61. Literaturliste Q:\diss\disstopo 2 Saunders Mac Lane and Ieke Moerdijk Sheaves in Geometry and Logic A FirstIntroduction to topos theory , 1992. 15 Johnstone, PT topos theory , 1977. http://fsmat.at/~weppens/p/disstopos.html

Saunders Mac Lane and Ieke Moerdijk Reals and Forcing with an Elementary Topos X Hauptbibio: DAT 440:528281I21; Logic for Computer Science , Moscovic; Saunders Mac Lane and Ieke Moerdijk Sheaves in Geometry and Logic: A First Introduction to Topos Theory X Inst 113 ; Springer-Verlag Michael Barr and Charles Wells Toposes, Theories, and Triples X ; Springer-Verlag Freyd, Peter The axiom of choice. X J. Pure Appl. Algebra Tierney, M. Sheaf Theory and the Continum Hypothesis X Mathe: 18A15T; Toposes, Algebraic Geometry and Logic , Lecture Notes ; Springer Moerdijk, I. Sets, topoi, and intuitionism. Colin McLarty Elementary Categories, Elementary Toposes ; Oxford Science Publications Oswald Wyler Topoi and Quasitopoi ; World Scientific Blass, Andreas and Scedrov, Andre Freyd's models for the independence of the axiom of choice. Mem. Am. Math. Soc. :134 p.,1989 Freyd, Peter Choice and well-ordering. Ann. Pure Appl. Logic Goldblatt, R. Topoi ; North-Holland Blass, A. The Interaction Between Category Theory and Set Theory UBG Mathe: 06P:G779; Math. App. of Category Theory

62. ISM Category Theory After a brief selected introduction to topos theory we will discussdistributions, complete spreads, and the symmetric topos. Special http://www.math.uqam.ca/ISM/english/ecat.html

Category Theory and Applications Category theory is a mathematical discipline that is characterized by its role in unifying mathematics as well as its foundational vocation. Since it was created by Eilenberg and MacLane, its influence has grown both in breadth and depth. The history of its development is intimately linked to that of contemporary mathematics. Montreal has been an important research centre in category theory for more than 20 years. The ISM offers a complete program of applications of category theory in the following areas: algebra and topology; logic and the foundation of mathematics; theoretical computer science; mathematical linguistics. The program includes training in general category theory and in the history of contemporary mathematics. Courses 2002-2003 Topics in Algebra III : Covering toposes with singularities McGill, MATH-724B-Marta Bunge-H/W The title of this course paraphrases that of a landmark 1957 paper by R.H. Fox, "Covering spaces with singularities''. The course will be centered around the connection which exists between complete spreads (with a locally connected domain) over a topos and distributions on the topos, and with a related factorization theorem for geometric morphisms. After a brief selected introduction to topos theory we will discuss distributions, complete spreads, and the symmetric topos. Special topics will be selected from the following: branched coverings and knot groupoids, exponentiable complete spreads, admissible KZ-doctrines, single universes for functions and distributions, categories of cosheaves. Notes of a preliminary version of a book in preparation by Marta Bunge and Jonathon Funk will be distributed as a basis for the course.

63. Sheaves In Geometry And Logic: A First Introduction To Topos Theory (Universitex Sheaves in Geometry and Logic A First Introduction to topos theory (Universitext),Sheaves in Geometry and Logic A First Introduction to topos theory http://www.wkonline.com/a/Sheaves_in_Geometry_and_Logic_A_First_Introduction_to_

Book > Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) by Authors: S. Mac Lane,I. Moerdijk,Saunders MacLane Released: April, 1992 ISBN: 0387977104 Paperback Sales Rank: List price: Our price: Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) > Customer Reviews: Average Customer Rating: Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) > Customer Review #1: Clear explicit descriptions This book is written in the best Mac Lane style, very clear and very well organized. It also benefits from Moerdijks extensive work organizing the theory of Grothendieck toposes by elementary means. The reader should have basic graduate knowledge of algebra and topology. The book is long because it gives very explicit descriptions of many advanced topicsyou can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field. Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) > Related Products Elementary Categories, Elementary Toposes (Oxford Logic Guides, 21)

