
1. I am fairly fond of the (awfully titled) book "Post-modern = Algebra"

http://www.amazon.com/Post-Modern-Algebra-= Applied-Mathematics-Wiley-Interscience/dp/0471127388
  by J. Smith and A. Romanwska, as a middle-ground between = full-fledged
use of categories and classical algebra in an introductory text.  = The
organization of the text is definitely non-classical and tries to
organize concepts according to mathematical criteria rather than
historical timelines.
2. I definitely believe that a proper encoding of modern mathematics
into a 'library' should be done top-done by specializing abstract
concepts.  The "Little Theories" method from theorem = proving is
essentially the recognition that a large body of mathematics is a = huge
diagram in an appropriate category of theories. These issues are all = too
frequently treated as meta-mathematical.  The constructive theory = of
"representations of theories" (via sketches or otherwise) can = play a
very important role in software construction.
3. Heuristics and examples are crucial for human understanding of
mathematics.  Thus, while a "mechanized mathematics = system" may be built
from highly abstract pieces, it needs to present itself to its users = in
as concrete a manner as possible.  This dichotomy between
system-building and usability is further explored in a paper = "High-Level
Theories" (available officially at
http://www= .springerlink.com/content/b1122523vtm88w73/, unofficially at
http://imps.mcmaster.ca/doc/= hlt.pdf )
4. I cannot over-emphasize my agreement with Joyal's statement that = "But
mathematics is naturally organised and simple!"
5. To a certain degree, some libraries (like those of the large = theorem
provers) attempt "partially unified presentations of = mathematics"; some
even have pseudo-databases.  These are unfortunately quite = classical in
their structure, with the exception of the work of the Kestrel = Institute
(in computer science) which is very categorical.  The algebra = library of
the programming language 'Aldor' (www.aldor.org) was definitely
influenced by categories.  Category theory is actively influencing = the
development of the language 'Haskell'.
6. What I propose to help with are the technical and online aspects = of
Andre Joyal's proposal:
> A special database for mathematics should be created
> (but I dont know how).
> Papers in the NB section of the arXves could be selected,
> modified and organised with a system of references,
> to give a partially unified presentation of mathematics.
> The same database could support different
> presentations realised by different competing teams.
> Each team could work like a mathematical journal,
> with an editor in chief and an editorial board.
>
using means like the ones which Andrej Bauer suggested.

------_=_NextPart_001_01C91D1E.004E3E13-- From rrosebru@mta.ca Tue Sep 23 21:14:02 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2008 21:14:02 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KiHuL-0004aD-Fq for categories-list@mta.ca; Tue, 23 Sep 2008 21:07:05 -0300 Date: Tue, 23 Sep 2008 14:01:58 -0400 From: jim stasheff MIME-Version: 1.0 To: Meredith Gregory , categories@mta.ca Subject: categories: Re: Bourbaki and Categories 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: 126 I would just add or emphasize if implicit here one thought: a view into the often very human process of interaction viz. several of the anecdotes about `life with Bott' at his recent memorial conference jim Meredith Gregory wrote: > All, > > i have been utterly delighted by this conversation. What i can't help but > think about, however, is that with the internet we have a different sort of > opportunity. Let me try to describe it. > > - What is missing in most mathematical presentations is a view into the > often very human and very messy process of getting to the presentation. What > young mathematicians need -- in my view -- is a view of mathematicians doing > mathematics. They need to see very top-down orientations rubbing elbows with > very bottoms-up orientations. They need to see highly inventive, unifying > viewpoints come up against skeptical viewpoints armed with vast arrays of > counter-examples. They need to see people desperately trying to organize > while others are desperately trying to de-construct. This is where the life > of mathematics is. This is how people bring mathematics to life. > - With the internet we have the opportunity to record not just the final > artifact, tractate or wiki, but the process. Ever since Andre Joyal > mentioned a 2nd life for Bourbaki i can't stop thinking about a Bourbaki > colloquium run in Second Life -- so that > whatever the outcome of a given process is in terms of artifact, people can > go back and look at the