81. SubjectOverview A topos is a category that is good enough for set theory. The results of topos theory(as in any theory) hinge on the interaction of objects and morphisms. http://mcs.open.ac.uk/cft36/SubjectOverview.htm

Subject Overview (Mathematical Research) For a less technical description of the work please read the essay: From Topology, Logic and Category to the Geometric Mathematical Framework: Research Explained Subject Background: Locales The Tychonoff theorem states that an arbitrary product of compact topological spaces is compact. My understanding is that the proof of a Tychonoff theorem without the axiom of choice by Johnstone in 1981 does not appear to have caused a huge response (certainly amongst mathematicians outside the confines of the study of the foundations). Apart from the complicated set-theoretic ordinal induction in the proof (later removed using techniques described below) one of the reasons for this lack of interest, I speculate, is the fact that most topologists would have dismissed the result as absurd. This is for the good reason that the Tychonoff result is well known to imply the axiom of choice and so how can one have a proof without choice? Of course Johnstone (and many before him, possibly back to Wallman 1938) was modelling spaces using locales (that is using complete Heyting algebras) and so had tampered sufficiently with the definition of a topological space to allow the result to go through without a choice axiom. Whether this is a 'good thing to do' really remains open to a certain extent. The logician's opening gambit could be that, yes, it is a good thing to do since it allows the development of a theory of spaces that is dependent on less axioms than the one proceeding it and so is, in an absolute sense, better. On the other hand we must satisfy ourselves that locales are capturing the correct notion of a space. This is a harder question since it is not precise. Locales are not a bad framework since a number of topological results have intuitively clear localic versions (e.g. Stone-Cech compactification) that reproduce the topological versions in the presence of the prime ideal theorem.

82. 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.

83. Ramifications Of Category Theory Workshop And Symposium Including The workshop paid attention not only to issues internal to category theory, andin particular to topos theory, but also to manifold relationships of category http://ramcat.scform.unifi.it/

84. Relational Databases And Indexed Categories - Rosebrugh, Wood large shared data banks (context) Codd - 1970 168 The Theory of Relational Databases(context) - Maier - 1983 73 topos theory (context) - Johnstone - 1977 38 http://citeseer.ist.psu.edu/rosebrugh92relational.html

85. OUP USA: Elementary Categories, Elementary Toposes: Colin McLarty . Thebook covers elementary aspects of category theory and topos theory....... Series and Imprint Information. Oxford Logic Guides, 21. http://www.oup.com/us/catalog/general/subject/Mathematics/Logic/~~/cGY9OTAmcHI9M

Check Out Help Search More Search Options Browse Logic Logic Browse Related Subjects Computer Science Philosophy Never Miss an OUP Sale! Free Email Alerts! More Oxford email lists Enter Sales Promo Code Related Links: Table of Contents Subjects Mathematics Logic Elementary Categories, Elementary Toposes Colin McLarty paper 280 pages Feb 1996 In Stock Price: $90.00 $5.00 (US) $10.00 (INTL) Series Reviews Product Details About the Author(s) ... Features and Fluents Erik Sandewall Price: $245.00 Modalities Ruth Barcan Marcus Price: $30.00 Browse More Titles In Literary Theory Electronic Text Kathryn Sutherland Price: $74.00

86. 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

87. 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

88. CMS/CAIMS Summer 2004 Meeting Printer friendly page, topos theory / Théorie des topos (Org Myles Tierney,Rutgers University and/et University of Quebec at Montreal). http://www.cms.math.ca/Events/summer04/abs/tt.html

Org: Myles Tierney, Rutgers University and/et University of Quebec at Montreal) Higher topological Galois theory Associating to any groupoid its category of representations (i.e. of presheaves), the 2-category of small groupoids can be seen as a full subcategory of topoi. Moreover, a topos X is Galois (i.e. is the category of representations of a small groupoid) if and only if it is locally connected and if any sheaf on X is locally constant. The classical topological Galois theory can then be stated as follows: The full inclusions of Galois topoi into locally simply connected topoi (i.e. locally connected topoi which admits a generating family on which any locally constant sheaf is constant) has a left adjoint: it is defined sending a topos X to the topos p X ) of locally constant sheaves on X If we think of homotopy types as some kind of -groupoids (whatever it means), then there should be some analog of this setting replacing groupoids by n -groupoids for n (and working with an adequate notion of n -topos). This has been done by B. Toen in some way for topoi which have the homotopy type of a CW-complexe. We shall give another proof of this which allows us to consider in a unified way an arbitrary

