41. Re: SUO: Re: IFF Comments Requested Are you meaning to suggest that topos theory is likely to provide a goodfoundation for a psychological theory of performance of novices? http://suo.ieee.org/email/msg06674.html

Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index Re: SUO: Re: IFF Comments Requested To josiane.caron@mshs.univ-poitiers.fr Subject : Re: SUO: Re: IFF Comments Requested From phayes@ai.uwf.edu Date : Mon, 8 Oct 2001 13:40:09 -0500 Cc standard-upper-ontology@ieee.org In-Reply-To 4.2.0.58.20011006103931.009f48a0@pop.mshs.univ-poitiers.fr References 4.2.0.58.20011006103931.009f48a0@pop.mshs.univ-poitiers.fr Reply-To phayes@ai.uwf.edu Sender owner-standard-upper-ontology@majordomo.ieee.org http://www.coginst.uwf.edu/~phayes References Re: SUO: Re: IFF Comments Requested From: Prev by Date: Re: SUO: Re: IFF Comments Requested Next by Date: SUO: Re: IFF Comments Requested Prev by thread: Re: SUO: Re: IFF Comments Requested Next by thread: Re: SUO: Re: IFF Comments Requested Index(es): Date Thread

42. Fwd: Re: SUO: Re: IFF Comments Requested You may be right, but (after trying to teach topos theory to undergraduates) Ifind it so wildly unlikely that I would like to see some case made for it. http://suo.ieee.org/email/msg06687.html

Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index Fwd: Re: SUO: Re: IFF Comments Requested To standard-upper-ontology@ieee.org Subject : Fwd: Re: SUO: Re: IFF Comments Requested From josiane.caron@mshs.univ-poitiers.fr Date : Tue, 09 Oct 2001 02:16:48 +0200 Reply-To josiane.caron@mshs.univ-poitiers.fr Sender owner-standard-upper-ontology@majordomo.ieee.org Prev by Date: Re: SUO: Re: IFF Comments Requested Next by Date: [Fwd: [Fwd: SUO: John Sowa's motion]] Prev by thread: SUO: RE: IFF Comments Requested Next by thread: SUO: Re: IFF Comments Requested Index(es): Date Thread

43. SUO: Re: IFF Comments Requested however. These are distinctions in the foundations of topos theory,not the usual fodder of the working mathematician. PH Most http://grouper.ieee.org/groups/suo/email/msg06683.html

Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index SUO: Re: IFF Comments Requested To jawbrey@oakland.edu Subject : SUO: Re: IFF Comments Requested From phayes@ai.uwf.edu Date : Mon, 8 Oct 2001 15:36:21 -0500 Cc standard-upper-ontology@ieee.org In-Reply-To 3BC20300.3F9ADAF0@oakland.edu References 3BBCF80F.15BC2FE2@oakland.edu 3BC20300.3F9ADAF0@oakland.edu Reply-To phayes@ai.uwf.edu Sender owner-standard-upper-ontology@majordomo.ieee.org http://www.coginst.uwf.edu/~phayes Follow-Ups SUO: Re: IFF Comments Requested From: References SUO: Re: IFF Comments Requested From: SUO: Re: IFF Comments Requested From: SUO: Re: IFF Comments Requested From: SUO: Re: IFF Comments Requested From: Prev by Date: Re: [Fwd: [Fwd: SUO: John Sowa's motion]] Next by Date: SUO: Modularity (and conformance) Prev by thread: SUO: Re: IFF Comments Requested Next by thread: SUO: Re: IFF Comments Requested Index(es): Date Thread

44. Siberian Toposes Below we gave the papers in which applications of topos theory to themodern Theoretical Physics are discussed discussed. AXIOMATIC http://users.univer.omsk.su/~topoi/appl_e.html