process, itself. They can see how people argued and > counter-argued. There is getting to be a precendent for this, from > Harvardto > Intel , to run > serious technical conversation in Second Life. > > Perhaps this idea is too far out, but i would urge those who seriously > consider a second life for Bourbaki to remember to record the living part as > well as the outcome. After all, looking over the last many emails to > categories so much of it is an attempt to recover process -- how things got > to be where they are. > > Best wishes, > > --greg > From rrosebru@mta.ca Tue Sep 23 21:14:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Sep 2008 21:14:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KiHv0-0004d7-QN for categories-list@mta.ca; Tue, 23 Sep 2008 21:07:46 -0300 Date: Tue, 23 Sep 2008 22:06:00 +0200 (CEST) Subject: categories: Re: are sketches math objects? From: pierre.ageron@math.unicaen.fr To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 127 The ontological status of "presentations of structures" is a key issue in the history of mathematics in the 20th century. I wrote a number of historical material about it. The following (in French) are available on the Internet : http://people.math.jussieu.fr/~burroni/mapage/Burroni.pdf http://www.univ-nancy2.fr/poincare/colloques/symp02/abstracts/ageron.pdf Pierre Ageron From rrosebru@mta.ca Wed Sep 24 14:47:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 24 Sep 2008 14:47:52 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KiYMy-00012t-Gk for categories-list@mta.ca; Wed, 24 Sep 2008 14:41:44 -0300 Mime-Version: 1.0 Date: Wed, 24 Sep 2008 18:17:10 +0200 To: categories@mta.ca From: Ugo Montanari Subject: categories: PhD Program at IMT, Lucca 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: 128 PhD in Lucca, Italy Call for applications: Computer Science and Engineering Deadline December 5th 2008 Classes start in March 2009 Students can apply if they obtain their degrees before December 24, 2008 PhD in Computer Science and Engineering The doctoral Program aims to prepare researchers and professionals with broad training in the foundations of informatics as well as applications to a variety of cutting-edge systems and disciplines. The frontiers of informatics are influencing the paths of other disciplines as well as ordinary life, and are the target of active research on the international scene. Research doctors may work in universities and research centers. They may also take on professional roles and high-profile tasks and responsibilities in both the private and public sectors. 15 positions and 8 scholarships 15 students: no tuition fees + free access to IMT Canteen 8 students: full grants (annual grant amounts +/- 13640 Euros gross) 8 students: free accommodation IMT Distinctive Features -IMT is an international graduate school that promotes cutting-edge research in areas with clear practical relevance, contributing to form international professional elites for business and institutions. -Research and teaching programs at IMT focus on institutional and technological change, the role of organizations and markets in economic systems, the analysis of complex systems in social sciences, computer science and engineering. -IMT aims to recruit students with high potential in a fast-moving global environment where research institutes and universities compete to attract resources and human capital. To do so, IMT uses international selection standards and seeks candidates from all around the globe. -The IMT community is the result of a lively interaction of students and scholars, building upon the campus system and residential services provided by the Lucca Foundation for Higher Education and Research. -PhD Courses are held in English and student curricula and performance is continually assessed through rigorous evaluation processes. CSE PhD Program Board Paolo Ciancarini - Bologna Rocco De Nicola - Firenze Carlo Ghezzi - Milano Luciano Lenzini - Pisa Ugo Montanari - Pisa, Coordinator Antonio Prete - Pisa IMT http://www.imtlucca.it/http://www.imtlucca.it/ is an International Graduate School Founded by: LUISS Guido Carli, Rome Politecnico di Milano Sant'Anna School of Advanced Studies University of Pisa Lucca Foundation for Higher Education and Research Online applications only at: http://www.imtlucca.it/phd_programs/call_for applications/index.php -- @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ Prof. Ugo Montanari Phone: +39 050 2212721 Dipartimento di Informatica Fax: +39 050 2212726 Universita' di Pisa Email: ugo@di.unipi.it Largo Bruno Pontecorvo, 3 Address changed recently I-56127 Pisa, Italy http://www.di.unipi.it/~ugo/ugo.html @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ From rrosebru@mta.ca