Date: Wed, 2 Aug 1995 09:18:56 -0300 (ADT) Subject: Montreal Category Theory Meeting Date: Tue, 1 Aug 95 14:13:22 EDT From: "Thomas F. Fox" CENTRE de RECHERCHE en THEORIE des CATEGORIES CATEGORY THEORY RESEARCH CENTER CATEGORY THEORY OCTOBERFEST: FIRST ANNOUNCEMENT McGill University, Montreal Saturday-Sunday, October 14-15, 1995 Dear Colleague, Once again this October the Category Theory Research Center will sponsor a weekend meeting at McGill University. This annual event brings together mathematicians interested in the categorical aspects of logic, computer science, combinatorics, universal algebra, homological algebra, topology, analysis, and theoretical physics. If you wish to speak, please contact Michael Barr as soon as possible. We particularly wish to encourage advanced students and new PhDs to use this forum to disseminate their work. The final schedule of talks be drawn up the morning of Oct 14. This being the era of fiscal responsibility, there will be a registration fee of $40 for faculty, $20 for students. As well, the university bureaucracy insists that we have a firm list of participants before the meeting, so please let us know if you intend to join us by sending a short email to fox@triples.math.mcgill.ca. Below you will find a list of hotels and tourist rooms close to McGill. If you have any further questions, please contact Tom Fox. CRTC Dept of Mathematics and Statistics McGill University 805 Sherbrooke West Montreal, Quebec CANADA H3A 2K6 Michael Barr barr@triples.math.mcgill.ca Tom Fox fox@triples.math.mcgill.ca ---------------------------------------------------- Hotels: L'Appartement, 455 Sherbrooke W, 284-3634, $79 Howard Johnson Plaza, 475 Sherbrooke W, 842-3961, $79 Citadelle, 410 Sherbrooke W, 844-8851, $88 Westen Mont-Royal, Sherbrooke W, 284-1110, $135 Holiday Inn, 420 Sherbrooke W, 842-6111, $99 Tours Richelieu, 2045 Peel, 844-3381, $75-95 Quality Hotel, 3440 Park Ave, 849-1413, $82 Hotel du Parc, 3625 Park Ave, 288-6666, $90 Versailles*, 1659 Sherbrooke W, 933-3611, $109 (B&B) Tourist Rooms: Ambrose, 3422 Stanley, 288-6922, $45-50 Casa Bella, 258 Sherbrooke W, 849-2777, $40-75 Armor*, 151 Sherbrooke E, 285-0140, $40-70 Centre Ville B&B*, 3458 Laval, 289-9749, $40-55 Pierre*, 169 Sherbrooke E, 288-8519, $40-60 Argoat**, 524 Sherbrooke E, 842-2046, $65 Bienvenue B&B** 3950 Laval, 844-5897, $45-80 *15 minute walk from McGill **30 minute walk from McGill Date: Sat, 12 Aug 1995 23:53:15 -0300 (ADT) Subject: separate continuity Date: Fri, 4 Aug 95 15:32:33 EDT From: Michael Barr Does anyone know anything about topological spaces for which the topological spaces for which the topology of separate continuity on their square coincides with the product topology? No non-discrete T_1 topology is possible; any order topology has the property and we have at least one other space and that about exhausts our knowledge of the subject. Michael Date: Tue, 15 Aug 1995 14:04:55 -0300 (ADT) Subject: Re: separate continuity Date: Mon, 14 Aug 95 19:58:55 +0200 From: Reinhold Heckmann Concerning the question "Which are the spaces X such that the topology of separate continuity on X x X coincides with the product topology?" I know of the following result: for a T0-space X, the following are equivalent: 1) for every T0-space Y, the topology of separate continuity coincides with the product topology on X x Y, 2) whenever a point x of X is in an open set U of X, there are a finite set F and an open set V of X such that x in V subset up F subset U. Here, up F is { b in X | b is above some a in F in the specialization preorder }. To prove 1 => 2, let Y be the space of open sets of X, where the topology is generated by assuming all neighborhood filters of points of X as open. Every continuous dcpo with its Scott topology satisfies condition (2) with F being a singleton. There are non-continuous dcpo's which also do; called quasi- or multi-continuous by some authors. With T1, up F = F holds. Thus, the only T1-spaces satisfying (2) are the discrete spaces. The above condition (2) probably is only sufficient, but not necessary for the special case of X x X. Hence, this is only a partial answer to the original question. Reinhold Heckmann Universitaet des Saarlandes Date: Tue, 22 Aug 1995 08:38:08 -0300 (ADT) Subject: update to (and correction of) Complexity Doctrines ( Date: Sun, 20 Aug 95 20:39:50 EDT From: James Otto Dear People, The following update to (and correction of) my thesis, Complexity Doctrines, is also linked to (either of) ftp://triples.math.mcgill.ca/ctrc.html ftp://triples.math.mcgill.ca/pub/otto/otto.html Regards, Jim Otto Update to Complexity Doctrines J. Otto August 20, 1995 In this note we 1. point out an (annoying) error (of ours) in background material on G"odel's system T, and thus pose an question, 2. improve the definition of tier 0, 3. improve the definition (and name) of sketches theories, 4. reduce presheaves to (Makkai) sketches, and 5. propose a definition of resolutions. We thus update the June 13, 1995 version of Complexity Doctrines. That version is currently linked to ftp://triples.math.mcgill.ca/pub/otto/otto.html By the way, Springer LNCS 953 contains a slightly earlier (with an X which clearly should be a 1) and abridged version of Chapter 2 of Complexity Doctrines. 1. System T We consider NNO (= natural numbers objects) in SMC (=symmetric monoidal closed) and CC (= cartesian closed) categories. Write (as this is not TeX) * for tensor and I for the unit. Define -o by _ * X -| X -o _. Write S for the category of SMC categories having NNO and of functors preserving chosen structure, and T for the category of CC categories having NNO and of functors preserving chosen structure. With J and K initial categories in S and T and with standard structure on set (the category of small sets), we have the (unique) S and T functors j : J --> set and k : K --> set. Statement 5 of Proposition 1.2.4.1 claims (which we now doubt) that j and k represent the same numeric functions (= functions between finite powers of N). The purported proof of Statement 5 fails (as we finally saw after kind remarks by P. Scott at the 1995 Category Theory and Computer Science meeting) as while the diagonal diag : N --> N * N is definable in J [Par'e Rom'an], it is doubtful that e.g. the diagonal diag : N -o N --> (N -o) * (N -o N) is definable in J. As both j and k represent the primitive recursive numeric functions, they both represent Turing machines modulo how long the machines run. Thus which numeric functions j and k represent is a matter of how fast such functions can grow. It is known [Rose, Subrecursion] that the (set of) numeric functions represented by k is the extended Grzegorczyk hierarchy below epsilon_0. At least a naive attempt to obtain such growth of numeric functions represented by j fails. In J we can define, by commuting diagrams, 0 * N s * N I * N ----> N * N ----> N * N | | | | l_N | + | + v v v N --------> N --------> N e_2 = + diag 0 s I ------> N ------> N | | | | id | e_n+1 | e_n+1 v v v I ------> N ------> N s 0 e_n Now we would like to diagonalize the e_n to get beyond primitive recursion. But this may be undefinable in J. However in K, by using diag : N -o N --> (N -o N) * (N -o N) and now writing _ x X -| X => _ and 1 rather than _ * X -| X -o _ and I, we can define 0 x (N => N) s x (N => N) 1 x (N => N) ------> N x (N => N) ------> N x (N => N) | | | | p_1 | p_1, ' | p_1, ' v v v N => N ---------> (N => N) x N ------> (N => N) x N id, s 0 ! p_0, @ 0 s 1 ------> N --------> N | | | | id | f | f v v v 1 ----> N => N ---> N => N f_2 " Here _ x X -| X => _ defines @ : (X => Y) x X --> Y terminal in the comma category (_ x X)/Y and e_2 : N --> N ------------------ f_2 : 1 --> N => N ' : N x (N => N) --> N ------------------------- " : (N => N) --> (N => N) e : N x N --> N ---------------- f : N --> N => N Further ! is the unique map to 1. Then we can diagonalize: e diag. Thus we pose the question Which numeric functions are represented by j? 2. Tier 0 Tier 0 can be defined as the pseudo-equalizer (aka iso-inserter) of T C ----> C ----> I ! (with I ! taking objects to the unit I and maps to the identity on I). 3. Sketches Theories Sketches theories, formerly sketch theories, are theories of (Makkai rather than Ehresmann) sketches. Definition. With a a cardinal, an a-sketches theory is a category S such that 1. S is small, 2. S is well-founded, and 3. S has fan-out < a. Here that S is well-founded is that for all S objects X all chains of composable non-identity maps starting from X X --> --> ... have finite length. Note that well-founded implies acyclic (aka 1-way) and skeletal. Further, S having fan-out < a is that for all S objects X the cardinality of the set (indeed, cone) of maps starting from X is < a. We are mainly interested in the finitary case, which is when a is countable. 