August 2002 - Categories - categories.org.au

Chu(Set,K) without sets
by Vaughan Pratt 31 Aug '02

31 Aug '02

TITLE: Chu(Set,K) without sets ABSTRACT: We define the case V=Set of Chu(V,K) without reference to Set. Somewhat analogously to toposes and abelian categories as 1-categorical abstractions of Set and Ab, we propose rigid couple categories, RCOUP, as a uniform 2-categorical abstraction of the Chu_K categories. Theorem 1 obtains the Chu_K's as the 2-final couple categories of the connected components of RCOUP. Theorem 2 obtains RCOUP as closure of the discrete 2-category of Chu_k's (k the canonical K of that cardinality) under equivalences, subobjects, and 2-cell duplication. One-page extended abstract at http://boole.stanford.edu/pub/couple.pdf The idea for this struck me while preparing my slides for CONCUR last week (any excuse for a break) and so is still in a very preliminary and fluid state. Comments on the perspective welcome. In particular what would you see as being the more natural definition of (ordinary) Chu space: sets A,X and a function r:AxX->K, or an object of a 2-final rigid couple category? Is the latter "what's really going on," or is it just a view of reality distorted by a horribly thick lens, category theory misapplied? Vaughan Pratt

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Bunched Logic (BI) Monograph
by David J. Pym 30 Aug '02

30 Aug '02

Readers may be interested to know that the monograph, "The Semantics and Proof Theory of the Logic of Bunched Implications'', by David J. Pym, published by Kluwer Academic Publishers in the Applied Logic Series, is now available: see http://www.wkap.nl/prod/s/APLS and follow the link or see http://www.wkap.nl/prod/b/1-4020-0745-0. Table of Contents. List of Figures. List of Tables. Preface. Acknowledgments. Foreword; Dov M. Gabbay. Introduction; David J. Pym. Part I: Propositional BI. 1. Introduction to Part I. 2. Natural Deduction for Propositional BI. 3. Algebraic, Topological, Categorical. 4. Kripke Semantics. 5. Topological Kripke Semantics. 6. Propositional BI as a Sequent Calculus. 7. Towards Classical Propositional BI. 8. Bunched Logical Relations. 9. The Sharing Interpretation, I. Part II: Predicate BI. 10. Introduction to Part II. 11. The Syntax of Predicate BI. 12. Natural Deduction & Sequent Calculus For Predicate BI. 13. Kripke Semantics for Predicate BI. 14. Topological Kripke Semantics for Predicate BI. 15. Resource Semantics, Type Theory & Fibred Categories. 16. The Sharing Interpretation, II. Bibliography. Index. -- Prof. David J. Pym Telephone: +44 (0)1 225 38 3246 Professor of Logic & Computation Facsimile: +44 (0)1 225 38 3493 University of Bath Email: d.j.pym(a)bath.ac.uk Bath BA2 7AY, England, U.K. Web: http://www.bath.ac.uk/~cssdjp

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SGL changes, identification of
by Keith Harbaugh 28 Aug '02

28 Aug '02

The message of Robert McGrail on 2002-08-23 raises a question: How did the second (1994) printing of Mac Lane and Moerdijk's "Sheaves in Geometry and Logic" (SGL) differ from the first (1992)? The difference was evidently nontrivial, as its review (96c:03119) in MR is specifically of the "Corrected reprint of the 1992 edition.", and the review was evidently delayed until such a corrected reprint could appear. I believe the question is worth asking, as many individuals and libraries have the first printing, and may not have the financial wherewithal to purchase another copy unless it is really vital. (My efforts to locate an errata on the web at all the likely (to me) places have been unavailing.) A (less significant) follow-up question: Did the recently announced third printing contain any further changes? In case it matters, here is the relevant part from the cited email: >You will find this in MacLane and Moerdijk, Sheaves in Geometry and Logic. >Check the index for "heyting algebras: of subobjects". I have the first >printing of the first edition in which this result is Proposition 1 of the >8th section of chapter III. Best wishes, Keith Harbaugh _________________________________________________________________ Join the world=92s largest e-mail service with MSN Hotmail. http://www.hotmail.com

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Preprint: Galois theory and a homotopy double groupoid
by Ronnie Brown 28 Aug '02

28 Aug '02

A preprint is available: R. Brown, G.Janelidze Galois theory and a new homotopy double groupoid of a map of spaces ABSTRACT: The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat^1-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle. Available from arXiv math.AT/0208211 or pdf file at http://www.bangor.ac.uk/~mas010/dgpsmap5.pdf Ronnie Brown School of Informatics, Mathematics Division, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 http://www.bangor.ac.uk/~mas010/ 28-Aug-2002 13:27:55 -0300,2098;000000000000-00000000