Fri Sep 26 15:40:04 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 26 Sep 2008 15:40:04 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KjI8t-0000Hx-TA for categories-list@mta.ca; Fri, 26 Sep 2008 15:34:15 -0300 To: categories@mta.ca Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit Mime-Version: 1.0 (Apple Message framework v929.2) Subject: categories: PhD studentships at Sussex Date: Fri, 26 Sep 2008 11:27:00 +0100 From: Bernhard Reus Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 129 Please distribute widely to students with interest in a PhD: ++++++++++++++++++++++++++++++++++++++++++++++++++++ 2 Phd Studentships in Program Semantics and Verification ++++++++++++++++++++++++++++++++++++++++++++++++++++ available at the University of Sussex (Brighton, UK) to work on an EPSRC funded project the objectives of which is to develop program logics for function pointers and reflective programming principles. Candidates need tp have a good BSc/MSc in Computer Science or Mathematics and should have some interest in programming languages and formal methods. Background in Logic, Semantics, Category Theory, or Type Theory may be useful but is not essential. On the more practical side, experience with theorem provers would be an asset. Students can take on theoretical problems as well as participate in concrete tool development or extension. The studentship (funded by the EPSRC, grant EP/G003173/1) will cover all tuition fees (for EU citizens) and a yearly maintenance grant at the standard rate (currently GBP 12,940). Students will be working under the supervision of Dr Bernhard Reus (PI) and in collaboration with a Post-Doc embedded in the Foundations of Computation Group of the Department. The University is situated in Brighton which is a famous seaside resort at the English south coast, about 50 miles from London (and half an hour from Gatwick Airport). It is renowned for its nightlife, fabulous shops, and cosmopolitan vibe. For further information about the project consult Candidates interested in the studentships should contact: Dr Bernhard Reus (bernhard@sussex.ac.uk) From rrosebru@mta.ca Fri Sep 26 15:40:04 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 26 Sep 2008 15:40:04 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KjI8M-0000F3-GX for categories-list@mta.ca; Fri, 26 Sep 2008 15:33:42 -0300 Date: Thu, 25 Sep 2008 21:46:39 -0700 (PDT) From: Bill Rowan To: categories@mta.ca Subject: categories: Group and abelian group objects in the category of Kelley spaces 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: 130 Hi all, Does anyone know of a good place where someone has written down the basic properties of such objects? As an example, if we have an (abelian, say) topological group, there is a natural uniform topology on the group such that the operations are uniformly continuous. Does the same hold for abelian group objects in the category of Kelley spaces? But anything would be helpful. Bill Rowan From rrosebru@mta.ca Fri Sep 26 15:40:04 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 26 Sep 2008 15:40:04 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KjI7x-0000DM-Vx for categories-list@mta.ca; Fri, 26 Sep 2008 15:33:18 -0300 MIME-Version: 1.0 Subject: categories: Re: BourWiki-iBourbaki-FunctorWiki? Date: Thu, 25 Sep 2008 13:55:27 -0400 Thread-Topic: categories: BourWiki-iBourbaki-FunctorWiki From: Andre Joyal To: "Ellis D. Cooper" , 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: 131 Dear Ellis Cooper, >Since this project entails substantial programming effort, I would=20 >like to measure the >level of support the project could expect from the category-oriented=20 >research community. >Your judgement is much appreciated. In addition to the peoples who have expressed their support in public many have done it privately. None have opposed it yet. Some peoples have expressed doubts about the feasability. In a letter to some of my correspondants I wrote: >The iBourbaki project is by nature a very long term project. >Mathematics is vast, and it may take a few generations to fully = integrate >the main body of mathematics to iBourbaki. >Also the potential contributors will refuse to work seriously >on something that may disappear a few years later. >They should have the possibility of making lasting contributions. >Hence we should make sure that iBourbaki can last for more than one = generation. >It will be like building a cathedral! >Of course, iBourbaki should evolve, since mathematics and the = technology evolve. >But mathematics is also remarquably stable. >The mathematics of Euclid is still valid today, inspite of the >fact that we now have non-euclidian geometry. >I expect that the main corpus of iBourbaki will not change very much = over long period of times. >We need to conciliate long term duration with the rapid developpement = of technology. The realisation of iBourbaki depends on the collaboration of the = mathematical community in large, not only of category theorists. It also depends on the = collaboration and support of a group of computer scientists. My task at this point is to convince = peoples that the=20 project is worthy and feasable.