From cat-dist Thu Oct 1 16:55:55 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA23829 for categories-list; Thu, 1 Oct 1998 15:22:51 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Marek Golasinski Message-Id: <199810011742.TAA11530@Waldemar.mat.uni.torun.pl> Subject: categories: center of a category To: categories@mta.ca Date: Thu, 1 Oct 1998 19:42:09 +0200 (MET DST) X-Mailer: ELM [version 2.4 PL25 PGP3 *ALPHA*] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Dear Categories, I would greatly appreciate getting some references on the CENTER of a category and related topics. Many thanks in advance for your kind attention. With my best wishes. Marek Golasinski From cat-dist Fri Oct 2 14:02:16 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA23556 for categories-list; Fri, 2 Oct 1998 12:20:38 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: Oege de Moor Message-Id: <199810021506.QAA02641@icarus.comlab> Subject: categories: postdoc funded by Microsoft Date: Fri, 2 Oct 1998 16:06:30 +0100 (BST) To: categories@mta.ca X-Mailer: fastmail [version 2.4ME+ PL27 (25)] Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: OXFORD UNIVERSITY COMPUTING LABORATORY IN COLLABORATION WITH MICROSOFT RESEARCH OFFICER INTENTIONAL PROGRAMMING PROJECT Oxford University Computing Laboratory has recently started a three-year research project in collaboration with Microsoft Research Laboratories. The goal is to develop a new kind of environment for transformational programming that permits software to be composed from a set of independent design decisions or "intentions", using domain-specific notations and optimization strategies. The specific aim of the Oxford component of the work is to design a meta-language for the environment, within which domain-specific abstractions can be described, implemented and reused. We now have a vacancy for an additional research officer to join the project with immediate effect, for an initial period of one year, but with the expectation of an extension to cover the three-year life of the project, depending on continuation of the research contract. The research officer will work at Oxford University Computing Laboratory, together with three research students and three academics, namely Oege de Moor, Michael Spivey and Bernard Sufrin. The research officer's specific tasks will include: (a) identifying suitable features of current meta languages in compiler construction and automated theorem proving. (b) designing and building a prototype implementation of a suitable meta language. (c) experimenting with the use of that meta language in case studies. The successful candidate will * have demonstrated research ability in a Computing-related discipline, * have experience of programming language design and implementation, transformational programming, or automated theorem proving, * actively enjoy the challenge of collaborating with engineers working in industry. Familiarity with formal methods of program construction will be an advantage. Salary will be on the age and experience related RS1A grade (currently 15,735 to 23,651 p.a.). Applications should clearly state the post title and be in the form of a full curriculum vitae plus application letter, together with the names of two referees. Further details and selection criteria are available from http://www.comlab.ox.ac.uk/oucl/jobs.html or on request to The Administrator of the Computing Laboratory. The completed application should be sent to arrive before the closing date of Friday 6th November 1998 and be addressed to: The Administrator, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD. (Email: Mike.Field@comlab.ox.ac.uk) . Oxford University is an Equal Opportunities Employer Informal email enquiries about the academic aspects of this project and this post are welcome, and should be directed to one of the academics: oege@comlab.ox.ac.uk (Oege de Moor) mike@comlab.ox.ac.uk (Michael Spivey) sufrin@comlab.ox.ac.uk (Bernard Sufrin) Oxford University Computing Laboratory is a full academic department of the University and at present has twenty-seven academic staff, thirty-five research officers and approximately sixty doctoral students, engaged in teaching and carrying out research in computer science and numerical analysis. From cat-dist Wed Oct 7 17:26:15 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA04272 for categories-list; Wed, 7 Oct 1998 16:11:36 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 7 Oct 1998 15:08:05 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: ftp site Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: >From now on, all the ftp sites here have been amalgamated to ftp.math.mcgill.ca. The material from triples, and a couple of other servers, have been amalgamated, but my papers are still found in pub/barr and Seely's in /pub/rags and so on. In a few days, you will be informed of this site if you try to sign on anonymously to triples. Sorry for the interruption. Michael From cat-dist Wed Oct 7 17:26:35 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA03081 for categories-list; Wed, 7 Oct 1998 16:12:45 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 7 Oct 1998 19:40:07 +0200 (MET DST) From: "Pedicchio M. Cristina" To: categories@mta.ca cc: "Pedicchio M. Cristina" Subject: categories: PSSL IN TRIESTE Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: The 69th Peripatetic Seminar on Sheaves and Logic November 28-29, 1998 TRIESTE - ITALY The 69th meeting of the PSSL will be held at the University of Trieste -Italy- the 28-29 of November 1998. The following colleagues have already announced their participation: J.Adamek, A.Carboni, G.Janelidze, P.Johnstone, W.Lawvere, J.Rosicky W.Tholen. If you intend to participate, please send me not later than the 31 of October your name, address, fax and e.mail, title of your talk (if you intend to speak) and dates of arrival and departure. I will send you information about hotels as well as a preliminary schedule of talks. Looking forward to seeing you in Trieste M.Cristina Pedicchio Dipartimento di Matematica Universita' di Trieste 34100 Trieste - ITALY tel. 39 40 6763270/3727 fax 39 40 6763256 pedicchi@univ.trieste.it From cat-dist Mon Oct 12 17:21:18 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA28109 for categories-list; Mon, 12 Oct 1998 15:55:15 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Mon, 12 Oct 1998 15:46:28 -0300 (ADT) From: Bob Rosebrugh To: categories Subject: categories: Mac Lane on Eilenberg Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: I haven't noticeed anyone on the list point out this article as yet... www.unipissing.ca/topology/t/o/p/c/52.htm It is on the following site which contains many interesting items: www.unipissing.ca/topology/topcom.htm regards, Bob Rosebrugh From cat-dist Tue Oct 13 17:57:17 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA09620 for categories-list; Tue, 13 Oct 1998 16:27:28 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 13 Oct 1998 12:14:08 +1300 (NZDT) From: Paul Bonnington Message-Id: <199810122314.MAA21261@aitken.scitec.auckland.ac.nz> To: categories@mta.ca Subject: categories: DMTCS'99 and CATS'99 Call for Participation Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Call for Participation: DMTCS'99, Discrete Mathematics and Theoretical Computer Science and CATS'99, Computing: The Australasian Theory Symposium University of Auckland and CDMTCS, Auckland, New Zealand 18-21 January 1999 --------------------------------------------------------------------------- DMTCS'99 and CATS'99 will be part of the Australasian Computer Science Week (ACSW'99) which is being held in the beautiful New Zealand city of Auckland (City of Sails, Museums, Cafes, Restaurants, Polynesian and Maori Culture and more). See http://www.tcs.auckland.ac.nz/~acsw99 for more information. [ Note