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[ANN] MBase 0.8 Released (Mathematical Knowledge Base)
by Michael Kohlhase 26 Aug '02

26 Aug '02

[Apologies for multiple copies] We are pleased to announce the first official release of The MBase Mathematical Knowledge Base MBase 0.8 is available from the MBase web page: http://www.mathweb.org/mbase/ The MBase system is distributed under the GNU General Public License, binary distributions are provided for Linux and Solaris. An online demo is can be found at http://mbase.mathweb.org:8080/ Version 0.8 is the first official release of the system. It should be considered as alpha quality code. ABOUT MBASE The MBase system is an ongoing development of a web-based knowledge base of formalized mathematics (collections of definitions, theorems, or proofs). The system specializes an SQL-based database system for mathematical structures in the document format OMDoc [1], which is an XML language. Primary applications of MBase are deduction systems, mathematical assistants, content presentation systems, and learning environments, which can interact with MBase via the MathWeb system, an implementation of a mathematical software bus for distributed theorem proving and symbolic computation [2]. MAJOR FEATURES -- import & export OMDoc content [1] -- object persistence by standard SQL database system back-end -- pattern-searching (e.g.... search for statements of commutativity) -- web browser interface -- xml-rpc interface for software clients -- multi-platform support (Linux, Solaris, MacOS X) PLANNED FEATURES (contact us if you want to help) -- distributed cross-referencing and caching -- content versioning management Comments, suggestions or bug reports are welcomed and should be directed to the MBase development team <mbase(a)mathweb.org>. Have fun, The MBase Team (Andreas Franke, Michael Kohlhase, Markus Moschner) --- [1] http://www.mathweb.org/omdoc [2] http://www.mathweb.org/mathweb 25-Aug-2002 12:43:46 -0300,6013;000000000000-00000000

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Categorical Monodromy?
by ARRINDELLZ＠aol.com 24 Aug '02

24 Aug '02

Hi,Prof.(I having trouble posting msg on the net) so maybe you can help, a Categorist whan-na-be. I read somewhere on the web that "Monodromy is a general concept in category theory involving the globalization of local morphism." I cannot find any work in Categorical Monodromy. Do you know of any Categorical Monodromy paper I can read? Thanks 23-Aug-2002 16:18:15 -0300,1357;000000000001-00000000

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Pro(Set) (and more generally Pro(C)
by Bill Rowan 24 Aug '02

24 Aug '02

Hello, I have been studying a construction that turns out to be equivalent to the full subcategory of Pro(Set) consisting of diagrams such that (1) The diagram category is a directed set (2) All the maps in the diagram are one-one. This turns out to have some uses that I am quite interested in, and I am preparing a writeup of the category, which I call Grm (the category of sets with subdomain system, and germs of partial functions). I would like to know if anyone has studied this before so that I can avoid reinventing the wheel any more than I already have. Bill Rowan 28-Aug-2002 13:27:55 -0300,1637;000000000000-00000000

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Correction
by Walter Tholen 24 Aug '02

24 Aug '02

Steve Lack's sharp eyes detected a kind of subtle error in Theorem 4.2 of my recently announced paper with A. Pultr on "Free Quillen Factorization Systems". I have posted a corrected version of the paper on my homepage http://www.math.yorku.ca/Who/Faculty/Tholen/research.html. Regards, Walter Tholen. 25-Aug-2002 12:43:46 -0300,2541;000000000000-00000000

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Topos and Heyting algebra question
by Galchin Vasili 24 Aug '02

24 Aug '02

Dear CT Community, I am new to Topos theory. Is it true that Hom[1->omega] in a topos defines a Heyting algebra? If so, please give me a reference where this is proved. Thanks and regards, Bill Halchin 22-Aug-2002 19:50:49 -0300,874;000000000000-00000000

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ICIAM Sydney 7-11 July 2002
by Ross Street 22 Aug '02

22 Aug '02

The 6th Australian-New Zealand Mathematics Convention is to be held as an "embedded meeting" within the International Congress on Industrial and Applied Mathematics to be held in Sydney from 7 to 11 July 2003. See the website <http://www.iciam.org/> operated by Macquarie University's Ross Moore. I have been asked to organize a minisymposium. Would anyone interested in giving a 25 minute talk at the minisymposium please contact me quickly. I need to fix on a theme (something like "categories, operads and homotopy theory") and make a submission by 30 August 2002. ---- Ross <street(a)math.mq.edu.au> 23-Aug-2002 16:27:06 -0300,1525;000000000000-00000000

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