=20 Andr=E9 -------- Message d'origine-------- De: Ellis D. Cooper [mailto:xtalv1@netropolis.net] Date: mar. 23/09/2008 09:33 =C0: Joyal, Andr=E9; categories@mta.ca Objet : categories: BourWiki-iBourbaki-FunctorWiki =20 Dear Andre Joyal and categorists, Various networking models for collaborating on mathematics have been=20 alluded to or suggested on this list (which is itself based on the listserv networking model). *************************************************************************= ******************************** Patrik Eklund on 7 Oct 2007 wrote: "Sound-video" is nothing but Skype, but adding whiteboards, that can be saved and worked with also offline, you have very good possibilities. The whiteboard mainly accepts non-formatted text, drawings and images. = You can read doc and ppt file which are "pasted" as bitmaps on the = whiteboard. They include desktop sharing if that would be required. Mathematical = text I add through LaTeX, compiling, converting to pdf, and using the = snapshot tool to paste bitmapped formulas on the whiteboard. *************************************************************************= ******************************** Dusko Pavlovic on 15 Sep 2008 wrote: an improved process, combining the integrity, and perhaps the structure of the categories@mta community with the available wiki-methods may bring categorical methods into a dynamic environment, perhaps more natural for them than books and papers. *************************************************************************= ******************************** Andrej Bauer on 16 Sep 2008 wrote: the usual kind of wiki is not suitable for collaborative science, but recently there has been news of a special wiki for scientists which has good support for references, keeps track of who said what, and has a rating system. *************************************************************************= ******************************** Meredith Gregory on 21 Sep 2008 wrote: - What is missing in most mathematical presentations is a view into = the often very human and very messy process of getting to the = presentation. Ever since Andre Joyal mentioned a 2nd life for Bourbaki i can't=20 stop thinking about a Bourbaki colloquium run in Second Life *************************************************************************= ******************************** Jacques Carette on 22 Sep 2008 wrote: I hereby volunteer my time and the technical resources I have at my disposal to 'host' a "New Bourbaki based on CT" project. *************************************************************************= ******************************** John Baez on 22 Sep 2008 wrote: Mathematics may be too much in a state of foundational flux for a systematic approach to be successful right now. Maybe the best we can hope for is something a bit more like Wikipedia, where different people contribute different portions of text, and they don't cohere in a polished whole. But presumably anyone calling for a new Bourbaki wants something different from Wikipedia. I would like to see lots of people try lots of different things. *************************************************************************= ******************************** A network model not yet mentioned is the multiple-thread technical=20 support forum. See, for example http://www.theimagingsourceforums.com/forumdisplay.php?f=3D6 http://www.essentialobjects.com/Forum/Default.aspx http://www.codeproject.com/KB/ajax/ http://www.codeproject.com/Lounge.aspx?msg=3D2730104#xx2730104xx Technical support forums centralize expertise, whereas collaborative=20 mathematics would best have mutually supportive distributed expertise. Each thread=20 would be moderated to maintain topic and civility, like the categories listserv. Crucial to much mathematical collaboration is sharing of diagrams=20 alongside mathematical expressions and natural language text. A webcam over a pad provides=20 this important function. I myself have used Skype to discuss one-sided limits with=20 my son who is away at college, where instead of looking at each other's face we aimed=20 our webcams at pads. One step beyond separate pads beneath webcams is the idea of a=20 "virtual drawing" which per discussion thread would overlay appropriately transparent images=20 from multiple collaborators: a shared blackboard in your web