41. Subject Classification topos theory. 11. Quantifiers and Sheaves, Proceedings of the International Comments on the Development of topos theory, Development of Mathematics http://www.acsu.buffalo.edu/~wlawvere/subject.html

F. William Lawvere Subject Classification of Articles HOME Chronological list Functorial Semantics of Algebraic Theories Proceedings of the National Academy of Science 50 , No. 5 (November 1963), 869-872. Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models ; North-Holland, Amsterdam (1965), 413-418. Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories Springer Lecture Notes in Mathematics No. 61 , Springer-Verlag (1968), 41-61. Review of P. M. Cohn's Universal Algebra , 2nd Edition, American Scientist (May-June 1982), p. 329. 42. with J. Adamek and J. Rosicky, How algebraic is algebra? Theory and Applications of Categories (2001) 253-283 (electronic). 44. with J. Adamek and J. Rosicky: On the duality between varieties and algebraic theories, Algebra Universalis, Topos Theory Quantifiers and Sheaves Proceedings of the International Congress on Mathematics , (Nice 1970), Gauthier-Villars (1971) 329-334. Introduction to the Proceedings of the Halifax Conference, Toposes, Algebraic Geometry and Logic Springer Lecture Notes in Mathematics No. 274

42. Casual Category Theory - Fall 2000 (Luigi Santocanale), Introduction to topos theory (continuation) Some properties of elementary toposes. The subobject classifier in a presheaf topos. http://www.brics.dk/~varacca/CCT/cct-fall00.html

Casual Category Theory - Fall 2000 Events in time-directed order Tuesday 12:15 (Luigi Santocanale) Introduction to topos theory [MM] (Other possible references: McLarty [CML] [BW] Tuesday 12:15 (Luigi Santocanale) Introduction to topos theory (continuation) Some properties of elementary toposes. The subobject classifier in a presheaf topos. Beck's theorem. Tuesday 9:15 (Pawel Sobocinski) Introduction to topos theory (continuation) Tutorial: proving properties in toposes. An alternative definition of topos. Tuesday 12:15 (Luigi Santocanale) Categorical logic. The internal language of a topos. [LS] Tuesday 12:15 (Luigi Santocanale) The internal language of a topos (continuation). The Kripke-Joyal semantics. Tuesday 12:15 (Daniele Varacca) Tutorial: using categorical logic. The Heyting algebra of subobjects. The extension of a formula of the internal language. Tuesday 12:15 (Daniele Varacca) Tutorial: using categorical logic II.

43. [math/0207028] Homotopical Algebraic Geometry I: Topos Theory Homotopical Algebraic Geometry I topos theory. Authors Bertrand Toen, Gabriele Vezzosi Comments 71 pages. Final version to appear in Adv. Math http://arxiv.org/abs/math.AG/0207028

Mathematics, abstract math.AG/0207028 From: Gabriele Vezzosi [ view email ] Date ( ): Tue, 2 Jul 2002 20:44:00 GMT (86kb) Date (revised ): Tue, 8 Oct 2002 11:49:17 GMT (88kb) Date (revised ): Tue, 10 Feb 2004 17:04:03 GMT (84kb) Date (revised v4): Sun, 20 Jun 2004 17:14:18 GMT (85kb) Homotopical Algebraic Geometry I: Topos theory Authors: Bertrand Toen Gabriele Vezzosi Comments: 71 pages. Final version to appear in Adv. Math Subj-class: Algebraic Geometry; Algebraic Topology MSC-class: This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts (for part II, see math.AG/0404373 Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv math find abs

44. [gr-qc/9607069] Topos Theory And Consistent Histories: The Internal Logic Of The topos theory and Consistent Histories The Internal Logic of the Set of all Consistent Sets. Authors CJ Isham Comments 28 pages, LaTeX http://arxiv.org/abs/gr-qc/9607069

General Relativity and Quantum Cosmology, abstract gr-qc/9607069 From: Chris Isham [ view email ] Date: Sun, 28 Jul 1996 09:00:34 GMT (28kb) Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets Authors: C.J. Isham Comments: 28 pages, LaTeX Report-no: Imperial/TP/9596/55 Journal-ref: Int.J.Theor.Phys. 36 (1997) 785-814 Full-text: PostScript PDF , or Other formats References and citations for this submission: SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted); CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers? Links to: arXiv gr-qc find abs

45. Category Theory (For more on the history of topos theory, see McLarty 1992.) Finally, from the 1980s to this day, category theory found new applications. On the one hand, http://setis.library.usyd.edu.au/stanford/archives/fall1997/entries/category-the

This is a file in the archives of the Stanford Encyclopedia of Philosophy Stanford Encyclopedia of Philosophy A B C D ... Z 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