Below we gave the papers in which applications of topos theory to the modern Theoretical Physics are discussed discussed. AXIOMATIC THEORY OF RELATIVITY Guts, A.K. A topos-theoretic approach to the foundations of Relativity theory Soviet Math. Dokl.- 1991.-V.43, No.3.-P.904-907. GENERAL THEORY OF RELATIVITY and THEORY OF GRAVITY Trifonov, V. Linear Solution of the Four-Dimensionality Problem Europhys. Lett. 1995. V.32, N.8, P.621-626. Grinkevich, E.B. Synthetic Differential Geometry: A Way to Intuitionistic Models of General Relativity in Toposes Preprint gr-qc/9608013 Guts, A.K., Grinkevich, E.B. Toposes in General Theory of Relativity. Preprint gr-qc/9610073 Guts, A.K. Intuitionistic Theory of Space-time (russian) // Zvyagintsev, A.A. Pseudo-Euclidean space in model of Synthetic Differential Geometry (russian) // Mathematical Structures and Modeling. 1998. No. 2. P.34-38. Guts, A.K., Zvyagintsev, A.A. Solution of the vacuum Einstein equations in Synthetic Differential Geometry of Kock-Lawvere . - Paper: physics/9909016. Guts, A.K., Zvyagintsev, A.A.

45. Prof. P.T. Johnstone Research Interests I have been involved in the development of elemntary topos theorysince its infancy in the early 1970s, and have written two books on the http://www.dpmms.cam.ac.uk/site2002/People/johnstone_pt.html

Department of Pure Mathematics and Mathematical Statistics DPMMS People Prof. P.T. Johnstone Prof. P.T. Johnstone Title: Professor of the Foundations of Mathematics Email: P.T.Johnstone@dpmms.cam.ac.uk College: St John's College Room: C1.07 Tel: +44 1223 337985 Research Interests: I have been involved in the development of elemntary topos theory since its infancy in the early 1970s, and have written two books on the subject: "Topos Theory" (Academic Press, 1977) and "Sketches of an Elephant: a Topos Theory Compendium" (Oxford U.P., 2002). My interests focus particularly on the way in which topos theory provides a means of integrating geometric and logical ideas in the foundations of mathematics and of theoretical computer science. Information provided by webmaster@dpmms.cam.ac.uk

46. Dpmms: Johnstone I am interested in categorytheoretic aspects of the foundationsof mathematics, particularly those which may be modelled using topos theory....... Long http://www.dpmms.cam.ac.uk/site2000/Staff/johnstone01.html

STAFF LIST: PURE MATHEMATICS Peter T. Johnstone Title: Reader in the Foundations of Mathematics Email: p.t.johnstone@dpmms.cam.ac.uk Telephone (direct line): (+44) (0)1223.337985 Fax: (+44) (0)1223.337920 College: St. John's College Personal Home Page Courses Given, 2000-01 Graduate Students Research Interests: Keywords : Category, topos, locale, categorical logic, semantics of computation. Long Description I am interested in category-theoretic aspects of the foundations of mathematics, particularly those which may be modelled using topos theory. I am the author of Topos Theory (Academic Press, 1977), which was the first book-length account of elementary topos theory, and which is still widely quoted as a standard reference for the subject. My interest in toposes has led me to study the 'pointless' locale-theoretic approach to topology, as described in my book Stone Spaces (Cambridge U.P., 1983). I am also interested in the application of categorical, and particularly topos-theoretic, methods to problems in the foundations of theoretical computer science. Return to Departmental Staff List Email: office@dpmms.cam.ac.uk

47. OUP USA: Elementary Categories, Elementary Toposes Elementary Toposes COLIN MCLARTY, Case Western Reserve University, ClevelandThe book covers elementary aspects of category theory and topos theory. http://www.oup-usa.org/isbn/0198514735.html

Mathematics or Browse by Subject paper In Stock Standard Oxford Logic Guides 21 Table of Contents Elementary Categories, Elementary Toposes COLIN MCLARTY, Case Western Reserve University, Cleveland The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout, rather than beginning with set theoretical foundations. It works with key concepts such as Cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis. The intended readership consists of graduate-level students in mathematics, computer science, logic, and category theory. "An exceptionally clearly written and wide-ranging introduction to category and topos theory. . . . packed with things interesting to the expert, yet presented in a manner intelligible to the beginner. . . . contains a wealth of thought-provoking exercises." Journal of Symbolic Logic "Provides a first introduction to the theory of categories and functors, and to topos theory. McLarty manages to cover a considerable range of topics in a clear and elegant fashion, thus giving these readers a good first impression of what the subject is about. McLarty's perspicuous treatment provides a good first introduction."