browser. A web-site for collaborative mathematics could combine the secure=20 multiple-thread forum model with virtual drawing. Every participant would have the option to=20 record part or all of a thread for off-line review. Combined with video-conferencing via Skype or=20 the like, real-time collaboration is an option. Since this project entails substantial programming effort, I would=20 like to measure the level of support the project could expect from the category-oriented=20 research community. Your judgement is much appreciated. Ellis D. Cooper From rrosebru@mta.ca Sat Sep 27 11:14:15 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 27 Sep 2008 11:14:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KjaSd-0007Fj-G6 for categories-list@mta.ca; Sat, 27 Sep 2008 11:07:51 -0300 Date: Fri, 26 Sep 2008 17:09:33 -0400 (EDT) From: Robert Seely To: Categories List Subject: categories: Octoberfest - Proposed schedule of talks 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: 132 We have a proposed schedule of talks for the Octoberfest meeting next weekend (4-5 Oct). Please be aware that changes may still happen, and a final schedule will be given out at the meeting itself. An html version of this document is available at the meeting website, and you can check there for any last-minute changes, as we know them. http://www.math.mcgill.ca/triples/octoberfest08.html Some talks are still "TBA" - I hope the speakers will give me titles asap! We have some abstracts - those we have will be available on the webpage (and I'll put others there as I get them). In addition, I suggest folks review the instructions on the website re location of the talks (at Concordia's Faubourg building), parking in the area (the university has indoor parking from 9am - a bit of a tight squeeze in terms of time, but possible - and there is metered parking on the streets around the area), and do check the weather (there are links to two weather sites on the webpage). -= rags =- ----------- Octoberfest08 Category Theory Octoberfest Concordia University, Montreal Saturday - Sunday, October 4 - 5, 2008 Proposed Schedule of Talks Room FGB040 (Faubourg building) Saturday, 4 October - morning session 9:15-9:45 M Makkai Revisiting the coherence theory for bicategories and tricategories 9:45-10:15 V Harnik TBA coffee break 10:45-11:15 W Tholen Towards an enriched understanding of Hausdorff and Gromov metrics 11:15-11:45 C Hermida TBA 11:45-12:15 M Warren Types and groupoids lunch afternoon session, in honour of P.J. Scott 2:00-2:30 P Hofstra From poset to quantifier 2:30-3:00 R Cockett Differentiation, linear matters, and other discussions from the Green Door 3:00-3:30 J Lambek In praise of quaternions coffee break 4:00-4:30 B Redmond Safe recursion revisited 4:40-4:50 J Morton 2-Vector Spaces and Finite Groupoids 4:50-5:10 T Kusalik The Continuum Hypothesis in topos theory and algebraic set theory 5:10-5:30 R Lucyshyn-Wright TBA 5:30-6:00 G Lukacs TBA 6:00-6:20 C Nourani Functorial parallel worlds model computations free evening Sunday, 5 October 9:00-9:30 N Yanofsky On the algorithmic informational content of categories 9:30-10:00 S Niefield Par-valued lax functors and exponentiability 10:00-10:30 A Joyal TBA coffee break 11:00-11:30 G Seal Kock-Zoberlein monads from monads on SET 11:30-12:00 D Pronk Translation groupoids and orbifold homotopy theory 12:00-12:30 J Kennison Spectra for Symbolic Dynamics 12:30-1:00 P Freyd TBA From rrosebru@mta.ca Sun Sep 28 18:51:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 28 Sep 2008 18:51:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kk448-0000YK-VG for categories-list@mta.ca; Sun, 28 Sep 2008 18:44:32 -0300 Date: Sun, 28 Sep 2008 10:19:18 -0400 (EDT) From: Michael Barr To: Bill Rowan , categories@mta.ca Subject: categories: Re: Group and abelian group objects in the category of Kelley spaces 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: 133 I have not thought deeply on this, but it strikes me that the basic problem is that such a group might not have a uniform topology. Such a group will have, I think, a separately continuous multiplication and hence, if U is a neighborhood of the identity {xU|x \in G} will be a cover, but there would seem no obvious reason for it to have a *-refinement. A continuous homomorphism would be uniformly continuous for those covers, if they do form a uniformity, it seems to me. If only John Isbell were still around to answer this kind of question, a wish I have wished many times since and well before his demise. But have you looked in his uniform spaces book? That is the sort of thing he might well have considered. If I were around the math library, I would look. Michael On Thu, 25 Sep 2008, Bill Rowan wrote: > Hi all, > > Does