46. Categories His main lines of research, besides model theory, are topos theory and categorical logic, applications of topos theory to Differential Geometry (Synthetic http://www.polimetrica.com/categories/01cat.html

Polimetrica S.a.s. INTERNATIONAL SCIENTIFIC PUBLISHER home about us catalog proposals ... to buy Generic figures and their glueings - A constructive approach to functor categories by Marie La Palme Reyes, Gonzalo E. Reyes, Houman Zolfaghari Download: table of contents introduction chapter Information 290 Pages Price: 30 Euro (Italy); 37 Euro (UE); 40 Euro (Extra-UE) Forwarding and delivery charges are included in the price ISBN 88-7699-004-6 Series Categories Short description This book is a "missing link" between the elementary textbook of Lawvere and Schanuel "Conceptual Mathematics" and the much more advanced textbooks such as the one by MacLane and Moerdijk "Sheaves in Geometry and Logic." Avoiding the complicated, fully fledged notion of a Grothendieck topos, whose very formulation presupposes a good deal of mathematical experience, this book introduces topos theory through presheaf toposes, i.e., readily visualizable categories whose objects result from glueing simpler ones, the "generic figures". Several phenomena which distinguish toposes from the ordinary category of sets appear already at this simpler level. Six easy to understand examples accompany the reader through the whole book, illuminating new material, interpreting general results and suggesting new theorems.

47. McGill Mathematics And Statistics - People Marta Bunge Galois groupoids and Covering Morphisms in topos theory Fields Marta Bunge topos theory and Souslin s Hypothesis J. Pure and Appl. Alg 4 http://www.math.mcgill.ca/department/display_people.php?id=103

48. Sets, Logic And Categories For topos theory S. MacLane I. Moerdijk, Sheaves in Geometry and Logic A first introduction to topos theory, Springer 1990. (Suggested by Steve Awodey. 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.

49. Fields Institute Audio - Bunge Covering Morphisms in topos theory Marta Bunge McGill University Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories. http://www.fields.utoronto.ca/audio/02-03/galois_and_hopf/bunge/

LECTURE AUDIO September 16, 2005 Home About Us Mailing List Publications ... 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

50. Prof. P.T. Johnstone 1977) and Sketches of an Elephant a topos theory Compendium (Oxford UP, 2002). My interests focus particularly on the way in which topos theory 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 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

51. People In DPMMS PT Johnstone Category theory, especially topos theory and locale theory; Prof. FP Kelly, FRS Random processes, networks and optimization http://www.dpmms.cam.ac.uk/site2002/people.html

Department of Pure Mathematics and Mathematical Statistics DPMMS People People at DPMMS University Academic Staff (Professors, Lecturers, etc.) Resident Academics (College Fellows etc.) Visitors Research Students M.Phil. Students Administrative Staff ... CMS Staff University Academic Staff (Professors, Lecturers, etc.) Dr P.M.E. Altham Analysis of discrete data, multivariate analysis, generalized linear modelling, computational statistics (R and S-Plus), graphical methods Prof. A. Baker, FRS Number theory, transcendence, logarithmic forms, effective methods, Diophantine geometry, Diophantine analysis Dr M.B. Batchelor Measuring coalgebras, comodules and their applications in geometry, physics and algebra Prof A.F. Beardon Complex analysis, hyperbolic geometry, discrete groups Dr C.J.B. Brookes Groups, non-commutative algebra and geometry, homological algebra Dr S.P. Brooks Stochastic simulation, optimisation, Markov chain Monte Carlo, Bayesian statistics, applications in ecology, sociology, archaeology, medicine and biology Dr T.K. Carne

52. Sheaves In Geometry And Logic (MacLane)-Springer K-Theory Book This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions http://www.springeronline.com/sgw/cda/frontpage/0,11855,5-10051-72-1352462-0,00.

Please enable Javascript in your browser to browse this website. Select your subdiscipline Algebra Analysis Applications Mathematical Biology Mathematical Physics Probability Theory Quantitative Finance Home Mathematics Applications Select a discipline Biomedical Sciences Chemistry Computer Science Economics Education Engineering Environmental Sciences Geography Geosciences Humanities Law Life Sciences Linguistics Materials Mathematics Medicine Philosophy Popular Science Psychology Public Health Social Sciences Statistics preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,5-0-17-900180-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,5-0-17-900170-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,5-0-17-900190-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,5-0-17-900200-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,5-0-17-900369-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,5-0-17-900344-0,00.gif');

53. Categorical Logic topos theory, as would now be understood, is the intuitionistic replacement for set The founders of elementary topos theory were Lawvere and Tierney. http://www.algebra.com/algebra/about/history/Categorical-logic.wikipedia