that a Java Teaching Day was appended to the conference for Friday January 22nd, immediately after the ACSW'99 conferences. ] For the DMTCS/CATS'99 satellite conference there will be five invited papers plus 19 contributed papers. Invited Speakers: R. Downey (U. Victoria, NZ). Parametric Complexity After (almost) Ten Years: Review and Open Questions J. Goguen (UCSD, USA) Hidden Algebra for Software Engineering. J. Pach (Hungarian Academy of Sciences). Crossing Numbers of Graphs. A. Restivo and F. Mignosi (U. Palermo, Italy) On Negative Informations in Language Theory. G. Walsh (U. Ottawa, Canada). Efficiency vs. Security in the Implementation of Public-Key Cryptography. Contributed Papers: N.H. Arai. No feasible monotone interpolation for cut-free Gentzen type propositional calculus with permutation inference. M.D. Atkinson and R. Beals. Permuting mechanisms and closed classes of permutations. J.A. Bergstra and A. Ponse. Process algebra. V. Brattka. A stability theorem for recursive analysis. A. Cherubini, S. Crespi-Reghizzi and P. San Pietro. Languages based on structural local testability. A.S. Elkjaer, M. Hoehle, H. Huettel and K.O. Nielsen. Towards automatic bisimilarity checking in the Spi calculus. M. Gobel. Three remarks on SAGBI bases for polynomial invariants of permutation groups. R. Greenlaw and R. Petreschi. Computing Prufer codes efficiently in parallel. M. Hamada, A. Middeldorp and T. Suzuki. Completeness results for a lazy conditional narrowing calculus. T. Hasunuma. The pagenumber of de Bruijn and Kautz digraphs. H. Hinrichsen, H. Eveking and G. Ritter. Formal synthesis for pipeline design. C.S. Iliopoulos and L. Mouchard. An O(n log n) algorithm for computing all maximal quasiperiodicities in strings. T. Knapik and H. Calbrix. The graphs of finite monadic semi-thue systems have decidable monadic second-order theory. L. Kristiansen. Low_n, high_n, and intermediate subrecursive degrees. K-W. Lih, K D.-F. Liu and X. Zhu. Star-extremal circulant graphs. L. Parida. On the approximability of physical map problems using single molecule methods. A. Schonegge and D. Kempe. On the weakness of conditional equations in algebraic specification. N. Shibata, K. Okano, T. Higashino and K. Taniguchi. A decision algorithm for prenex normal form rational Presburger sentences by means of combinatorial geometry. S-C Sung and K. Tanaka. Lower bounds on negation-limited inverters. Cost of Participation: The registration fee is NZ$500 (which includes the dinner, excursion and proceedings), or NZ$150 for students (including only the proceedings). ACS discount NZ$20 and early discount NZ$40 (excluding students). [Note current exchange: $1 US = $2 NZ ] For More Information: See the home-page of the conference http://www.tcs.auckland.ac.nz/~acsw99/, or contact the local chair Bakh Khoussainov at bmk@cs.auckland.ac.nz. --------------------------------------------------------------------------- Conference Committee: C.P. Bonnington C.S. Calude (general chair) E. Calude R. Coles, P.B. Gibbons U. Guenther B. Khoussainov (local chair) ACSW'99 Contact Members: R.W. Doran (general chair) P. Fenwick Programme Committee: R.J. Back, TUCS, Finland M. Conder, U. Auckland, NZ B. Cooper, U. Leeds, UK M.J. Dinneen, U. Auckland, NZ (chair) R. Goldblatt, Victoria U., NZ S. Goncharov, Novosibirsk U., Russia J. Harland, RMIT, Australia R.E. Hiromoto, UTSA, USA H. Ishihara, JAIST, Japan M. Ito, Kyoto S.U., Japan M. Li, U. Waterloo, Canada X. Lin, UNSW, Australia R. Shore, Cornell U., USA T. Tokuyama, IBM, Japan D. Wolfram, ANU, Australia Proceedings Editors: C.S. Calude and M.J. Dinneen --------------------------------------------------------------------------- Important Dates: Registration Date (for authors): 06 Nov. 1998 (for others): January 1999 --------------------------------------------------------------------------- From cat-dist Wed Oct 14 12:34:54 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA24965 for categories-list; Wed, 14 Oct 1998 10:48:06 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f From: "Lutz Schroeder" Message-Id: <9810141242.ZM15583@i20.informatik.uni-bremen.de> Date: Wed, 14 Oct 1998 12:42:36 -0600 X-Mailer: Z-Mail (3.2.3 08feb96 MediaMail) To: categories@mta.ca Subject: categories: Paracategories Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: There seems to be a weakened concept of category called 'paracategory'; are there any references on this (or does anybody know what it is)? Thanks, Lutz Schroeder From cat-dist Wed Oct 14 15:23:35 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA24287 for categories-list; Wed, 14 Oct 1998 14:09:44 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <3.0.32.19981014121032.007a6210@pop.apk.net> X-Sender: cwells@pop.apk.net X-Mailer: Windows Eudora Pro Version 3.0 (32) Date: Wed, 14 Oct 1998 12:10:39 -0400 To: categories@mta.ca From: Charles Wells Subject: categories: Natural numbers objects and free algebras Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: I am vaguely aware that there are theorems to the effect that in a topos (pretopos?) with nno you have free algebras in some sense. Where can I read about that? What I want to know is what role the nno plays. Thanks, Charles Wells, Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-7058, USA. EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893. FAX: 216 368 5163. HOME PHONE: 440 774 1926. HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html From cat-dist Thu Oct 15 20:54:21 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA06749 for categories-list; Thu, 15 Oct 1998 19:53:27 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <3.0.5.32.19981015094256.007bf780@pop.doc.ic.ac.uk> X-Sender: sjv@pop.doc.ic.ac.uk X-Mailer: QUALCOMM Windows Eudora Light Version 3.0.5 (32) Date: Thu, 15 Oct 1998 09:42:56 +0100 To: categories@mta.ca From: S Vickers Subject: Re: categories: Natural numbers objects and free algebras In-Reply-To: <3.0.32.19981014121032.007a6210@pop.apk.net> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: At 12:10 14/10/98 -0400, Charles Wells wrote: >I am vaguely aware that there are theorems to the effect that in a topos >(pretopos?) with nno you have free algebras in some sense. Where can I >read about that? What I want to know is what role the nno plays. Johnstone and Wraith "Algebraic Theories in a Topos". The nno provides internally an infinite object. Once it is there, the elementary topos structure can be used to construct other free algebras. However, the constructions use exponentiation and subobject classifier, and pretopos structure is not enough. Adam Eppendahl and I are working on a conjecture that the structure of Joyal's Arithmetic Universes is sufficient to construct general free algebras, that structure comprising (following ideas of Joyal and Wraith) pretopos structure + free categories over graphs + free category actions over graph actions. Steve Vickers. From cat-dist Mon Oct 19 16:08:33 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA14443 for categories-list; Mon, 19 Oct 1998 14:15:21 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <1.5.4.32.19981019161440.0067b7fc@goliat.ugr.es> X-Sender: bullejos@goliat.ugr.es X-Mailer: Windows Eudora Light Version 1.5.4 (32) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Mon, 19 Oct 1998 17:14:40 +0100 To: categories@mta.ca From: Manuel Bullejos Subject: categories: Comma categories Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Does any body know if comma categories have been defined in enriched contexts? I have an idea of how they can be defined in some particular contexts, such as Cat-categories or Simplicial-categories, but I don't know if there is a general definition or even if a definition in the above two contexts can be found in the literature. Thanks Manuel Bullejos From cat-dist Mon Oct 