4. Reducing Presheaves to (Makkai) Sketches. [Ad'amek Rosick'y] elegantly present the accessible categories, in various classes, as the full subcategories in categories of presheaves set^C of the objects [cone] (orthogonal | injective) relative to small sets A of (cones | maps). We reduce, by orthogonality, the categories of presheaves set^C to categories of a-sketches set^S. Proposition. Given a small category C there is a 4-sketches theory S and a (small) set A of finitely presentable set^S maps such that set^C is equivalent to the full subcategory in set^S of objects orthogonal to A. Proof. As S objects, take the objects and maps of C. As non-identity S maps, indexed by the C maps f : X --> Y, take the (formal) maps c_f : f --> Y, d_f : f --> X. Define a functor G : S --> C by c_f |-> f, d_f |-> id_X. Then G^* : set^C --> set^S by F |-> F G is faithful full. We recover the image of G^*, up to equivalence, by defining A through the following 3 schemes (using notation as in Complexity Doctrines). For C maps f : X --> Y {! i_f x : f d_f i_f x = x : X [x : X]} For C identity maps j : X --> X {c_j a = d_j a : X [a : j]} For C compositions h ------------> X ----> Y ----> Z f g {c_h c = c_g b : Z [d_h c = d_f a : X c_f a = d_g b : Y a : f b: g c : h]} QED 5. Resolutions Consider a finitary sketches theory S and a (small) set A of finitely presentable maps in set^S. Again (as in [Ad'amek Rosick'y]) reduce orthogonality to injectivity. (Thus, for non-epi maps a in A, add to A the a^* : P --> X induced by the push-out of a along a: a a ------> ------> | | | | | a | | a | id v po v v v ------> P ------> X .) Roughly following [Makkai], define deductions d as compositions of push-outs (in set^S) of maps a (thought of as axioms) in A. We propose defining resolutions as cospans (in set^S) | | v ----> d with the left map a deduction. Then resolutions compose as cospans: | | d' v ----> | | | | v po v ----> ----> d As the initial sketch orthogonal to the (original) A is the colimit of the diagram of deductions from the empty sketch 0, one may wish to resolve to 0: | | v 0 ----> d In a resolution X | | f v ----> d think of X as a query and of f as a partial answer. End of note. Date: Thu, 24 Aug 1995 00:17:14 -0300 (ADT) Subject: Logic/Category Theory POSITIONS Date: Wed, 23 Aug 1995 09:15:32 -0500 (CDT) From: fzalamea@bacata.usc.unal.edu.co Dear Logic/Category Theory Colleagues: WE WOULD BE VERY GRATEFUL IF YOU COULD CIRCULATE THE FOLLOWING MESSAGE: The Department of Mathematics, National University, BOGOTA, COLOMBIA is looking for two full-time positions at the level of associate professor, in order to push forward its newly formed Ph.D. program in Mathematics. Some of the areas of interest are MATHEMATICAL LOGIC and CATEGORY THEORY. The most needed subexpertises are categorical logic, non-classical logics and/or mathematical theory of computation. Candidates are expected to have their Ph.D., some international publications, willingness to supervise Ph.D. thesis, strong commitment to research and some beginning interest in doing research/teaching in spanish. Annual salary is expected to be around $25.000 (USA). Nevertheless, the real adcquisitive power of the salary goes beyond its face value. It should be noted that in Bogota, for example, a fair lease of a 3-bedroom apartment costs aprox. $600 (USA) monthly. Candidates should send an e-mail to FERNANDO ZALAMEA : fzalamea@bacata.usc.unal.edu.co with a short resume of their curriculum vitae and a proposal of activities to be carried on in Colombia (advanced courses/seminars, plans about future research are encouraged). FURTHER INSTRUCTIONS WOULD THEN BE SENT TO CANDIDATES WHO REPLY TO THIS COMMUNICATION. Date: Sat, 26 Aug 1995 12:17:04 -0300 (ADT) Subject: AMS classification Date: Fri, 25 Aug 1995 13:14:00 -0400 From: James Stasheff Apologies for using overlapping mailing lists. Date: 23 Aug 1995 08:23:26 -0400 (EDT) From: Arthur Greenspoon We're starting to gather suggestions for classification revisions/additions for 2000--major changes are contemplated because of mushrooming of cross-pollination and subsequent germinations-- so I ask you to send any suggestions and pass the word along to anyone you like. Date: Mon, 28 Aug 1995 16:03:43 -0300 (ADT) Subject: Peripatetic Seminar on Sheaves and Logic Date: Mon, 28 Aug 1995 14:32:36 +0100 From: Tracy Combe PERIPATETIC SEMINAR ON SHEAVES AND LOGIC 59th meeting - Preliminary announcement The 59th meeting of the seminar will be held at the University of Edinburgh's Computer Science Department over the weekend of 7-8 October 1995. This particular meeting is dedicated to Peter Freyd in celebration of his 60th birthday. Peter of course will attend. The seminar welcomes talks using or addressing category theory or logic, either explicitly or implicitly, in the study of any aspect of mathematics or science. Unusually for this seminar, we hope to publish a proceedings of the meeting as a special issue of the Journal of Pure and Applied Algebra, dedicated to Peter. Obviously, it is impractical for many scientists from outside western Europe to attend the meeting, so we expect to allow submissions to the proceedings by people who cannot attend but would like to participate. There are frequent rail and air connections to Edinburgh. We will send further information on the location of the seminar, along with local travel details to those who register, a week or so before the meeting. For accommodation, please either see the department's world wide web page and make your own arrangements, or submit the enclosed request form to Tracy Combe, our very efficient secretary. Marcelo Fiore Mike Fourman John Power Alex Simpson ................................................................................ Please return to Tracy Combe, Dept of Computer Science, University of Edinburgh, The King's Buildings, Edinburgh EH9 3JZ, Scotland. email: tlc@dcs.ed.ac.uk. I intend to come to the 59th meeting of the PSSL * I intend to give a talk entitled ..................................... * Please reserve accommodation for Friday/Saturday/Sunday night(s) Name ....................................... Address .......................................................... ............................................................................ ................................................... Email ...................................... Tel No .............................................................. *Delete if inapplicable Date: Tue, 29 Aug 1995 11:20:43 -0300 (ADT) Subject: Macquarie Math Chair Date: Tue, 29 Aug 1995 11:06:36 +1000 From: Ross Street The Macquarie University Mathematics Department expects soon to advertise a "permanent" Chair in Mathematics. The current holder, the Number Theorist John Loxton, has become Deputy Vice-Chancellor (Academic) at Macquarie. The other Chairs in the Department are held by Alf van der Poorten (Number Theory), Alan McIntosh (Functional Analysis & PDEs), and myself. I would certainly like the vacant Chair to be filled by someone in a field closely allied with Category Theory. So I am looking for expressions of interest. Please feel free to contact me by email if you are interested or have suggestions. /\ / \ M --/ Co \--> MQ / A C T\ /________\ Centre of Australian Category Theory Mathematics Department, Macquarie University New South Wales 2109, AUSTRALIA Telephone: (61-2-)850-8921 Facsimile: (61-2-)850-8114 email: street@mpce.mq.edu.au Date: Wed, 30 Aug 1995 17:02:40 -0300 (ADT) Subject: my new address Date: Wed, 30 Aug 1995 02:51:10 -0400 From: Hongde Hu My new address is Hongde Hu, Departement de mathematiques, Univ. du Quebec a Montreal, C.P.8888, Succursale A, Montreal, QC, Canada, H3C 3P8 e-mail address: hu@math.uqam.ca Date: Thu, 31 Aug 1995 09:31:35 -0300 (ADT) Subject: Re: AMS classification Date: 31 Aug 95 11:46:21 +0200 From: Dr. Reinhard Brger Dear colleagues and friends, I have the following suggestions for the revision of the AMS classification scheme: - In 18A20 the word "bicategories" should be cancelled because nowadays itusually has a different meaning. Maybe, bicategories in the present sense should be included somethere under 18D. - In 18A32 facorizations of cones and cocones (or sources and sinks) with diagonalizaion should be included as well as the orthogonality relations between objects and morphisms, maybe also injective and projective objects with respect to some class of morphisms. - The formulation "functors commuting with limits" in 18A35 should be changed to "functors preserving limits" or "preservation of limits (by functors)" - Generators (and cogenerators) should be included somewhere under 18A. - Accessible and locally presentable categories should be included under 18B. Maybe topoi deserve their own section (the topos people should decide that). Greetings Reinhard