Categorical logic Regular View Dictionary View (all words explained) Algebra Help my dictionary with pronunciation , wikipedia etc Categorical logic Categorical logic is a branch of category theory within mathematics , adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science . In broad terms, it is a theory about the transition from a type theory , understood to be within an intuitionistic logic or constructive mathematics setting, to a category , by means of a translation that respects both the syntax and the intended computational meaning of type-theoretic constructions. The subject has been recognisable in these terms since about , when the needs of domain theory started to call on category theory. The earlier history is relatively complex, and contains some ironies. Categorical logic originated within sheaf theory , as a suitable version of Kripke semantics one can say with hindsight, and emerged as a theory with a character of its own only in shedding the necessary connection with sheaves. This can be traced in a number of stages, from onwards: the formulation of the Grothendieck topos , and then of the elementary topos , giving rise first to topos theory . Topos theory, as would now be understood, is the intuitionistic replacement for

54. Siberian Toposes Toposophy, 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. http://www.univer.omsk.su/omsk/Sci/topoi/

55. ? Isham, CJ topos theory and Consistent Histories The Internal Logic of the Set Isham, CJ, Butterfield J. Some Possible Roles for topos theory in Quantum http://www.univer.omsk.su/omsk/Sci/topoi/appl.html

56. Category Theoretic Perspectives On The Foundations Of Mathematics topos theory is the platform from which category theory approaches set theory. This solves some of the problems which category theorists may have with the http://www.rbjones.com/rbjpub/philos/maths/faq004.htm

Category Theoretic Perspectives on the Foundations of Mathematics See also the Categories Home Page I don't pretend to understand what category theory has to say about the foundations of mathematics, but I would rather like to understand. To that end these notes are compiled recording my impressions, such as they are, about the why Category theorists have misgivings about classical set theory as a foundation for mathematics and what (if anything) they would like to offer in its stead. problems solutions links Problems for Category Theory in Classical Set Theoretic Foundations Doing Category Theory in Set Theory Many categories are categories of all examples of a particular kind of mathematical structure (or of all models of a particular theory). Such collections are problematic in set theory in the same way as the set of all sets. In consistent set theoretic foundations based on the iterative conception of set they can be proven not to exist. Though there are techniques which can be employed to mitigate these difficulties, the solutions are not entirely satisfactory, and the view that set theory is not a natural context in which to do category theory may persist. For more on this kind of problem, see: Chapter 8 Introduction and Section 3 . Furthermore, certain kinds of construction which are important in category theory cannot reliably be done in set theory, for similar reasons, see

57. Notes On: The Logical Foundations Of Mathematics Section 8.6 topos theory topos theory provides an alternative approach to axiomatisation of the category of sets. This is generalised in the sense that the http://www.rbjones.com/rbjpub/philos/bibliog/hatch82.htm

by on The Logical Foundations of Mathematics by William S. Hatcher Chapter 1 - First Order Logic A presentation of first order logic including a general treatment of "variable binding term operators", such as set comprehension, which are often required in foundation systems. Chapter 2 - The Origin of Modern Foundational Studies Chapter 3 - Frege's System and the Paradoxes Chapter 4 - The Theory of Types Chapter 5 - Zermelo Fraenkel Set Theory Chapter 7 - The Foundational Systems of W.V.Quine Chapter 8 - Categorical Algebra Chapter 2 - The Origin of Modern Foundational Studies Section 1 - Mathematics as an Independent Science Section 2 - The Arithmetisation of Analysis Section 3 - Constructivism Section 4 - Frege and the notion of a Formal System Section 5 - Criteria of Foundations "What must a foundation be, and what must it do?" Chapter 8 - Categorial Algebra Introduction In the introduction, Hatcher describes the relevance of category theory for foundations. Section 8.1 The notion of a category An informal introduction to category theory. Section 8.2

58. Index Of /~jesper/seminarier/0405/050413 topos theory has led to elegant proofs of various metamathematical results (independence and consistency results, derived rules). Many of them depend on the http://www.math.su.se/~jesper/seminarier/0405/050413/

Index of /~jesper/seminarier/0405/050413 Name Last modified Size Description ... Parent Directory 01-Sep-2005 21:14 - anslag.dvi 08-Apr-2005 09:00 5k anslag.pdf 08-Apr-2005 09:00 19k

59. Citebase - Homotopical Algebraic Geometry I: Topos Theory Cisinski, Theories homotopiques dans les topos, JPAA 174 (2002), 4382. G/A, Del1 P. Deligne, Categories Tannakiennes, in Grothendieck Festschrift Vol. http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:math/0207028

60. Citebase - Topos Theory And Consistent Histories: The Internal Logic Of The Set Sheaves in Geometry and Logic A First Intro duction to topos theory. SpringerVerlag, London. G/A, Omn` R. (1988a). Logical reformulation of quantum http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:gr-qc/9607069

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