48. OUP USA: ToC: Sketches Of An Elephant Sketches of an Elephant A topos theory Compendium 2 Volume Set PeterT. Johnstone CONTENTS. VOLUME ONE A. Toposes as Categories A1. http://www.oup-usa.org/toc/tc_019852496X.html

Sketches of an Elephant A Topos Theory Compendium 2 Volume Set Peter T. Johnstone CONTENTS VOLUME ONE A. Toposes as Categories A1. Regular and cartesian closed categories A2. Toposesbasic theory A3. Allegories A4. Geometric morphismsbasic theory B. 2-Categorical Aspects of Topos Theory B1. Indeexed categories and fibrations B2. Internal and locally internal categories B3. Toposes over a base B4. BTop/S as a 2-Category Bibliography Index of notation General index VOLUME TWO C. Toposes as Spaces C1. Sheaves on a locale C2. Sheaves on a site C3. Classes of geometric morphisms C4. Local compactness and exponentiability C5. Toposes as groupoids D. TOPOSES AS THEORIES D1. First-order categorical logic D2. Sketches D3. Classifying toposes D4. Higher-order logic D5. Aspects of finiteness Bibliography Index of notation General index General Catalog Information Publication dates and prices are subject to change without notice. Prices are stated in US Dollars and valid only for sales transacted through the US website. Please note: some publications for sale at this website may not be available for purchase outside of the US.

49. Music And Mathematics From Topos Theory To Music Software Music and Mathematics From topos theory To Music Software . Prof.Guerino Mazzola. Universität Zürich ­ Duebendorf Institut für http://www.di.fc.ul.pt/~llf/etaps98/mazzola.html

"Music and Mathematics: From Topos Theory To Music Software" Prof. Guerino Mazzola We give an overview of mathematical music theory and first describe and exemplify topoi of denotators. These are special set-valued functors on the category of modules; they realize a general concept of music objects which meets requirements of mathematics and of data base management systems. We then outline the theory of transformation processes of symbolic music objects, as they are codified in traditional scores, into objects of physical reality, as they appear in artistic performance. Such processes rely on performance vector fields and involve mathematical music analyses of the given symbolic data, as well as performance grammars to shape performance fields as an expression of analytical facts. We discuss the theory, known results, and related problems. We conclude with an outlook on future research in mathematical music theory.

50. Excerpts From Cat-dist: Re: Subtypes become subtypes. I have been aware of this (standard) constructionthat can be found in most textbooks on topos theory. One does http://www.disi.unige.it/aila/Notizie/da-cat-list/0058.html

Re: Subtypes categories ( cat-dist@mta.ca Thu, 4 Apr 1996 14:14:52 -0400 (AST) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: categories: "Re:subtypes etc." Previous message: categories: "Subtypes" Maybe in reply to: categories: "Subtypes" Date: Thu, 04 Apr 1996 20:08:28 MESZ I definitely appreciate the following answer to my mail by Jim Lambek though I am afraid I wanted to make emphasise another aspect I have been aware of this (standard) construction that can be found in most textbooks on topos theory. One does not only have to extend the language by the subset types but ALSO by constants for a lot of new morphism (where the domain or the codomain is a new type). The problem is not a mathematical one but a question of notational economy : provided I use the extension of the language formally. For a precise formulation of the syntax of a language with subtypes see e.g. top of page 134 W. Phoa An Introduction to Fibrations, Topos Theory, the Effective ToposW. Phoa An Introduction to Fibrations, Topos Theory, the Effective Topos and Modest Sets (Univ. Edinburgh LFCS Report, obtainable by ftp from the

51. Philosophia Mathematica Abstracts: Accepted Papers Category theory and topos theory have been seen (by Mac Lane, Bell, Awodey, et al.)as providing a structuralist framework for mathematics autonomous vis à http://www.umanitoba.ca/publications/philosophia_mathematica/abstracts.html

