Geometry.Net - Pure_And_Applied

Topos
topos theory in a Nutshell January 3, 2001 Okay, you wanna know what a topos is? First I'll give you a handwavy vague explanation, then an actual definition, then a few consequences of this definition, and then some examples.
http://math.ucr.edu/home/baez/topos.html

Extractions: Okay, you wanna know what a topos is? First I'll give you a hand-wavy vague explanation, then an actual definition, then a few consequences of this definition, and then some examples. I'll warn you: despite Chris Isham's work applying topos theory to the interpretation of quantum mechanics, and Anders Kock and Bill Lawvere's work applying it to differential geometry and mechanics, topos theory hasn't really caught on among physicists yet. Thus, the main reason to learn about it is not to quickly solve some specific physics problems, but to broaden our horizons and break out of the box that traditional mathematics, based on set theory, imposes on our thinking. Around 1963, Lawvere decided to figure out new foundations for mathematics, based on category theory. His idea was to figure out what was so great about sets, strictly from the category-theoretic point of view. This is an interesting project, since category theory is all about objects and morphisms. For the category of sets, this means SETS and FUNCTIONS. Of course, the usual axioms for set theory are all about SETS and MEMBERSHIP. Thus analyzing set theory from the category-theoretic viewpoint forces a radical change of viewpoint, which downplays membership and emphasizes functions. Even earlier, this same change of viewpoint was also becoming important in algebraic geometry, thanks to the work of Grothendieck on the Weil conjectures. So topos theory can be thought of as a merger of ideas from geometry and logic - hence the title of this book, which is an excellent introduction to topos theory, though not the easiest one:
An Introduction To Fibrations, Topos Theory, The Effective Topos And Modest Sets
An introduction to fibrations, topos theory, the effective topos and modest sets Abstract A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical `universe' of recursive mathematics.
http://www.lfcs.informatics.ed.ac.uk/reports/92/ECS-LFCS-92-208

Extractions: Wesley Phoa Abstract: A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical `universe' of recursive mathematics. Among its objects are the modest sets , which form a set-theoretic model for polymorphism. More precisely, there is a fibration of modest sets which satisfies suitable categorical completeness properties, that make it a model for various polymorphic type theories. These lecture notes provide a reasonably thorough introduction to this body of material, aimed at theoretical computer scientists rather than topos theorists. Chapter 2 is an outline of the theory of fibrations, and sketches how they can be used to model various typed lambda-calculi. Chapter 3 is an exposition of some basic topos theory, and explains why a topos can be regarded as a model of set theory. Chapter 4 discusses the classical PER model for polymorphism, and shows how it `lives inside' a particular topos - the effective topos - as the category of modest sets. An appendix contains a full presentation of the internal language of a topos, and a map of the effective topos. Chapters 2 and 3 provide a sampler of categorical type theory and categorical logic, and should be of more general interest than Chapter 4. They can be read more or less independently of each other; a connection is made at the end of Chapter 3.
OUP: Sketches Of An Elephant: A Topos Theory Compendium: Johnstone
topos theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and
http://www.oup.co.uk/isbn/0-19-852496-X

Extractions: VIEW BASKET Quick Links About OUP Career Opportunities Contacts Need help? oup.com Search the Catalogue Site Index American National Biography Booksellers' Information Service Children's Fiction and Poetry Children's Reference Dictionaries Dictionary of National Biography Digital Reference English Language Teaching Higher Education Textbooks Humanities International Education Unit Journals Law Medicine Music Oxford English Dictionary Reference Rights and Permissions Science School Books Social Sciences World's Classics UK and Europe Book Catalogue Help with online ordering How to order Postage Returns policy ... Table of contents Peter T. Johnstone , Reader in the Foundations of Mathematics, University of Cambridge, Cambridge, UK
Topos Theory And Quantum Theory
topos theory and Quantum Theory. Chris In Section 2, we introducetopos theory, especially the idea of a topos of presheaves. In
http://www.mmsysgrp.com/QIS/topos.htm

Extractions: Physicist Chris Isham discusses Topos Theory and Quantum Theory in this multimedia presentation given at the Newton Institute for Mathematical Sciences, Cambridge University, England. Chris Isham (Imperial College, London) and Jeremy Butterfield (All Souls College, Oxford): Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Section 1, we introduce these problems. In Section 2, we introduce topos theory, especially the idea of a topos of presheaves. In Section 3, we discuss several possible applications of topos theory to the problems in Section 1. In Section 4, we draw some conclusions. The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.
Steven Vickers
Imperial College, London Geometric logic, topos theory, quantales and semantics of programming languages.
http://mcs.open.ac.uk/puremaths/pmd_department/pmd_vickers/pmd_vickers.html
Topos Theory
Quantum Gravity and The Theory of Everything topos theory and Quantum Theory. Chris Isham (Imperial College, London) Quantum Theory and Reality
http://www.mmsysgrp.com/QuantumGravity/topos.htm

Extractions: Physicist Chris Isham discusses Topos Theory and Quantum Theory in this multimedia presentation given at the Newton Institute for Mathematical Sciences, Cambridge University, England. Chris Isham (Imperial College, London) and Jeremy Butterfield (All Souls College, Oxford): Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Section 1, we introduce these problems. In Section 2, we introduce topos theory, especially the idea of a topos of presheaves. In Section 3, we discuss several possible applications of topos theory to the problems in Section 1. In Section 4, we draw some conclusions. The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.
The Theory Of Everything
Algebra.MacMillan, 1967; MacLane, Saunders and Moerdijk,I. Sheaves inGeometry and Logic A First Introduction to topos theory. Springer
http://www.mmsysgrp.com/QIS/category.htm