anyone know of a good place where someone has written down the basic > properties of such objects? As an example, if we have an (abelian, say) > topological group, there is a natural uniform topology on the group such > that the operations are uniformly continuous. Does the same hold for > abelian group objects in the category of Kelley spaces? But anything > would be helpful. > > Bill Rowan > > From rrosebru@mta.ca Sun Sep 28 20:49:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 28 Sep 2008 20:49:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kk5ua-0006Sg-Iw for categories-list@mta.ca; Sun, 28 Sep 2008 20:42:48 -0300 From: Martin Escardo MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Mon, 29 Sep 2008 00:10:56 +0100 To: Michael Barr , categories@mta.ca Subject: categories: Re: Group and abelian group objects in the category of Kelley spaces Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 134 It is interesting that an answer to this question, before it was asked here in the categories list, came up two weeks ago in a discussion I had with Jimmie Lawson and Matthias Schroeder. Schroeder showed recently that N^(N^N), where N is discrete and the exponential is calculated in k-spaces, is not regular, and hence not zero-dimensional either, which was an open problem (notice that the compact-open topology on N^(N^N) is easily seen to be zero-dimensional). Lawson observed that this is isomorphic to Z^(N^N), which, with the pointwise operations, is an abelian group in the category of k-spaces. This gives your counter-example. Lawson also said that counter-examples to complete regularity of k-groups where previously known among the experts in the subject, but were more complicated and/or artificial. (I don't know references.) I hope this helps. Martin Escardo Michael Barr writes: > I have not thought deeply on this, but it strikes me that the basic > problem is that such a group might not have a uniform topology. Such a > group will have, I think, a separately continuous multiplication and > hence, if U is a neighborhood of the identity {xU|x \in G} will be a > cover, but there would seem no obvious reason for it to have a > *-refinement. A continuous homomorphism would be uniformly continuous for > those covers, if they do form a uniformity, it seems to me. > > If only John Isbell were still around to answer this kind of question, a > wish I have wished many times since and well before his demise. But have > you looked in his uniform spaces book? That is the sort of thing he might > well have considered. If I were around the math library, I would look. > > Michael > > On Thu, 25 Sep 2008, Bill Rowan wrote: > > > Hi all, > > > > Does anyone know of a good place where someone has written down the basic > > properties of such objects? As an example, if we have an (abelian, say) > > topological group, there is a natural uniform topology on the group such > > that the operations are uniformly continuous. Does the same hold for > > abelian group objects in the category of Kelley spaces? But anything > > would be helpful. > > > > Bill Rowan > > > > > > From rrosebru@mta.ca Sun Sep 28 20:49:13 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 28 Sep 2008 20:49:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kk5vO-0006VT-O4 for categories-list@mta.ca; Sun, 28 Sep 2008 20:43:38 -0300 Date: Sun, 28 Sep 2008 19:33:43 -0400 To: Andre Joyal ,categories@mta.ca From: "Ellis D. Cooper" Subject: categories: BourWiki-iBourbaki-FunctorWiki? Mime-Version: 1.0 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: 135 Dear Andr=E9 Joyal, At 01:55 PM 9/25/2008, you wrote: (i) Some peoples have expressed doubts about the feasability. (ii) we should make sure that iBourbaki can last for more than one= generation. (iii) We need to conciliate long term duration=20 with the rapid developpement of technology. (iv) My task at this point is to convince peoples=20 that the project is worthy and feasable. If it will be useful to distinguish between (a)=20 mathematicians informally doing mathematics at a=20 blackboard or on a shared napkin or pad, and (b)=20 formally composing mathematical exposition in,=20 say, LaTeX supplemented by software technology=20 for diagrams (e.g., xy-pic and Xfig), then my=20 project is relevant solely to (a). It seems to=20 me that both (a) and (b) are necessary for the Subject. Imagine (a) a mathematics discussion forum with a=20 conversation like that on the categories=20 listserv, except augmented by the ability to=20 draw and edit all kinds of mathematical=20 expressions and diagrams (it has been said that=20 "The Bourbaki were Puritans, and Puritans are=20 strongly opposed to pictorial representations of=20 truths of their faith."), and (b) a Wiki-like=20 