64. Siberian Toposes The topos theory was created by Lawvere. Topos is a category that hasmany properties of category Set, which is known as theory of sets. http://www.univer.omsk.su/omsk/Sci/topoi/

The Topos Theory was created by Lawvere. Topos is a category that has many properties of category Set, which is known as theory of sets. Development of topos-theoretic mathematics is conjugated with many technical complexities. The bulky diagrams that are used in category theory is one of them. The construction of formal theory models of which are toposes implies an intuitionistic logic instead of classical one. The Synthetic Differential Geometry of Kock-Lawvere is such formal theory. The topos-theoretic mathematics open new perspectives in Theoretical Physics, and in theory of space-time in particular. This site contains scientific and educational materials, important links that can be useful for Russian students who are wishing to study the Topos Theory and its applications. Web-master: guts@univer.omsk.su Omsk State University, Auguest 30, 1999 Since October 5, 1999.

65. ? ? . Isham, CJ topos theory and Consistent HistoriesThe Internal Logic of the Set of all cosistent sets // Int. Jour. Theor. http://www.univer.omsk.su/omsk/Sci/topoi/appl.html

66. TCD CS Foundations And Methods Group Robert, BA (Mod) Postgraduate Student (Ph.D. by research) Research Areas Foundationsof Computing, Formal Methods, Category Theory, topos theory Research Group http://www.cs.tcd.ie/dept/personnel/fmg.html

Computer Science Department Trinity College Foundations and Methods Group A ... Z If dialing from outside college add the prefix to the extension number given, outside Dublin add to the extension number. + for international access. Butterfield, Andrew , BA, BAI, PhD Lecturer Course Director of BA ICT, Tutor Research Areas Mathematical Modelling; , Formal Methods; Functional Languages; Design Automation Research Group FMG ) Foundations and Methods Group, ( CAG ) Computer Architecture Group Location : Oriel.3.10 Telephone ext e-mail Andrew.Butterfield@cs.tcd.ie Comments : Also a member of (DSG) Distributed Systems Group Byrne, Robert , B.A. (Mod) Postgraduate Student (Ph.D. by research) Research Areas Foundations of Computing, Formal Methods, Category Theory, Topos Theory Research Group FMG ) Foundations and Methods Group Location : Oriel.4.09 e-mail Robert.Byrne@cs.tcd.ie Doherty, Gavin , B.A.(Mod), D.Phil Lecturer Research Areas Human-Computer Interaction Research Group DSG ) Distributed Systems Group, ( FMG ) Foundations and Methods Group Location : Oriel.4.17

67. Web Pages/ME_FEIN/references.html ISBN 1871408-05-9. (Johnstone 1977), PT Johnstone topos theory Academic Press,London, 1977 ISBN 0-12-387850-0 Definitely not for the faint-hearted. http://www.cs.tcd.ie/Micheal.MacanAirchinnigh/Web Pages/ME_FEIN/references.html

REFERENCES (Allen 1990) Robert Edward Allen (ed). The Concise Oxford Dictionary of Current English (Eighth Edition), Clarendon Press, Oxford, [1911] 1990. ISBN 19 861200 1. (Barr and Wells 1995) Barr, Michael and Wells, Charles. Category Theory for Computing Science, Second ed., Prentice Hall, London, 1995. (Biggs 1989) Norman L. Biggs. Discrete Mathematics (Revised Edition), Clarendon Press, Oxford, [1985] 1989. ISBN 19 853427 2. This is the primary text for the discrete mathematics of 1ICT5. (Bird and de Moor 1997) Richard Bird and Oege de Moor. Algebra of Programming, Prentice Hall, London, 1997. ISBN 0-13-507245-X, EAN 9 780135 072455. (Boyer 1990) Ernest L. Boyer. Scholarship Reconsidered, Priorities of the Professoriate, Josey-Bass, Inc., Publishers, 350 Sansome Street, San Francisco, California 94104, 1990. ISBN 0-7879-4069-0. This is the seminal work/report on the re-evaluating and re-structuring of the concept of research (narrowly the Scholarship of Discovery) in the (Research) Universities (initially in the United States of America, and consequently world-wide). (Courant and Robbins 1996) Richard Courant and Herbert Robbins What is Mathematics? An Elementary Approach to Ideas and Methods