19 17:45:24 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA06673 for categories-list; Mon, 19 Oct 1998 16:38:07 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199810191819.LAA20517@coraki.Stanford.EDU> To: Manuel Bullejos cc: categories@mta.ca Subject: Re: categories: Comma categories In-reply-to: Your message of "Mon, 19 Oct 1998 17:14:40 BST." <1.5.4.32.19981019161440.0067b7fc@goliat.ugr.es> Date: Mon, 19 Oct 1998 11:19:50 -0600 From: Vaughan Pratt Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: >Does any body know if comma categories have been defined in >enriched contexts? I'm not sure if it's what you have in mind, but combining comma categories and enrichment is the theme of Casley, R.T., Crew, R.F., Meseguer, J., and Pratt, V.R., ``Temporal Structures'', Mathematical Structures in Computer Science, Volume 1:2, 179-213, July 1991. The abstract is at my web page as http://boole.stanford.edu/chuguide.html#P2 Vaughan Pratt From cat-dist Tue Oct 20 21:11:44 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA00897 for categories-list; Tue, 20 Oct 1998 20:19:18 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 20 Oct 1998 08:13:01 +0200 (DFT) From: "jean-pierre-C." To: categories@mta.ca Subject: categories: category theory and probability theory Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Bonjour. I am a statistician and I should be interested in a categorical framework for probability and statistical theory. Does anyone know references (books, articles, websites...) about applications of categories and functors to probability or even measure theory ? Thank you. Very truly yours, Jean-Pierre Cotton. From cat-dist Tue Oct 20 21:11:46 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA01274 for categories-list; Tue, 20 Oct 1998 20:22:21 -0300 (ADT) Message-Id: <199810202322.UAA01274@mailserv.mta.ca> X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 20 Oct 1998 17:11:48 -0400 (EDT) From: F W Lawvere Reply-To: wlawvere@ACSU.Buffalo.EDU To: categories@mta.ca Subject: categories: Re: Comma categories Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: A crucial point is whether the recipient of the enriching is cartesian or not. Note that fully internalising always must involve a cartesian aspect since one must diagonalize on the parametrizers of families of objects (at least) in order to explain eg natural transformations, even if the parametrizers for individual homs are not cartesian (eg linear or metric). One can envisage replacing individual "comma" categories by families of categories parametrized by (commutative) coalgebras, which seems just a way of constructing a cartesian category for the purpose, to which it may or may not be adequate. Symmetric monoidal categories in which the unit object is terminal seem to have a special role, but that may be illusory.(After all "any" smc is covered by one with that additional property) . Perhaps the affine modules ( see my paper "Grassmann's dialectics and category theory") constitute a good test case for proposed constuctions Bill Lawvere. ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Mon, 19 Oct 1998, Manuel Bullejos wrote: > > Does any body know if comma categories have been defined in > enriched contexts? > > I have an idea of how they can be defined in some particular > contexts, such as Cat-categories or Simplicial-categories, but I > don't know if there is a general definition or even if a > definition in the above two contexts can be found in the > literature. > > Thanks > > Manuel Bullejos > > > From cat-dist Tue Oct 20 21:11:47 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA01048 for categories-list; Tue, 20 Oct 1998 20:20:26 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 20 Oct 1998 16:15:27 +0100 From: Lindsay Errington Message-Id: <199810201515.QAA21908@asterix.doc.ic.ac.uk> To: categories@mta.ca Subject: categories: twisted arrow categories Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: In Chapter IX, Sec 6, Ex 3 of CWM, MacLane defines the twisted arrow category of C such that objects are arrows f : a -> b of C and arrows are pairs of morphisms of C, (l,m) : f -> g, such that g = mfl. The construction is part of a proof of the reduction of ends to limits. I would be grateful for pointers to other occurrences of this construction in the literature. Lindsay Errington From cat-dist Tue Oct 20 21:11:50 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA00767 for categories-list; Tue, 20 Oct 1998 20:19:44 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Tue, 20 Oct 1998 11:33:58 +0200 From: Philippe Gaucher Message-Id: <199810200933.AA00901@irmast1.u-strasbg.fr> To: categories@mta.ca Subject: categories: cogenerator in omegaCat ? Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Content-Md5: qqa5a//AKrw7cruPMIHqDw== Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear all, Does it exist a cogenerator in the category of (strict) omega-categories ? pg. From cat-dist Tue Oct 20 21:12:28 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id UAA00763 for categories-list; Tue, 20 Oct 1998 20:18:23 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199810200024.KAA27351@macadam.mpce.mq.edu.au> X-Sender: street@macadam.mpce.mq.edu.au Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 20 Oct 1998 10:26:56 +1000 To: categories@mta.ca From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: Comma categories Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: >Does any body know if comma categories have been defined in >enriched contexts? Lawvere's La Jolla paper, where general comma categories were introduced, showed how to construct them from pullbacks and a "cylinder" (or "arrow object") construction. John Gray (SLNM p. 254) showed that cylinder is a universal notion which a 2-category may or may not have. I pointed out [Fibrations and Yoneda's lemma in a 2-category, Lecture Notes in Math. 420 (1974) 104-133; MR53#585] that finite completeness for a 2-category should mean that it have pullbacks, a terminal object, and cylinders (a similar idea was in my PhD thesis for differential graded categories which are finitely complete when they admit pullbacks, a zero object and "suspension"). Finite completeness for 2-categories is further analysed in [Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8 (1976) 149-181; MR53#5695]. More generally, finite completeness for a V-category A (= a category with homs enriched in V) means that its underlying category has finite ordinary limits, which are preserved by representables A(a,-) into V, and that it admits cotensoring by the "finite" objects of V. There is some choice about what you mean by "finite" object in V however "finitely presentable" is often the right thing. Sometimes, as in the case of V = Cat, the finite objects are generated by a few finite objects - that is why "cylinder" plays the important role in 2-categories (it is the finite generating object, cotensor with which is cylinder). So why am I going on about finite limits in 2-categories? Well, Lawvere's construction shows that comma objects exist in any finitely complete 2-category. Comma objects are particular finite limits just like pullbacks. In particular, there is a 2-category V-Cat of V-categories, V-functors and V-natural transformations. It is certainly complete (as a 2-category) for any decent V. So, indeed, it is well known that comma objects (or comma V-categories) exist. They have their uses but NOT for the wonderful use that Lawvere put them to: Lawvere provided a formula for left (right) Kan extensions of ordinary functors which involves taking a colimit (limit) over a comma category. [Indeed, more is true; see my definition of "pointwise Kan extension" in "Fibrations and Yoneda's lemma in a 2-category".] However, this formula does not work even for additive categories (= categories enriched in the monoidal category of abelian groups). Regards, Ross http://www.mpce.mq.edu.au/~street/ From cat-dist Wed Oct 21 12:41:39 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA27054 for categories-list; Wed, 21 Oct 1998 10:41:24 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199810210001.KAA24706@macadam.mpce.mq.edu.au> X-Sender: street@macadam.mpce.mq.edu.au Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 21 Oct 1998 10:03:33 +1000 To: categories@mta.ca From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: twisted arrow categories Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: >In Chapter IX, Sec 6, Ex 3 of CWM, MacLane defines the twisted arrow >category of C such that objects are arrows f : a -> b of C and arrows >are pairs of morphisms of C, (l,m) : f -> g, such that g = mfl. The >construction is part of a proof of the reduction of ends to limits. > >I would be grateful for pointers to other occurrences of this construction >in the literature. The twisted arrow category of A is the category of elements of the hom functor A(-,-) : A^op x A --> Set. In internal and variable category theory there is a sense in which the twisted arrow fibration comes first and then can be used to define hom functors and local smallness. For example, see page 292 of my paper Cosmoi of internal categories, Transactions American Math. Soc. 258 (1980) 271-318; MR82a:18007 Regards, Ross www.mpce.mq.edu.au/~street/ From cat-dist Wed Oct 21 12:43:58 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA29870 for categories-list; Wed, 21 Oct 1998 10:58:28 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <023a01bdfcd2$2cb17140$aec488c1@acs.math.ist.utl.pt> From: "Amilcar Sernadas" To: "jean-pierre-C." , Subject: Re: categories: category theory and probability theory Date: Wed, 21 Oct 1998 10:06:42 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: We are working on a related problem. It seems that it is necessary to work with a relaxed notion of category, namely where the compostion of f:a->b and g:b->c is not always defined. You should look at relaxed notions of category such as composition graphs, paracategories, precategories and the like. On our own preliminary results look at the working paper P. Mateus, A. Sernadas and C. Sernadas. Combining Probabilistic Automata: Categorial Characterization. Research Report, April 1998. Presented at the FIREworks Meeting, Magdeburg, May 15-16, 1998 that you can fetch from http://www.cs.math.ist.utl.pt/s84.www/cs/pmat.html Amilcar Sernadas -----Original Message----- From: jean-pierre-C. To: categories@mta.ca Date: Quarta-feira, 21 de Outubro de 1998 0:19 Subject: categories: category theory and probability theory > > Bonjour. I am a statistician and I should be interested in a categorical >framework for probability and statistical theory. Does anyone know >references (books, articles, websites...) about applications of categories >and functors to probability or even measure theory ? Thank you. > > Very truly yours, > > Jean-Pierre Cotton. > From cat-dist Wed Oct 21 13:25:01 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA03991 for categories-list; Wed, 21 Oct 1998 11:37:10 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <3.0.32.19981021102501.00881e60@pop.apk.net> X-Sender: cwells@pop.apk.net X-Mailer: Windows Eudora Pro Version 3.0 (32) Date: Wed, 21 Oct 1998 10:25:09 -0400 To: categories@mta.ca From: Charles Wells Subject: categories: Re: twisted arrow categories Cc: le@doc.ic.ac.uk Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: The twisted arrow category is discussed in my paper "Extension Theories for Categories", available at http://www.cwru.edu/artsci/math/wells/pub/papers.html (it has never been published). > >In Chapter IX, Sec 6, Ex 3 of CWM, MacLane defines the twisted arrow >category of C such that objects are arrows f : a -> b of C and arrows >are pairs of morphisms of C, (l,m) : f -> g, such that g = mfl. The >construction is part of a proof of the reduction of ends to limits. > >I would be grateful for pointers to other occurrences of this construction >in the literature. Charles Wells, Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-7058, USA. EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893. FAX: 216 368 5163. HOME PHONE: 440 774 1926. HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html From cat-dist Wed Oct 21 13:50:12 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA10564 for categories-list; Wed, 21 Oct 1998 12:16:27 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 21 Oct 1998 10:54:19 -0400 (EDT) From: F W Lawvere Reply-To: wlawvere@ACSU.Buffalo.EDU To: categories@mta.ca Subject: categories: Re: cogenerator in omegaCat ? In-Reply-To: <199810200933.AA00901@irmast1.u-strasbg.fr> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: No, it seems not since a co-generator for omega cat would surely give rise to one for cat in particular, but such does not exist. This contrasts with the situation for the "larger" universe of simplicial sets. A category of "small" sets is a kind of approximation to a co-generator, but each enlargement of the meaning of "small" creates new categories which are not co-generated. Bill ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Tue, 20 Oct 1998, Philippe Gaucher wrote: > Dear all, > > Does it exist a cogenerator in the category of (strict) omega-categories ? > > > pg. > > From cat-dist Wed Oct 21 15:36:12 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA32690 for categories-list; Wed, 21 Oct 1998 14:30:58 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 21 Oct 1998 18:45:26 +0200 From: Philippe Gaucher Message-Id: <199810211645.AA06093@irmast1.u-strasbg.fr> To: categories@mta.ca Subject: categories: Re: cogenerator in omegaCat ? Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: > No, it seems not since a co-generator for omega cat > would surely give rise to one for cat in particular, but such > does not exist. This contrasts with the situation for the > "larger" universe of simplicial sets. A category of "small" > sets is a kind of approximation to a co-generator, but each > enlargement of the meaning of "small" creates new categories > which are not co-generated. The argument sounds reasonable. Before this question, I was convinced of the existence of this cogenerator. I have to find something else for the lemma I would like to prove... Since it does not exist, I have another questions (I suppose well- known) and any reference abou the subject would be welcome : How does one prove the cocompleteness of omegaCat (small & strict) ? The only idea of proof I had in mind until this question was : omegaCat is obviously complete (and the forgetful functor towards the category of Sets preserves projective limits), and well-powered and a cogenerator => the cocompleteness (Borceux I, prop 3.3.8 p 112). Without cogenerator, how can one prove the cocompleteness ? The explicit construction of the colimit seems to be very hard : the forgetful functor towards Set does not preserve colimits because the underlying set of the colimit might be bigger than the colimit of the underlying sets. Every time two n-morphisms are identified in the colimit of the underlying sets, p-morphisms (with p>n) might be "created" by the colimit. Thanks in advance for any answer. pg. From cat-dist Thu Oct 22 11:22:19 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA24395 for categories-list; Thu, 22 Oct 1998 08:54:31 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 21 Oct 1998 15:36:24 -0400 (EDT) From: Michael Barr To: categories@mta.ca Subject: categories: Re: cogenerator in omegaCat ? In-Reply-To: <199810211645.AA06093@irmast1.u-strasbg.fr> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: I imagine that omega-categories, however defined, will be a locally