peer-generated-and-reviewed presentation of=20 research, report, and tutorial material=20 with links back to the discussions, arguments,=20 and diagrams in (a). All students of mathematics=20 should be allowed to visit the forum, perhaps=20 like a virtual Gallery of the Senate Chamber=20 combined with the MIT OpenCourseWare initiative. ( (i), (iv) ) Regarding feasibility of (a), last year I=20 sketched a possible algorithm intended to=20 address basic issues of presence and=20 registration in a paper available at=20 http://distancedrawing.com/Napkin071215a.pdf . A revision of that algorithm substitutes (1)=20 webcam USB technology for digital video camera=20 technology, and (2) the browser protocol for instant messaging. Perhaps the AJAX design pattern called HTTP=20 Streaming -- a long-lived HTTP connection with=20 each collaborator -- might be worth=20 investigating. A merging algorithm for multiple=20 independently generated images to create a single=20 "virtual drawing" is described in the aforementioned paper. The open source project I am proposing would=20 combine forum software like=20 http://www.yetanotherforum.net/ with webcam software. ( (ii), (iii) ) With regard to (b) there will be not only=20 development of computer technology, but also=20 development of mathematical technology. Someday=20 might there be something that is as much beyond=20 category theory as category theory is beyond Bourbaki? Strictly speaking, the Subject and this project=20 are not about categories. I wonder how you think=20 this discussion should be diverted to a separate=20 stream and joined just by those keenly interested in (i)-(iv). Ellis From rrosebru@mta.ca Mon Sep 29 09:49:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Sep 2008 09:49:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KkI6P-0007Ka-M4 for categories-list@mta.ca; Mon, 29 Sep 2008 09:43:50 -0300 Date: Mon, 29 Sep 2008 03:36:27 -0700 (PDT) From: Jeff Egger Subject: categories: Re: Group and abelian group objects in the category of Kelley spaces To: categories@mta.ca, Bill Rowan 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: 136 > if we have an (abelian, say) > topological group, there is a natural uniform topology on > the group such > that the operations are uniformly continuous. Does the > same hold for > abelian group objects in the category of Kelley spaces? As others have already noted, the answer is no. One possible solution (assuming you regard this as a defect) is to apply the idea implicit in the definition of Kelley space, not to the category of all topological spaces, but to that of all Tychonov (=uniformisable) spaces. What results is a cartesian closed category (that of "k_R-Tychonov spaces") with somewhat different properties; a group in this category is tautologously uniformisable and, if I recall correctly, is also true that the operations are uniformly continuous. Gabor Lukacs has studied these things and spoken about them at several conferences. Cheers, Jeff. __________________________________________________________________ Yahoo! Canada Toolbar: Search from anywhere on the web, and bookmark your favourite sites. Download it now at http://ca.toolbar.yahoo.com. From rrosebru@mta.ca Mon Sep 29 09:49:54 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Sep 2008 09:49:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KkI6x-0007Rx-MO for categories-list@mta.ca; Mon, 29 Sep 2008 09:44:23 -0300 Date: Mon, 29 Sep 2008 13:40:15 +0100 (BST) From: "Prof. Peter Johnstone" To: Categories mailing list Subject: categories: Subtoposes of Eff 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: 137 Readers of Jaap van Oosten's new book on realizability will have come across the statement on p. 235 that the question "Is Set the only Grothendieck topos which occurs as a subtopos of the effective topos?", which Ieke Moerdijk asked him at his Ph.D. viva in 1991, is still unanswered. In fact I've known the answer to this question for at least five years: I referred to it in my talks at the European category theory meeting in Haute-Bodeux (September 2003) and the meeting on Ramifications of Category Theory in Firenze (November 2003). However, the proof hasn't been published anywhere (it will of course be in volume 3 of the Elephant when that is published -- and please don't ask me to predict when that will be!), so I thought I should let people know about it. We work in the realizability topos over an arbitrary pca A (all unexplained notation can be found in Jaap's book). Andy Pitts showed in his thesis that local operators on this topos correspond to maps j: PA --> PA whch "are nuclei in the logic of A-realizability", and that the corresponding subtopos of sheaves is equivalent to Set (or degenerate) iff the intersection \bigcap_{a \in A} j({a}) is inhabited. Suppose this condition fails; we shall show that the subtopos of j-sheaves is not complete. Let A_1 denote the assembly with underlying set A and [[a \in A_1]] = {a} for all a, and A_2 the assembly with the same underlying set and [[a \in A_2]] = j({a}). Note that A_2 is the j-closure of A_1 in \nabla A, and hence its associated j-sheaf. Also, by the assumption we've made, it is not the whole of \nabla A. We claim that the product of A copies of A_2 cannot exist in the topos of sheaves: if it did, it would obviously be a subobject of \nabla(A^A) (the latter being the product of A copies of \nabla A) and hence an assembly; moreover, its underlying set would be the whole of A^A since the forgetful functor from assemblies to Set (is representable, and hence) preserves all products which exist. Writing P for the product, for each a \in A the a^th projection A^A --> A must be trackable as a morphism P --> A_2; let e_a be an element of A which tracks it. Since A_2 is not the whole of \nabla A, we can find f(a) such that e_aa \not \in [[f(a) \in A_2]] (if e_aa is undefined, choose f(a) to be any element of A). Then f: A --> A has no realizers as a member of P. Peter Johnstone From rrosebru@mta.ca Mon Sep 29 15:37:29 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Sep 2008 15:37:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KkNVq-00022K-26 for categories-list@mta.ca; Mon, 29 Sep 2008 15:30:26 -0300 MIME-Version: 1.0 To: , "Bill Rowan" , "Jeff Egger" Content-Type: text/plain; charset="utf-8" Date: Mon, 29 Sep 2008 11:19:12 -0400 Subject: categories: Re: Group and abelian group objects in the category of Kelley spaces From: wlawvere@buffalo.edu Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 138 The term "Kelley space" is a misnomer (due to Gabriel & Zisman ?), resulting from a misinterpretation of the prefix in "k-space". JLK's excellent 1955 textbook, from which=20 many of us learned mathematics, was not doing subtle self-promotion when he used that term in=20 his clear exposition. In fact, "k" stands for kompakt, and the term was used by Hurewicz in his 1949 lectures at Princeton where he introduced these spaces. I have that from telephone discussions with the late David Gale, who had mentioned Hurewicz's k-spaces in his=20 1950 PAMS paper (as noticed by Horst Herrlich). The same implicit idea is used in RH Fox's 1945 paper (except based on countable=20 compact spaces instead of all), which was directly incited by a letter from Hurewicz. Bill On Mon 09/29/08 6:36 AM , Jeff Egger jeffegger@yahoo.ca sent: > > if we have an (abelian, say) > > topological group, there is a natural uniform > topology on> the group such > > that the operations are uniformly continuous.=20 > Does the> same hold for > > abelian group objects in the category of Kelley > spaces? > As others have already noted, the answer is no. One possible > solution (assuming you regard this as a defect) is to apply > the idea implicit in the definition of Kelley space, not to > the category of all topological spaces, but to that of all > Tychonov (=3Duniformisable) spaces. What results is a cartesian > closed category (that of "k_R-Tychonov spaces") with somewhat > different properties; a group in this category is tautologously > uniformisable and, if I recall correctly, is also true that the > operations are uniformly continuous. Gabor Lukacs has studied > these things and spoken about them at several conferences. >=20 > Cheers, > Jeff. From rrosebru@mta.ca Mon Sep 29 16:53:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Sep 2008 16:53:26 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KkOiD-0000nc-Mr for categories-list@mta.ca; Mon, 29 Sep 2008 16:47:17 -0300 Date: Mon, 29 Sep 2008 09:37:20 -0400 From: jim stasheff MIME-Version: 1.0 To: Categories list Subject: categories: George Janelidze 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: 139 Contact has been established - thanks to all those many who helped jim From rrosebru@mta.ca Tue Sep 30 21:30:11 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Sep 2008 21:30:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KkpUa-0006FP-0G for categories-list@mta.ca; Tue, 30 Sep 2008 21:23:00 -0300 Date: Tue, 30 Sep 2008 11:20:44 -0500 From: "Charles Wells" To: catbb Subject: categories: mathematical objects 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: 140 Readers of this list may be interested in my new blog post, Mathematical objects in Wikipedia . -- Charles Wells professional website: http://www.cwru.edu/artsci/math/wells/home.html blog: http://www.gyregimble.blogspot.com/ abstract math website: http://www.abstractmath.org/MM//MMIntro.htm personal website: http://www.abstractmath.org/Personal/index.html