68. LECTURE NOTES ON TOPOI AND QUASITOPOI Fuzzy Sets and Systems, 1991. The present book is the first coherent account ofthe theory of quasitoposes, stressing the similarity with topos theory; in fact http://www.wspc.com/books/mathematics/1047.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Keyword Author Concept ISBN Series New Titles Editor's Choice Bestsellers Book Series ... Join Our Mailing List LECTURE NOTES ON TOPOI AND QUASITOPOI by Oswald Wyler (Carnegie-Mellon University, USA) Quasitopoi generalize topoi, a concept of major importance in the theory of Categoreis, and its applications to Logic and Computer Science. In recent years, quasitopoi have become increasingly important in the diverse areas of Mathematics such as General Topology and Fuzzy Set Theory. These Lecture Notes are the first comprehensive introduction to quasitopoi, and they can serve as a first introduction to topoi as well. Contents: Basic Properties Examples of Topoi and Quasitopoi Logic in a Quasitopos Topologies and Sheaves Geometric Morphisms Internal Categories and Diagrams Topological Quasitopoi Quasitopoi and Fuzzy Sets Readership: Mathematicians and theoretical computer scientists. "This book is excellently and clearly written ... Every topos theorist and every fuzzy set theorist interested in topoi and foundations will find it both valuable and enjoyable ... Highly recommended." Fuzzy Sets and Systems, 1991

69. RE: Applied Vs. Theoretical clear in some of my earlier posts on this, mostly this past summer, I'm not makingany grandiose claims for category theory and topos theory as being the http://www.mail-archive.com/everything-list@eskimo.com/msg04178.html

everything-list Chronological Find Thread RE: Applied vs. Theoretical From: Marchal Bruno Subject: RE: Applied vs. Theoretical Date: Thu, 05 Dec 2002 05:32:17 -0800 RE: Applied vs. Theoretical Marchal Bruno Chronological Thread Reply via email to

70. Constructive Topology (L24) theory (as provided by the Michaelmas Term Part III course) is therefore desirable,although no previous knowledge of sheaf theory or topos theory will be http://www.maths.cam.ac.uk/CASM/courses/descriptions/node28.html

Next: Number Theory Up: Logic Previous: Set Theory (M24) Constructive Topology (L24) P.T. Johnstone Desirable Previous Knowledge The course will concentrate on the constructive nature of locale theory, and in particular on the equivalence between locales over a given base locale X and internal locales in the category of sheaves on X . Some previous knowledge of category theory (as provided by the Michaelmas Term Part III course) is therefore desirable, although no previous knowledge of sheaf theory or topos theory will be assumed. The only other prerequisites are some acquaintance with the basic notions of general topology (compactness, connectedness and so on), and with the basic ideas of (classical) propositional logic. Level: General Appropriate Books A good idea of the flavour of the subject is given by [1], which is recommended for preliminary reading. [2] has been the standard textbook on locales for the past two decades, but the approach taken in the course will be closer in spirit to Part C of [3]. (However, [3] is not recommended for preliminary reading.)