c-presentable category (c=cardinal of the continuum) in the sense of Gabriel-Ulmer, equivalently complete and c-accessible in the sense of Makkai-Pare and hence cocomplete. In other words, the colimit will grow but not by much. Actually, aleph_1 is all you are really going to need. On Wed, 21 Oct 1998, Philippe Gaucher wrote: > > > > No, it seems not since a co-generator for omega cat > > would surely give rise to one for cat in particular, but such > > does not exist. This contrasts with the situation for the > > "larger" universe of simplicial sets. A category of "small" > > sets is a kind of approximation to a co-generator, but each > > enlargement of the meaning of "small" creates new categories > > which are not co-generated. > > > The argument sounds reasonable. Before this question, I was > convinced of the existence of this cogenerator. I have to find > something else for the lemma I would like to prove... > > Since it does not exist, I have another questions (I suppose well- > known) and any reference abou the subject would be welcome : > > How does one prove the cocompleteness of omegaCat (small & strict) ? > The only idea of proof I had in mind until this question was : omegaCat > is obviously complete (and the forgetful functor towards the category of Sets > preserves projective limits), and well-powered and a cogenerator > => the cocompleteness (Borceux I, prop 3.3.8 p 112). > > Without cogenerator, how can one prove the cocompleteness ? The explicit > construction of the colimit seems to be very hard : the forgetful > functor towards Set does not preserve colimits because the > underlying set of the colimit might be bigger than the colimit of the > underlying sets. Every time two n-morphisms are identified in the > colimit of the underlying sets, p-morphisms (with p>n) might be "created" > by the colimit. > > Thanks in advance for any answer. pg. > > From cat-dist Thu Oct 22 16:46:30 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA18344 for categories-list; Thu, 22 Oct 1998 15:47:30 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 22 Oct 1998 10:03:49 +1000 (EST) Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: categories@mta.ca From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: comma? Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Dear Carlos > Hi. I'm interested in following the discussion of enriched categories and >``comma objects'' on the categories list, and particularly in what you just >wrote. But...what is a ``comma category'' or more generally ``comma object''? Comma categories are explained, for example, in Mac Lane's book "Categories for the working mathematician" Grad Texts in Math #5 (Springer). In a sense they are "lax pullbacks" defined when given two functors with the same codomain. Hence, just as we can carry over the notion of pullback in Set to any category, we can carry over, by representability, the notion of comma category from Cat to any 2-category (I call them comma objects in SLNM 420). > Also, was your PhD thesis published? I used the notion of differential >graded category in a paper about vector bundles a while ago; then found out >that Kapranov et al had a few earlier papers about it, but I didn't know >that it came from even before that. Exaggerating only slightly, the whole reason enriched category theory got going in the 60s was to deal with the example of differential graded categories: these are categories enriched in chain complexes (Eilenberg-Kelly "Closed categories" LaJolla 1965). As I understand it, that example prompted Sammy and Max to their joint work: they had each used DG-categories before. My thesis [0] was not published in full. For the published papers [2], [5] on the thesis, I took out the stuff on DG-categories. However, [25] uses the DG-category approach very strongly and gives new proofs of more general results (I had some of these at the time of my thesis but hadn't included them). The papers provide universal coefficients (or homotopy classification) theorems for diagrams of chain complexes. 0. PhD Thesis: Homotopy Classification of Filtered Complexes, University of Sydney, August 1968. 2. Projective diagrams of interlocking sequences, Illinois J. Math. 15 (1971) 429-441; MR43#4881. 5. Homotopy classification of filtered complexes, J. Australian Math. Soc. 15 (1973) 298-318; MR49#5135. 25. Homotopy classification by diagrams of interlocking sequences, Math. Colloquium University of Cape Town 13 (1984) 83-120; MR86i:55025. Best regards, Ross www.mpce.mq.edu.au/~street/ From cat-dist Thu Oct 22 16:46:48 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA17573 for categories-list; Thu, 22 Oct 1998 15:42:39 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Wed, 21 Oct 1998 22:47:13 GMT Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Mailer: Eudora F1.5.4 To: categories@mta.ca From: carlos@picard.ups-tlse.fr (Carlos Simpson) Subject: categories: Re: cogenerator in omegaCat ? Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: In response to Ph. Gaucher's question: I (try to, at least...) treat this question for weak $n$-categories in my preprint ``Limits in $n$-categories'', available on the xxx preprint server as alg-geom 9708010. If I understand correctly, the set-theoretical problem you raise is the same as the one encountered in section 5 of my preprint. The conclusion is that the (weak) $n+1$-category $nCAT$ is closed under direct limits. It seems that coproducts of strict $n$-categories, if they exist, cannot actually be the ``right'' ones because in that case, every weak $n$-category would be equivalent to a strict one. I haven't made this argument rigorous, though. ---Carlos Simpson PS what is a ``comma category'' or ``comma object''? > >The argument sounds reasonable. Before this question, I was >convinced of the existence of this cogenerator. I have to find >something else for the lemma I would like to prove... > >Since it does not exist, I have another questions (I suppose well- >known) and any reference abou the subject would be welcome : > >How does one prove the cocompleteness of omegaCat (small & strict) ? >The only idea of proof I had in mind until this question was : omegaCat >is obviously complete (and the forgetful functor towards the category of Sets >preserves projective limits), and well-powered and a cogenerator >=> the cocompleteness (Borceux I, prop 3.3.8 p 112). > >Without cogenerator, how can one prove the cocompleteness ? The explicit >construction of the colimit seems to be very hard : the forgetful >functor towards Set does not preserve colimits because the >underlying set of the colimit might be bigger than the colimit of the >underlying sets. Every time two n-morphisms are identified in the >colimit of the underlying sets, p-morphisms (with p>n) might be "created" >by the colimit. > >Thanks in advance for any answer. pg. From cat-dist Thu Oct 22 16:46:55 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA18699 for categories-list; Thu, 22 Oct 1998 15:49:52 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 22 Oct 1998 12:11:50 +0100 (BST) From: Ronnie Brown To: categories@mta.ca Subject: categories: Cocompleteness of infinity categories and groupoids Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: The existence of colimits can be proved by adjoint functor type arguments: Lew Hardy and I wrote out details for the not too dissimilar situation of topological groupoids in ``Topological groupoids I: universal constructions'', {\em Math. Nachr.