ABSTRACTS OF PAPERS ACCEPTED TABLE OF CONTENTS Jeremy Avigad, Number theory and elementary arithmetic Helen Billinge, Did Bishop Have a Philosophy of Mathematics? Roy T. Cook, Aristotelian Logic, Axioms, and Abstraction Elizabeth F. Cooke, Peirce, Fallibilism, and the Science of Mathematics E. B. Davies, Empiricism in arithmetic and analysis Geoffrey Hellman, Does Category Theory Provide a Framework for Mathematical Structuralism? Sarah Hoffman, Kitcher, Ideal Agents and Fictionalism David Liggins, On Being Twice as Heavy Graham Priest, Meinongianism and the Philosophy of Mathematics Success by Default? Greg Restall, Just What Is Full-Blooded Platonism? Peter M. Schuster, Countable Choice as a Questionable Uniformity Principle William Stirton, Caesar Invictus Gabriel Uzquiano, Plural Quantification and Classes Papers in the completed volume are listed under Abstracts of papers in Vol. 10 (2002). Updated 2003 3 7 Back to the Philosophia Mathematica main page. JEREMY AVIGAD, avigad@cmu.edu Number theory and elementary arithmetic ABSTRACT. Elementary arithmetic (also known as ``elementary function arithmetic'') is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.

52. Theory And Semantics Group Centred around mathematical models of a variety of languages and logics, using techniques such as Category Computers Computer Science Theoretical Research Groups...... semantics of the lambda calculus, general recursion theory, sheaf models for intuitionistictheories, general categorical logic and topos theory, the effective http://www.cl.cam.ac.uk/Research/TSG/

Theory and Semantics Group University of Cambridge Computer Laboratory The work of the Theory and Semantics Group is centred around mathematical models of a variety of languages and logics. These models are intended to be used as a basis for specification and verification, and as a tool for clarifying programming concepts. We use techniques such as structural operational semantics, linear logic, domain theory and category theory. Work is in progress on the underlying mathematical structures of these, and on their application to the study of higher order typed programming languages such as Standard ML, to object-based languages, to foundational languages for concurrent, distributed and mobile computation, to hardware description languages, and to security problems. Work is also being undertaken on the analysis of programming languages in the setting of abstract interpretation and on practical optimising compilation for imperative and functional languages. Related research is undertaken within the Automated Reasoning Group . We also have links with the Logic Seminar at DPMMS (Dept of Pure Mathematics and Mathematical Statistics).

53. Toward A New Understanding Of Space, Time And Matter: Workshops Chris Isham. Thursday, June 17, 330 pm 530 pm, Category theory,topos theory, and topological quantum field theory, Lou Kauffman. http://axion.physics.ubc.ca/Workshop/Workshops.html

Toward a New Understanding of Space, Time, and Matter Workshop Schedule June 16-19, 1999 All workshops take place in the large Peter Wall Institute conference room on the third floor of the University Centre Wednesday, June 16 7:30 pm - 9:30 pm Post-talk discussion Steve Weinstein Thursday, June 17 9:30 am - 11:30 am What is a quantum theory? Chris Isham Thursday, June 17 3:30 pm - 5:30 pm Category theory, topos theory, and topological quantum field theory Lou Kauffman Thursday, June 17 7:30 pm - 9:30 pm Spacetime in string/M-theory Tamiaki Yoneya Friday, June 18 9:30 am - 11:30 am Time and space in physical theory and experience Larry Sklar Friday, June 18 3:30 pm - 5:30 pm Field-theoretic issues in quantum gravity Lee Smolin Saturday, June 19 9:30 am - 11:30 am Quantum gravity: physics, metaphysics, or mathematics? Simon Saunders Saturday, June 19

54. FOM: Orthogonal Roles Mathias wrote 2. Start from Wilson's remark that W4 topos theory can providean ontology that is just as abundant as the one set theory provides. http://www.cs.nyu.edu/pipermail/fom/2000-April/003943.html

FOM: Orthogonal roles Todd Wilson twilson@csufresno.edu Fri, 7 Apr 2000 18:04:56 -0700 (PDT) Previous message: FOM: British 2000 abstract Next message: FOM: Re: Simpson on Russell's paradox for category theory Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] As promised earlier, here is my response to the thoughtful posting of A.R.D.Mathias concerning my posting of 27 Feb 2000, entitled "A dual view of foundations". In short, I find myself in agreement with most of Mathias's points. On Fri, 3 Mar 2000, Andrian-Richard-David Mathias wrote: > 2. Start from Wilson's remark that W4 topos theory can provide an ontology that is just as abundant as the one set theory provides. To a set-theorist that is plainly untrue: taking topos theory to be a decorated version of Mac Lane set theory, it cannot, for example, prove the existence of an infinite set of infinite cardinals. [Mac Lane, be it noted, does not see the issue between topos theory and set theory as an ontological competition.] 3. But let us turn Wilson's remark around and take it to mean that