71. Category Theory This expository article is an entry in the Stanford Encyclopedia of Philosophy.Category Science Math Algebra Category Theory...... geometry. The 1970s saw the development and application of the concept.(For more on the history of topos theory, see McLarty 1992.). http://plato.stanford.edu/entries/category-theory/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z content revised JUL Category Theory Category theory is a general mathematical theory of structures and sytems of structures. It allows us to see, among other things, how structures of different kinds are related to one another as well as the universal components of a family of structures of a given kind. The theory is philosophically relevant in more than one way. For one thing, it is considered by many as being an alternative to set theory as a foundation for mathematics. Furthermore, it can be thought of as constituting a theory of concepts. Finally, it sheds a new light on many traditional philosophical questions, for instance on the nature of reference and truth. General Definitions Brief Historical Sketch Philosophical Significance Bibliography ... Related Entries General Definitions Category theory is a generalized mathematical theory of structures. One of its goals is to reveal the universal properties of structures of a given kind via their relationships with one another. Formally, a category C can be described as a collection Ob , the objects of C , which satisfy the following conditions: For every pair a b of objects, there is a collection

72. References As for topos theory there is, Johnstone, PT Sketches of an Elephanta topos theory Compendium. Oxford Univ. Press, to appear, 2002,. http://mcs.open.ac.uk/cft36/References.htm

References Introductory A very good introductory reference for locale theory is Peter Johnstone's account: Johnstone, P.T. Stone Spaces . Cambridge Studies in Advanced Mathematics . Cambridge University Press, 1982. This book is remarkable as it exposes the story of the subject while at the same time containing all the required proofs in detail. As for topos theory there is, Johnstone, P.T. Sketches of an Elephant: a Topos Theory Compendium . Oxford Univ. Press, to appear, 2002, which I have not yet fully reviewed but am confident will provide an excellent and rigorous introduction to the subject. The standard reference for category theory, MacLane, S. Categories for the Working Mathematician. Texts in Mathematics . Springer-Verlag, 1971, which you may see referred to as CWM, is still probably the best introductory account for category theory. This subject is now quite standard and so a number of texts are available. A good check as to whether a particular text really 'goes the distance' is to see whether an adjoint functor theorem (not just definition) is included. Key Papers A number of papers stand out in the literature as being of particular importance to our area of research.

73. Logics For Uncertainty category theory and topos theory;; constructive and substructural logics;;expert systems and diagnostic reasoning;; robotics applications. http://www.ladseb.pd.cnr.it/infor/logic/logic.html

Uncertainty in Autonomous Systems Group ISIB-CNR , Padova, Italy Research Topic: Logics for Uncertainty People involved: Gaetano Chemello , Research Scientist, ISIB-CNR Claudio Sossai , Research Scientist, ISIB-CNR Interest Areas: category theory and topos theory; constructive and substructural logics; expert systems and diagnostic reasoning; robotics applications. Current research: application of category theory and topos theory to the representation of uncertainty using a generalized set theory called U-Sets; algebraic description of conditional events in evidence theory, in collaboration with F. Esteva and L. Godo of IIIA-CSIC representation of space and uncertainty in intelligent systems, in collaboration with D. Dubois and H. Prade of IRIT-CNRS , Toulouse, France; applicability to robotics, with emphasis on the integration of different uncertainty models (stochastic, fuzzy); mathematical foundations of fuzzy control, in collaboration with D. Driankof and A. Saffiotti of the Center for Applied Autonomous Sensor Systems (AASS) Publications: C. Sossai and G. Chemello.

74. PLT Online (link). Wesley Phoa. An introduction to fibrations, topos theory, the effectivetopos and modest sets. gzipped PS. Andrew M. Pitts. Lecture Notes on Types. http://www.cs.uu.nl/people/franka/ref

PLT Online Programming language theory texts online This is a collection of programming language theory texts and resources, all of which are freely available over the Internet. Many valuable reference texts on programming language theory, previously only available in paper form, have in recent years become publicly accessible from the net. I list here the ones I know of; below that you will also find a much broader list of lecture notes and tutorials other interesting reading , plus a collection of related resources Administrative note An editable version of this page is now available here on the Utrecht University Software Technology Group WikiWiki server , as part of The Software Technologist , which is intended to become an electronic magazine. For now, both versions will coexist and I will probably add any updates to that page to this page as well, and vice versa. If the Wiki version sees a lot of activity and takes off, and does not evolve too far from my original purpose (outlined in the guidelines below), then I may trash this page and redirect page views to the Wiki. For researchers If you know of any other texts or resources which might belong here, please