} 71 (1976) 273-286. Identification of vertices of a groupoid (or category) gives what Philip Higgins called a universal groupoid or category, and of course more compositions are allowed. Philip constructed this explicitly, but it also follows from the general construction. There is a mention of the cocompleteness of the multiple situation in (with P.J. HIGGINS), ``On the algebra of cubes'', {\em J. Pure Appl. Algebra} 21 (1981) 233-260. (see p. 238). Analysis and computation of colimits of various forms of multiple groupoids is necessary for applying Generalised Van Kampen Theorems - for the groupoid case this is most conveniently done in the `small' model of crossed complexes. For cat^n-groups (= n-fold categories in groups) it is probably most convenient to work in Ellis-Steiner's crossed n-cubes of groups (generalising Guin-Walery/Loday's crossed squares). Analysing pushouts of crossed squares led Loday and me to the non-abelian tensor product of groups (which act on each other). The identification of p-cells in an n-category (n>p) to give a new n-category is discussed in relation to homotopy theory in my survey ``Homotopy theory, and change of base for groupoids and multiple groupoids'', {\em Applied categorical structures}, 4 (1996) 175-193. That is, it shows how complicated and interesting are even simple cases of this general idea. We show how the n-adic Hurewicz theorem can be seen as an example of this in (with J.-L. LODAY), ``Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces'', {\em Proc. London Math. Soc.} (3) 54 (1987) 176-192. (This was the first proof of even a triadic Hurewicz theorem. The relative case goes back to early homotopy theory.) The general idea is of universally constructing an n-fold category from lower dimensional information, or lower dimensional identifications. The computational aspect (how to compute the answers) seems really interesting. One expects to be able to be more explicit in the groupoid case. On the other hand, even general double groupoids are a bit mysterious. Ronnie Brown From cat-dist Thu Oct 22 16:47:40 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA18022 for categories-list; Thu, 22 Oct 1998 15:45:14 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 22 Oct 1998 09:36:19 +1000 (EST) Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" To: categories@mta.ca From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: cogenerator in omegaCat ? Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mailserv.mta.ca id UAA14996 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: As to the cocompleteness of Omega-Cat, it is a result of Harvey Wolff that V-Cat is cocomplete for decent V. By induction, it follows that n-Cat is cocomplete (since (n+1)-Cat = n-Cat). A limiting process gives that Omega-Cat is also cocomplete. However, a better approach is to use a result of Michael Batanin that Omega-Cat is finitarily monadic over globular sets (a presheaf category). It follows that Omega-Cat is cocomplete. The required monad on globular sets is beautiful: it involves plane trees. See: M. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Advances in Mathematics 136 (1998) 39-103. R. Street, The role of Michael Batanin's monoidal globular categories, Proceedings of the Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997 (to appear). M. Batanin, Computads for finitary monads on globular sets, Proceedings of the Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997 (to appear). M. Batanin and R. Street, The universal property of the multitude of trees, Macquarie Mathematics Report 98/233, March 1998 (submitted). R. Street, The petit topos of globular sets, Macquarie Mathematics Report 98/232 (March 1998; talk at the "Billfest" in Montréal, September, 1997; submitted). Regards, Ross From cat-dist Mon Oct 26 12:35:34 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA19492 for categories-list; Mon, 26 Oct 1998 10:36:42 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Sun, 25 Oct 1998 13:18:33 -0500 (EST) From: F W Lawvere Reply-To: wlawvere@ACSU.Buffalo.EDU To: categories@mta.ca Subject: categories: Re: category theory and probability theory Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: In reply to the query of Jean-Pierre Cotton, I would like to mention the following: In Springer LNM 915 (1982), an article by Michele Giry develops some aspects of "A categorical approach to probability theory".The key idea, which also was discussed in an unpublished 1962 paper of mine, is that random maps between spaces are just maps in a category of convex spaces between "simplices". There is a natural (semi) metric on the homs which permits measuring the failure of diagrams to commute precisely, suggesting statistical criteria. To make full use of the monoidal closed structure,as well as to account for convex constraints on random maps,it seems promising to consider also nonsimplices. (Noncategories is usually not a good idea). The central observation that the metrizing process is actually a monoidal functor was exploited in the unpublished doctoral thesis here at Buffalo by X-Q Meng a few years ago in order to clarify statistical decision procedures and stochastic processes as diagrams in a basic convexity category. She can be reached at : meng@lmc.edu Best wishes to those interested in pursuing this topic! Bill Lawvere ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Tue, 20 Oct 1998, jean-pierre-C. wrote: > > Bonjour. I am a statistician and I should be interested in a categorical > framework for probability and statistical theory. Does anyone know > references (books, articles, websites...) about applications of categories > and functors to probability or even measure theory ? Thank you. > > Very truly yours, > > Jean-Pierre Cotton. > > > From cat-dist Sun Oct 25 13:32:29 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA27599 for categories-list; Sun, 25 Oct 1998 12:17:17 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: categories: Please insert a d To: categories@mta.ca Date: Sun, 25 Oct 1998 14:58:52 +0000 (GMT) X-Mailer: ELM [version 2.4 PL25] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: "Dr. P.T. Johnstone" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Dear All, I have been unofficially informed that "pmms", which has hitherto been an acceptable alias for "dpmms" in the domain dpmms.cam.ac.uk, is liable to stop working at some point in the near future. So, if you have my e-mail address recorded as , or anything else containing the string "@pmms", please insert a "d" in it. Peter Johnstone P.S. -- The same applies to Martin Hyland, and to anyone else you may know in this Department. From cat-dist Tue Oct 27 20:11:00 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id TAA08322 for categories-list; Tue, 27 Oct 1998 19:01:15 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199810271453.KAA30624@mailserv.mta.ca> From: boerger Organization: FernUniversitaet - Gesamthochschule To: categories@mta.ca Date: Tue, 27 Oct 1998 15:52:52 +0100 MIME-Version: 1.0 Content-type: text/plain; charset=ISO-8859-1 Subject: categories: Re: category theory and probability theory In-reply-to: X-mailer: Pegasus Mail for Win32 (v2.54DE) Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from Quoted-printable to 8bit by mailserv.mta.ca id KAA31170 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: I like to add some references concerning applications of category theory to measure theory: - Fred Linton´s thesis and his paper "Functorial measure theory" in the Irvine Proceedings, Thompson , Washington D.C., !966. - Various papers by Mike Wendt, in particular his thesis on direct integrals of Hilbert spaces - My still unpublished long paper "Vector integration by universal properties" and its simplified version in the Bremen Proceedings, Heldermann, Berlin, 1991. Kind regards Reinhard From cat-dist Wed Oct 28 22:09:04 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id VAA02252 for categories-list; Wed, 28 Oct 1998 21:18:37 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <199810290049.NAA19472@ftp.paradigm.co.nz> Comments: Authenticated sender is From: "Ralph Loader" Organization: Paradigm Technology Limited To: Barry Jay Date: Thu, 29 Oct 1998 13:46:02 +0000 MIME-Version: 1.0 Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Subject: categories: Re: Update on FISh and shape Reply-to: ralph.loader@paradigm.co.nz CC: categories@mta.ca In-reply-to: <199810280822.TAA16653@algae.socs.uts.EDU.AU> X-mailer: Pegasus Mail for Win32 (v2.54) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: > > Polymorphic quicksort in FISh is faster than in C for quicksort (see below). Some comments. You're simply not comparing like-with-like. The differences in performance that you're seeing have nothing to do with the boxing/ unboxing issues that you claim are involved. (1) Your C code compares using two floating point operations (subtraction, compare), while the fish program uses just one (no subtraction). (2) The output of the fish-to-C translation uses statically allocated memory, and won't multi-thread correctly. Sun Solaris does support multi-threading, so while your compiler uses statically allocated memory here, the Solaris C library can't. [As a matter of interest, what happens if you call your fish quicksort with a comparison function that itself calls quicksort?]. (3) The C library qsort takes a three-valued comparison function, while your fish code uses a two-valued comparison. > Shape analysis in FISh allows polymorphic functions to be specialised > to the exact shape of their arguments, so that it is never necessary > to box data structures. This can have a significant impact when the > cost of pointer-chasing is high compared to the operation being > performed. ["never"? Show me an unboxed cyclic data structure.] (4) As you note, your fish code gets optimised by specialising to the datatype and comparison function used. This is the main performance issue, and has nothing to do with boxing/unboxing issues. Note that the C library qsort is a polymorphic function, is not specialised to particular arguments, and still takes an array of unboxed elements. Specialising functions to their arguments is nothing new to FISh - some (but not all) C compilers, for instance, on inlining a function will also specialise it. The reason why the qsort function is not specialised to its arguments is because it's a precompiled library function, not because it's written in C. > The reason is that the C program achieves polymorphism by requiring > the comparison function to act on pointers, rather than values. The > actual comparison function for floats is The reality is that the machine code generated by your fish program ends up dealing with pointers also: on a typical RISC processor, to compare two doubles held in memory, you need to compute the addresses of the doubles, load them into registers and do the comparison. Passing pointers to the comparison function just means that the values are loaded into registers in the function, rather than before the function call. Whether its faster to load the values into registers before the function call or in the function will depend on things like instruction scheduling in the compiler & CPU, & register allocation in the caller & in any case the difference will be tiny. > As well as slowing down the program, these pointers are a source of It's not very often that you want to sort an array of numbers. If you want to sort some records that contain both a sort-key and some extra data, then using pointers is faster. > program clutter, potential errors, and > type violations. > > > > Ralph. (not speaking on behalf of Paradigm) Ralph Loader Paradigm Technology, Level 13, Paxus House, 79 Boulcott Steet, Wellington, New Zealand Phone: +64 4 495 1004 Fax: +64 4 499 7762 From cat-dist Thu Oct 29 10:51:10 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id IAA32417 for categories-list; Thu, 29 Oct 1998 08:27:31 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 29 Oct 1998 11:24:26 +0100 (MET) Message-Id: <199810291024.LAA27480@brics.dk> From: Uffe Henrik Engberg To: categories@mta.ca Subject: categories: Research Positions at BRICS Research Centre and Int. PhD School Mime-Version: 1.0 (generated by tm-edit 7.106) Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: [Please accept our apologies if you receive this more than once] BRICS, Basic Research in Computer Science Universities of Aarhus and Aalborg Research Positions at BRICS Research Centre and International PhD School There are several research positions at BRICS starting next year, 1999. Applications are welcome by researchers in theoretical computer science and related areas, especially, but not exclusively, within the following areas: - Semantics of Computation, - Logic, - Algorithms and Data Structures, - Complexity Theory, - Data Security, - Programming Languages, - Distributed Computing, - Verification. Openings, while likely to start as postdoctoral positions, generally for 1-2 years, have the possibility of extension to longer-term positions. BRICS, Basic Research in Computer Science, is funded by the Danish National Research Foundation and consists of the BRICS Research Centre and the BRICS International PhD School. The Research Centre (Director Glynn Winskel) is a joint venture between the theoretical-computer-science groups at the universities of Aarhus and Aalborg, Denmark. The PhD School (Director Mogens Nielsen) is based in the Computer Science Department at the University of Aarhus. The BRICS Research Centre is based on a commitment to develop theoretical computer science and related mathematics, using a combination of long-term efforts and a number of short-term, intensive programmes, within carefully chosen scientific themes. The BRICS International PhD School offers a full PhD programme in computer science, including a wide range of courses, summer schools etc., and a number of PhD grants for Danish as well as foreign students. Further information on BRICS can be accessed by opening the URL: http://www.brics.dk/ The BRICS WWW entry contains information about activities, courses, PhD grants and researchers as well as access to electronic copies of information material and reports of the BRICS Series. Addresses: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK - 8000 Aarhus C Denmark. Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: BRICS@brics.dk How to apply -------------------------------------- Applications for positions should preferably be sent by e-mail (BRICS@brics.dk), preferably early in January 1999, and include - curriculum vitae, and - a description of your research interests, to include, on its first page, a short say 5-10 line summary of research interests, - two or three names of referees for recommendations with the referees' + regular mail addresses and, if possible, + e-mail addresses, as well as - an URL to your WWW home directory if available. The various parts of the application (application letter, CV, etc.) can be sent by e-mail as e.g. uuencoded PDF, PostScript, clear ASCII text or if it causes trouble, just as an URL in which case we will try to load the files. From cat-dist Thu Oct 29 17:23:32 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA04902 for categories-list; Thu, 29 Oct 1998 16:06:39 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Thu, 29 Oct 1998 14:00:03 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Reference? Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Can someone give me a reference for the fact that if the hom functor on a category factors through commutative monoids then finite products are sums and vice versa. Also conversely. Michael From cat-dist Fri Oct 30 13:10:07 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA28774 for categories-list; Fri, 30 Oct 1998 11:20:40 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Fri, 30 Oct 1998 08:24:52 -0400 To: categories@mta.ca From: Marta Bunge Subject: categories: Paper available Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: This is to announce a new paper, by Marta Bunge and Marcelo Fiore, "Unique factorization Lifting Functors and Categories of Processes". http://www.dcs.ed.ac.uk/~mf/CONCURRENCY/ufl.dvi http://www.dcs.ed.ac.uk/~mf/CONCURRENCY/ufl.ps The paper is organised as follows. After an Introduction, Section 1 presents background material motivated from the point of view of computer science. In Section 2, the category UFL of unique factorisation lifting (ufl) functors is recalled and its basic properties are studied. Section 3 explores applications of ufl functors to concurrency. In particular we show that they may be used in the study of interleaving models like transition systems. In Section 4, we introduce triangulated categories. Our main use for them is in Section 5 where, for C a triangulated category, we exhibit the category UFL/C as a sheaf topos. These toposes may be regarded as models of linearly-controlled processes. Some concluding remarks are provided in Section 6. Professor Marta Bunge McGill University Department of Mathematics & Statistics Burnside Hall 805 Sherbrooke Street West Montreal, QC Canada H3A 2K6 Fax: (514) 933 8741 Phone: (514) 933 6191 From cat-dist Fri Oct 30 13:11:19 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id LAA31103 for categories-list; Fri, 30 Oct 1998 11:24:24 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Subject: Re: categories: Reference? To: categories@mta.ca Date: Fri, 30 Oct 1998 09:26:24 +0000 (GMT) MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: "Dr. P.T. Johnstone" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: > > Can someone give me a reference for the fact that if the hom functor on a > category factors through commutative monoids then finite products are sums > and vice versa. Also conversely. > > Michael Mac Lane (Categories for the Working Mathematician) does one direction in Theorem 2 on page 190. (He assumes enrichment over abelian groups rather than commutative monoids, but a glance at the proof shows that the additive inverses are not used.) The converse is stated as Exercise 4 on page 194. Peter Johnstone From cat-dist Fri Oct 30 14:27:48 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id MAA13509 for categories-list; Fri, 30 Oct 1998 12:59:29 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 30 Oct 1998 11:02:31 -0500 (EST) From: F W Lawvere Reply-To: wlawvere@ACSU.Buffalo.EDU To: Categories list Subject: categories: Re: Reference? Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: Re: Mike Barr's question concerning two equivalent definitions of a class of categories Since the term 'additive' had already been established to refer to the special case where the homs are abelian groups, I called these 'linear categories' in my paper Categories of Space and of Quantity in The Space of Mathematics, Philosophical, Epistemological and Historical Explorations, de Gruyter, Berlin (1992) pp 14-30 because 'linear' is a term well known to engineers, statisticians and others, and because these categories form the natural environment for applications of Linear Algebra. Of course, the entries in the matrices are in general maps, not necessarily scalars, although scalars for which the addition is idempotent are an important special case. (Here by the scalars of such a category I mean the elements of the rig which is its center.) In that paper I referred to what I believe is the first reference to this theory, namely Saunders Mac Lane's 1950 paper Duality for Groups, Bull AMS vol 56, pp 485-516, (1950) expounding work he did in the late 40's. Bill Lawvere ****************************************************************** F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ****************************************************************** On Thu, 29 Oct 1998, Michael Barr wrote: > Can someone give me a reference for the fact that if the hom functor on a > category factors through commutative monoids then finite products are sums > and vice versa. Also conversely. > > Michael > > > From cat-dist Sat Oct 31 10:44:28 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id JAA01959 for categories-list; Sat, 31 Oct 1998 09:58:05 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-Id: <3.0.32.19981030144751.008c6e70@pop.apk.net> X-Sender: cwells@pop.apk.net X-Mailer: Windows Eudora Pro Version 3.0 (32) Date: Fri, 30 Oct 1998 14:47:56 -0500 To: categories@mta.ca From: Charles Wells Subject: categories: Unicode for math symbols (forwarded) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: We have a unique chance over the next few weeks to influence various international standards bodies in presenting a complete set of mathematical and other scientific characters for inclusion in standard character sets. Much of the work for this has already been done, but there remain some problem characters on which we really need the help of the author community to describe uses and distinctions from other similar characters. Further details are available from the math society web site at: http://www.ams.org/STIX/glyphs/proposal/newsub/challenge.html The AMS has set a November 20 deadline for receipt of this information, so there is not much time to act. Thanks for your help! Arthur Smith (apsmith@aps.org) Research & Development, The American Physical Society (516)591-4072 Charles Wells, Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106-7058, USA. EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893. FAX: 216 368 5163. HOME PHONE: 440 774 1926. HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html From cat-dist Sat Oct 31 10:44:47 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id JAA05418 for categories-list; Sat, 31 Oct 1998 09:58:44 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Fri, 30 Oct 1998 15:50:46 -0500 (EST) From: Michael Barr Reply-To: Michael Barr To: Categories list Subject: categories: Linear categories Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: I have checked Mac Lane's 1950 paper and I cannot find any such result. The converse, that a category in which finite sums are equivalent to products then the category takes homs in commutative monoids is sort of there, but the one I asked is not, or at least I didn't find it. However, CWM is a likely source and I will check that out. I just need some reference in any case. Michael From cat-dist Sat Oct 31 16:40:44 1998 Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA27328 for categories-list; Sat, 31 Oct 1998 15:40:17 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Date: Sat, 31 Oct 1998 14:19:05 -0500 (EST) From: F W Lawvere Reply-To: wlawvere@ACSU.Buffalo.EDU To: Categories list Subject: categories: Re: Linear categories Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: Dear Mike and everbody The converse result , as stated by Mac Lane, was what needed to be said in 1950, especially since it began to bring out that category theory has content. What it's converse to, namely that when maps can be added then the cartesian product has the mapping property now called coproduct, had already been folklore for years... or if it hadn't been, how else to explain the widespread terminological ambiguity concerning "direct sums"? Perhaps a much older reference needs to be adduced. Thanks for bringing this up. Best regards Bill ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Fri, 30 Oct 1998, Michael Barr wrote: > I have checked Mac Lane's 1950 paper and I cannot find any such result. > The converse, that a category in which finite sums are equivalent to > products then the category takes homs in commutative monoids is sort of > there, but the one I asked is not, or at least I didn't find it. However, > CWM is a likely source and I will check that out. I just need some > reference in any case. > > Michael > > >