55. FOM: Categorical Cardinals (in Two Senses) I asked whether the alternative foundational pictures embodied by NFand topos theory have any analogous ``categoricity properties''. http://www.cs.nyu.edu/pipermail/fom/2000-April/003947.html

FOM: categorical cardinals (in two senses) Stephen G Simpson simpson@math.psu.edu Wed, 12 Apr 2000 21:52:25 -0400 (EDT) Previous message: FOM: completeness theorem for stratification? Next message: FOM: Re: categorical cardinals (in two senses) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: completeness theorem for stratification? Next message: FOM: Re: categorical cardinals (in two senses) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

56. Category Theory topos theory. S. Mac Lane and I. Moerdijk, M. Barr and C. Wells, Toposes,Triples and Theories; P. Johnstone, topos theory; P. Johnstone, Stone Space. http://www.kyoto-su.ac.jp/~hxm/categorical/ct/

$B7wO@!J(BCategory Theory$B!K(B $B7wO@$K4X$9$k%a%b!%$^$@:n$j$+$1$G$9!%(B Books Basic and Advanced Texts S. Mac Lane, Categories for the Working Mathematician. F. Borceux, Handbook of Categorical Algebra 1-3. G. M. Kelly, Basic Concepts of Enriched Category Theory. Topos Theory S. Mac Lane and I. Moerdijk, ... M. Barr and C. Wells, Toposes, Triples and Theories P. Johnstone, Topos Theory P. Johnstone, Stone Space Categorical Logic and Type Theory J. Lambek and P. J. Scott, Introduction to Higher Order Categorical Logic. B. Jacobs, Categorical Type Theory. A. Torelstra, Lectures on Linear Logic M. Makkai and G. Reyes, ... Categorical Model Theory and Varieties J. Adamek and J. Rosicky, ... LNM, Computer Science J. Mitchell, Semantics of Programming Language C. Gunter, Semantics of Programming Language G. Winskel, Semantics of Programming Language J. Reynolds, Semantics of Programming Language M. Barr and C. Wells, Category Theory for Computing Science Synthetic Differential Geometory Models of Infinitesimal Analysis Basic Concepts of Synthetic Differential Geometory J. L. Bell, ...

57. Philosophy Of Physics In recent years, I have mostly worked on philosophical aspects of quantum gravityand on applying topos theory to quantum theory (especially the KochenSpecker http://users.ox.ac.uk/~ppox/research/fields.html

Home Research Teaching Local Events ... Coming to Oxford Research Fields of Research Research Seminar Discussion Group Workshop ... Archives and Other Fields of Research in Philosophy of Physics Harvey Brown My main areas of research are foundations of quantum mechanics and space-time structure, including the role and meaning of symmetries in physics. Most of my present work is related to a projected book on the philosophy of relativity theory-mostly special relativity. I defend the view (advocated by Lorentz, Pauli and Bell, inter alia) that relativistic kinematics are essentially dynamical in origin, arising out of the treatment of rods and clocks as 'structured' objects and thus involving the Lorentz covariance of all the fundamental non-gravitational interactions. I am also interested in the attempt to formulate gravitational and non-gravitational dynamics relationally, in the sense of the program of Julian Barbour. My work on symmetries engages with current disputes concerning the appropriate formulation of spacetime and internal symmetries. It also involves a collaborative effort with Katherine Brading on the history and philosophy of Noether's theorems. My most recent work on the foundations of quantum mechanics deals with the so-called topological approach to identical particles, in which the characteristic phase factor determining Fermionic, Bosonic or intermediate behaviour is seen as a configuration space analogue of the Aharonov-Bohm effect. Jeremy Butterfield My main research interests have long been philosophy of quantum theory and relativity. In recent years, I have mostly worked on philosophical aspects of quantum gravity and on applying topos theory to quantum theory (especially the Kochen-Specker 'no go' theorem). But I expect in coming years to work mainly on; a) philosophical aspects of quantum field theory. I am interested especially in the following topics: nonlocality, localization and the emergence of particles, renormalization, and the gauge structures of quantum field theories. b) philosophical aspects of quantum information.