75. News Letter Vol.5 No.3 a more general interpretation of fuzzy logic within the environment of other propercategories of fuzzy sets stemming either from the topos theory, or even http://www.erudit.de/erudit/newsletters/news_53/page10.htm

New Books / Journals MATHEMATICAL PRINCIPLES OF FUZZY LOGIC Vilem Novak, Irina Perfilieva, Jiri Mockor Kluwer Academic Publishers, Boston/DordrechtLondon 1999. ISBN 0-7923-8595-0 MATHEMATICAL PRINCIPLES OF FUZZY LOGIC provides a systematic study of the formal theory of fuzzy logic. The book is based on logical formalism demonstrating that fuzzy logic is a well-developed logical theory. It includes the theory of functional systems in fuzzy logic, providing an explanation of what, and how it can be represented by formulas of fuzzy logic calculi. It also presents a more general interpretation of fuzzy logic within the environment of other proper categories of fuzzy sets stemming either from the topos theory, or even generalizing the latter. This book presents fuzzy logic as the mathematical theory of vagueness as well as the theory of the common-sense human reasoning, which is based on the use of natural language, the distinguishing feature is vagueness of its semantics. MATHEMATICAL PRINCIPLES OF FUZZY LOGIC will be of interest to all researchers of fuzzy logic, including mathematicians and computer scientists interested in the mathematical aspects of fuzzy logic. It may be used as a text in advanced level courses on various fuzzy logic applications, artificial and computational intelligence, decision-making and more.

76. Abstracts The general mathematician, who regards category theory as `generalized abstractnonsense,' tends to regard topos theory as generalized abstract category theory http://grad.math.arizona.edu/~gradcolloq/fll01abs.html

August 29 Speaker - Alan Von Hermann Title - Von Neumann Regular Rings In this talk, I will give some classical examples of Von Neumann Regular Rings (VNRR) and discuss some interesting properties that such rings have. In particular, I will give a sufficient condition for a VNRR to be a division ring. Undergraduates with some background in Algebra are encouraged to attend. September 5 Speaker - Katrina Jimenez,Jennifer Christian Smith, and William Y. Velez Title - Outreach Opportunities for Graduate Students Outreach activities are a fun, rewarding way to add some spark to your resume during graduate school. We will present a variety of ways that you can get involved in outreach activities with local high school students through a developing program coordinated by graduate students in the department. Two types of outreach activities, the development and presentation of weekday mathematics workshops for high school students and "special guest" visits to local calculus classes, will be discussed. Students with VIGRE support are especially encouraged to get involved this year. Come and learn how you can get involved! September 12 Speaker - David Gay Title - How to Draw Pictures of 4-Manifolds This is partly designed as an introduction to some upcoming talks I will give in the geometry seminar, but it should also be entertaining all by itself. I will discuss "Kirby calculus", a technique for drawing pictures of 4-manifolds using knots and links; this is old-fashioned topology (no PDE's, no moduli spaces, no connections on weird bundles) but it continues to be a useful tool in modern work on 4-manifolds. I will also talk about how to add a little extra structure and use Kirby calculus to think about symplectic 4-manifolds.

77. Fields Institute Audio - Bunge Covering Morphisms in topos theory Marta Bunge McGill University.This web presentation contains the audio of a lecture given at http://www.fields.utoronto.ca/audio/02-03/galois_and_hopf/bunge/

LECTURE AUDIO March 18, 2003 Home About Us Prizes and Honours People ... Search Covering Morphisms in Topos Theory Marta Bunge McGill University This web presentation contains the audio of a lecture given at the Fields Institute on September 23, 2002 as part of the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories . RealPlayer 7 or later, or other software capable of playing streaming audio, is required. Start audio presentation

78. PSSL 79 is a locale 16.45 Sabine Koppelberg (Berlin) A lattice of isomorphism types ofBoolean algebras Sunday 09.45 Richard Josza (Oxford) topos theory in physics http://www.wraith.u-net.com/PSSL/1979.html