89. MaMuX 2002-2003 Translate this page Following a series of sometimes very technical talks at IRCAM, I shall this timerather concentrate on the adequacy of topos theory for music, dealing with the http://www.ircam.fr/equipes/repmus/mamux/documents/ResumesMars.html

Samedi 20 mars 2004 Ircam, Salle I. Stravinsky Moreno Andreatta Guerino Mazzola Franck Jedrzejewski Thomas Noll : Transformational Logics in Harmonic and Metric Analysis Charles Alunni Peer Bundgaard Table ronde (avec la participation de Peter Johnstone Guerino Mazzola Following a series of sometimes very technical talks at IRCAM, I shall this time rather concentrate on the adequacy of topos theory for music, dealing with the questions of (1) formalization of musical ontology (which concept framework? which topoi? which addresses for denotators?), (2) the synthesis of aesthetics and logic (is there a common ground for beauty and truth?), and (3) languages for creativity (in the mental and technological realm). This will include comments on prominent reactions (is music THAT complicated?) relating to the mathematics used in "The Topos of Music". Franck Jedrzejewski (CEA Saclay - INSTN/UERTI) Z/12Z Thomas Noll The paper presents investigations following the strands of ideas that Guerino Mazzola and I presented in a joint talk "Extending Set Theory to Harmonic Topology and Topos Logic" at the Set Theory Conference at Ircam in october 2003. "Transformational Logics" in the title of my paper refers to the topos logics of functor categories. Functors F: C -> Sets from a small category C generalize the concept of group- or monoid actions, i.e in my applications I depart from small categories C and interpret functors F: C -> Sets music-theoretically as families of transformations on musical objects.

90. IC Theory Group Postdoc List consistenthistories version of quantum theory. This enables the set theory ('topoi') in the context of the consistent histories programme where the internal logic of a topos seems http://theory.ic.ac.uk/staffresearch.html

Theoretical Physics Group Research Interests Staff Members: Professor Chris Isham Professor Chris Isham My primary research interests are: (i) all aspects of quantum gravity; (ii) the general problem of quantum theory, both technical and conceptual. My main current research programme is partly motivated by the 'problem of time' in quantum gravity which, in turn, is a special manifestation of the general questions in quantum gravity that have intrigued me for many years: (a) to what extent are standard space-time ideas (both mathematical and conceptual) applicable in quantum gravity; and (b) to what extent is the formalism (both mathematical and conceptual) of standard quantum theory applicable in quantum gravity. In particular, I have been developing a new approach to quantum gravity based on a quantum-logic extension of the consistent-histories version of quantum theory. This enables the standard ideas of quantum theory to be extended to situations where there is no normal notion of time, including the possibility that a variety of non-metrical aspects of space-time may also be subject to quantisation. I have become particularly interested in the use of generalised set theory ('topoi') in the context of the consistent histories programme where the internal logic of a topos seems to play an important role. I am also planning to use topos ideas in the development of new models for spacetime; in particular, there may be important links with topological quantum field theory, expecially the recent work involving ideas coming from the loopspace approach to the Ashtekar programme of canonical quantisation of gravity.

91. Categorical Logic And Type Theory of PERs and omegasets over the effective topos 4. Natural numbers in the effectivetopos and some associated principles Chapter 7 Internal category theory. http://www.cs.kun.nl/~bart/CLT/bookinfo.html

Categorical Logic and Type Theory B. Jacobs, Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, North Holland, Elsevier, 1999. ISBN 0-444-50170-3 BibTex entry This book gives a survey of categorical logic and type theory starting from the unifying concept of a fibration. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists. To get an impression, the prospectus from the book is made available. Reviews By Peter Johnstone, in the Zentralblatt fur Mathematik , vol. 905, 1999. By Robert Seely, to appear in The Bulletin of Symbolic Logic available on the web). By Andreas Blass, in Mathematical Reviews How to order This book can be ordered directly via the publisher Elsevier , or via Amazon , or (possibly) via your local bookstore. Contents Chapter 0: Prospectus 1. Logic, type theory and fibred category theory 2. The logic and type theory of sets Chapter 1: Introduction to fibred category theory 1. Fibrations 2. Some concrete examples: sets, omega-sets and PERs