58. Home_page_workshop There were three short courses on Set Theory (Philip Welch), topos theory(Ieke Moerdijk) and Constructive Type Theory (Giovanni Sambin). http://www.science.unitn.it/~baratell/ftm2.html

Mini-Workshop on Foundational Theories in Mathematics Department of Mathematics University of Trento September 1214, 2002 We invite you to participate in the mini-workshop on Foundational Theories in Mathematics that will be held in Trento, Italy, September 1214, 2002 The workshop will take place at the Department of Mathematics of the University of Trento. In September 2001 a workshop on "Foundational Theories in Mathematics" took place at the Department of Mathematics of the University of Trento. There were three short courses on Set Theory (Philip Welch), Topos Theory (Ieke Moerdijk) and Constructive Type Theory (Giovanni Sambin). During that workshop our sensation was confirmed that an essential part was missing: the one of tying the different theoretical threads of Set Theory, Topos Theory and Constructive Type Theory even more together. For this reason we have invited John Bell to lecture on a Synthesis of the three different foundational theories. We have also invited three leading specialists: Keith Devlin (Set Theory - to be confirmed), Giuseppe Rosolini (Topos Theory) and Jan Smith (Constructive Type Theory) to participate with John Bell in two panel discussions and to share with the participants their views on the foundations of Mathematics. This second workshop on "Foundational Theories in Mathematics" does not presuppose participation in the first one; it can be attended quite independently of the latter.

59. PUBLICATIONS 197 (1974) 355390. Marta Bunge, topos theory and Souslin's Hypothesis. J.Pure Applied Algebra 4 (1974) 159-187. Marta Bunge, Internal Presheaves Toposes. http://www.math.mcgill.ca/bunge/pub.html

PUBLICATIONS Marta Bunge, Categories of Set-valued Functors, . Ph.D. thesis, University of Pennsylvania, 1966. Marta Bunge, Relative Functor Categories and Categories of Algebras J.of Algebra 11 Marta Bunge, Relative Functor Categories and Categories of Algebras . Russian translation in : Mathematics: Periodical collections of Translations of Foreign Articles, Vol.16, Izdat. "Mir" , Moscow(1972) 11-46, MR 50, #12532 (Editors). Marta Bunge, Bifibration-Induced Adjoint Pairs . in: J. Gray (editor), Reports of the Midwest Category Seminar V - Zurich 1970. Lecture Notes in Mathematics 195 Marta Bunge, Coherent Extensions and Relational Algebras Trans. Amer.Math.Soc. 197 Marta Bunge, Topos Theory and Souslin's Hypothesis Marta Bunge, Internal Presheaves Toposes Cahiers de Top. et geo. Diff. XVIII-3 Marta Bunge, On the Relationship between Composite and Tensor Product Triples Marta Bunge and Robert Pare, Stacks and Equivalence of Indexed Categories Cahiers de Top. et Geo.Diff. XX-4 Marta Bunge, Stack Completions and Morita Equivalence for Category Objects in a Topos Cahiers de Top. et Geo.Diff. XX-4

60. DOWNLOADABLE FILES BACK. GALOIS GROUPOIDS AND COVERING MORPHISMS IN topos theory Submitted to the Proceedingsof the Fields Institute Workshop on Categorical Structures for Descent http://www.math.mcgill.ca/bunge/current-papers.html

GALOIS GROUPOIDS AND COVERING MORPHISMS IN TOPOS THEORY Submitted to the Proceedings of the Fields Institute Workshop on Categorical Structures for Descent, Galois Theory, Hopf Algebras and Semiabelian categories, Fields Institute Communications Series. Version posted 21/11/02. galcov.ps galcov.dvi galcov.pdf Abstract The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S S which generalizes that of Grothendieck (1971) and Moerdijk (1989), (4) explain the role of stack completions in distinguishing Galois groupoids from fundamental groupoids when the base topos S is arbitrary, (5) extend to the case of an arbitrary base topos S results of Bunge-Moerdijk (1997) concerning the comparison between the coverings and the paths fundamental group toposes, and (6) discuss pseudofunctorialy of the fundamental groupoid constructions and apply it to give a simple version (for locally paths simply connected toposes) of the (pushout version of the) van Kampen theorem for toposes given in Bunge-Lack (2001). VAN KAMPEN THEOREMS FOR TOPOSES (with S. Lack)

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