THE PERIPATETIC SEMINAR ON SHEAVES AND LOGIC Twelfth Meeting: Oxford, 34 March 1979 Saturday Peter Johnstone (Cambridge) Another factorization theorem for geometric morphisms John Haigh Wim Ruitenburg (Utrecht) Intuitionistic Galois theory Gavin Wraith (Sussex) The Galois group is a locale Sabine Koppelberg (Berlin) A lattice of isomorphism types of Boolean algebras Sunday Richard Josza (Oxford) Topos theory in physics Mike Brockway (Oxford) When are hyperconnected morphisms logical? Gavin Wraith (Sussex) Orderings of the field of rational functions Thirteenth Meeting: Sussex, 23 June 1979 Friday Ronnie Brown (Bangor) Higher-dimensional group theory Saturday Peter Johnstone (Cambridge) Open maps Ronnie Brown (Bangor) Topologising spaces of partial maps with open domain Anders Kock (Aarhus) Synthetic theory of differential forms Roswitha Harting Chris Mulvey (Sussex) Stone-Cech compactification in a topos, III Sunday Jon Zangwill (Bristol) Internal geometric theories Francis Borceux (Louvain) Localisations of module categories Gavin Wraith (Sussex) Freyd's work on independence of (AC) Fourteenth Meeting: Cambridge, 1718 November 1979

79. Sets, Logic And Categories Further references. For topos theory S. MacLane I. Moerdijk, Sheaves inGeometry and Logic A first introduction to topos theory, Springer 1990. http://www.maths.qmw.ac.uk/~pjc/slc/

Peter J. Cameron Sets, Logic and Categories This book is published by Springer-Verlag , in the Springer Undergraduate Mathematics Series , in February 1999. Another book in the series is Geoff Smith's Introductory Mathematics: Algebra and Analysis A PDF file of the preface and table of contents is available. Solutions to the exercises (PDF files): Chapter 1: Naive set theory Chapter 2: Ordinal numbers Chapter 3: Logic Chapter 4: First-order logic Others to be added! Here is a list of known misprints, together with comments and improvements from various readers. From the review by A. M. Coyne in The text is clearly written. It would make an excellent first course in foundational issues in mathematics at the undergraduate level. Further references Sheaves in Geometry and Logic: A first introduction to topos theory , Springer 1990. (Suggested by Steve Awodey A computer scientist's view: Paul Taylor, Practical Foundations of Mathematics , Cambridge University Press, 1999. A book about how our brains are wired to do mathematics: Brian Butterworth, The Mathematical Brain , Macmillan, London, 1999.

80. Past Category Theory Seminars Autumn term 1996. 15 October Peter Johnstone The nonclassifying topos of a first-ordertheory. 16 October Peter Johnstone Open/compact duality in topos theory. http://www.ihes.fr/~leinster/pastsems.html

Past Category Theory Seminars This is roughly speaking a list of the Category Theory seminars held at the Department of Pure Mathematics and Mathematical Statistics in Cambridge when I was there (October 1996 to September 2002). More exactly, it's a list of the ones at which I took notes and where these notes ended up in a file rather than a pile. Jump down to: My home page is here . Information on current Category Theory seminars in Cambridge is here Autumn term 1996 15 October Peter Johnstone The non-classifying topos of a first-order theory 22 October Martin Hyland Girard's completeness `theorem' for linear logic 29 October Jeff Egger Duality 5 November D Cubric Normalization via the Yoneda lemma 19 November Audrey Tan Some full completeness results for models of multiplicative linear logic 26 November Kim Wagner Enriched categories, adjoint bimodules, Cauchy completeness Spring term 1997 21 January Peter Johnstone Some remarks on Scott's category of equilogical spaces 28 January Martin Hyland What is the cyclic category? 4 February Jeff Egger Plonka's theorem 25 February Peter Johnstone The non-classifying topos, continued

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