92. Topos In Architectural Theory Translate this page Navigation topos in architectural theory (251.071). Seminar WS 3,0 oderSS 3,0. Im Lehrzielkatalog gibt es eine ausführliche Beschreibung http://info.tuwien.ac.at/histu/lehr/251071.html

Topos in architectural theory (251.071) Seminar WS 3,0 oder SS 3,0 Im Lehrzielkatalog Die Lehrveranstaltung wird am Institut f. Technik und Theorie (E 259) abgehalten. Orte und Zeiten Die Lehrveranstaltung wird im Wintersemester 2003 gelesen: Vortragende(r): FASCHINGEDER Kristian O.Univ.Prof. Dipl.-Ing. Dr.phil. JORMAKKA Kari Juhani Die Lehrveranstaltung wird im Sommersemester 2004 gelesen: Vortragende(r): FASCHINGEDER Kristian O.Univ.Prof. Dipl.-Ing. Dr.phil. JORMAKKA Kari Juhani Im Lehrzielkatalog Hilfe LVA-Suchen Suchen TU-Wien ... Kritik-Lob Quellen: TUWIS Last update June 6th, 2004 Only to report a technical problem, e.g. dangling links or if you have technical questions regarding this information system, you may use the following mail address HISTU-team

93. K-theory Preprint Archives Electronic preprint archives for mathematics research papers in Ktheory. 692 May 15, 2004, The Postnikov tower in motivic stable homotopy theory, by Marc Levine 2002, Homotopical Algebraic http://www.math.uiuc.edu/K-theory

K-theory Preprint Archives Welcome to the preprint archives for papers in K-theory. We accept submissions of preprints in electronic form for storage until publication. Storage after publication may be possible, too. Search with htdig: Search with google: Our mirror site in Germany Our home site in the USA Instructions for authors Instructions for joining the mailing list. Members of the mailing list receive announcements of preprints when they are deposited in the archives. (The list has more than 300 members.) Best Current Practices: Recommendations on Electronic Information Communication A calendar of K-theory events. A bibtex file (2342698 bytes), or the same thing gzipped (557172 bytes) containing 6454 citations to papers related to K-theory and instructions for enlarging bibtex to work with such a large file. MPRESS , a searchable index of preprints, including references to the preprints stored here. Some TeX fonts , stored in a tar image compressed with gzip, including the lams* and xy* fonts, which are needed for some of the preprints. The unprocessed recent submissions bibtex entries Source code for xdvi , a viewer under unix for dvi files.

94. Mazzola, G., The Topos Of Music now include topologies for rhythm, melody, and harmony, as well as a classificationtheory of musical objects that comprises the topostheoretic concept http://www.birkhauser.ch/books/math/5731.htm

The Topos of Music Geometric Logic of Concepts, Theory, and Performance 2002. 1368 pages. Hardcover. incl. CD-ROM ISBN 3-7643-5731-2 English Since the Greek antiquity it has been a tradition of European thinking to describe musical facts in a mathematical language. This formal apparatus has always mirrored the status quo of mathematical knowledge and the requirements of current sound technology. The Topos of Music is the upgraded and vastly deepened English extension of the seminal German . It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances.

95. Abstract:011010bunge The fundamental group of a (universal) branched covering in topostheory. Marta Bunge (10/10/01). Background Branched coverings in http://www.maths.usyd.edu.au:8000/u/stevel/auscat/abstracts/011010bunge.html

The fundamental group of a (universal) branched covering in topos theory Marta Bunge (10/10/01) Background Branched coverings in topos theory [BN] [Fu] may be described (in several ways with varying degrees of abstraction) in terms of the more general notion of a complete spread [BF]. The goal of this talk is to establish a theorem relating the fundamental group of a (universal) branched covering of a topos with that of the (universal) covering of its unbranched part. Motivation. The main motivation comes from theorem of R.H. Fox [Fox] (Section 7) for locally finite complexes, where, however, the fundamental group of a branched covering is directly given in terms of paths rather than by exploiting the Galois theory that is inherent in the given (branched and unbranched) coverings. I set out to prove one key ingredient of this theorem in the general context of an admissible KZ-doctrine in the sense of [BF2], satisfying an additional axiom of "interior" (or "density"). Definitions. Context: an admissible KZ-doctrine M satisfying an axiom of interior on a 2-category K.

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