Extractions: 'Categorification' is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called `coherence laws'. Iterating this process requires a theory of `n-categories', algebraic structures having objects, morphisms between objects, 2-morphisms between morphisms and so on up to n-morphisms. After a brief introduction to n-categories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. John C. Baez (University of California, Riverside): From Finite Sets to Feynman Diagrams This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' and `state', and similarly those of `spacetime' and `process', which we argue is to be expected in any background-free quantum theory. We sketch how higher-dimensional algebra provides the mathematical tools to make this link precise. Finally, we comment on attempts to formulate a theory of quantum gravity in 4-dimensional spacetime using `spin networks' and `spin foams'.
Week68
False. Let's call the set of truth values Omega, just to make it soundimpressive and because it's traditional in topos theory. So
http://math.ucr.edu/home/baez/week68.html

Extractions: Okay, now the time has come to speak of many things: of topoi, glueballs, communication between branches in the many-worlds interpretation of quantum theory, knots, and quantum gravity. 1) Robert Goldblatt, Topoi, the Categorial Analysis of Logic, Studies in logic and the foundations of mathematics vol. 98, North-Holland, New York, 1984. If you've ever been interested in logic, you've got to read this book. Unless you learn a bit about topoi, you are really missing lots of the fun. The basic idea is simple and profound: abstract the basic concepts of set theory, so as to define the notion of a "topos", a kind of universe like the world of classical logic and set theory, but far more general! For example, there are "intuitionistic" topoi in which Brouwer reigns supreme - that is, you can't do proof by contradiction, you can't use the axiom of choice, etc.. There is also the "effective topos" of Hyland in which Turing reigns supreme - for example, the only functions are the effectively computable ones. There is also a "finitary" topos in which all sets are finite. So there are topoi to satisfy various sorts of ascetic mathematicians who want a stripped-down, minimal form of mathematics. However, there are also topoi for the folks who want a mathematical universe with lots of horsepower and all the options! There are topoi in which everything is a function of time: the membership of sets, the truth-values of propositions, and so on all depend on time. There are topoi in which everything has a particular group of symmetries. Then there are *really* high-powered things like topoi of sheaves on a category equipped with a Grothendieck topology....
Topos Theory And Constructive Logic Papers Of Andreas R. Blass
topos theory and Constructive Logic Papers. Andreas Blass. We begin witha brief outline of the history and basic concepts of topos theory.
http://www.math.lsa.umich.edu/~ablass/cat.html

Extractions: Topos Theory and Constructive Logic Papers Andreas Blass Papers on linear logic are on a separate page An induction principle and pigeonhole principle for K-finite sets (J. Symbolic Logic 59 (1995) 11861193) PostScript or PDF We establish a course-of-values induction principle for K-finite sets in intuitionistic type theory. Using this principle, we prove a pigeonhole principle conjectured by Benabou and Loiseau. We also comment on some variants of this pigeonhole principle. Seven trees in one (J. Pure Appl. Alg. 103 (1995) 1-21) PostScript or PDF Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the bijection to a seven-tuple of trees involves case distinctions only down to a fixed depth (namely four) in the given seven-tuple. We clarify how this and similar bijections are related to the free commutative semiring on one generator X subject to X=1+X^2. Finally, our main theorem is that the existence of particularly elementary bijections can be deduced from the provable existence, in intuitionistic type theory, of any bijections at all. Topoi and Computation (Bull. European Assoc. Theoret. Comp. Sci. 36 (1988) 57-65)
K-theory Preprint Archives
Accepts submissions of preprints and offers a search tool for users to access its large archive of papers on Ktheory. With links to related topics. 579 July 2, 2002, Homotopical Algebraic Geometry I topos theory, by Bertrand Toen and Gabriele Vezzosi.
http://www.math.uiuc.edu/K-theory

Extractions: 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.
Re: Topos Theory For Physicists
Re topos theory for physicists. Subject Re topos theory for physicists;From baez@galaxy.ucr.edu (John Baez); Date Mon, 1 Jan 2001 054218 GMT;
http://www.lns.cornell.edu/spr/2001-01/msg0030351.html

Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index Subject : Re: Topos theory for physicists From : baez@galaxy.ucr.edu (John Baez) Date : Mon, 1 Jan 2001 05:42:18 GMT Approved : mmcirvin@world.std.com (sci.physics.research) Newsgroups : sci.physics.research Organization : UCR References Sender : mmcirvin@world.std.com (Matthew J McIrvin) Prev by Date: Re: A Philosophical Query and Comment about Quantizing Gravity Next by Date: Re: Hopf algebras, quantum groups Prev by thread: Re: A Philosophical Query and Comment about Quantizing Gravity Next by thread: Re: Topos theory for physicists Index(es): Date Thread
Re: Topos Theory For Physicists
Index Re topos theory for physicists. Subject Re topos theoryfor physicists; From Chris Hillman hillman@math.washington.edu ;
http://www.lns.cornell.edu/spr/2001-01/msg0030379.html

Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index http://www.math.washington.edu/~hillman/papers.html http://www.math.washington.edu/~hillman/ Prev by Date: Re: Scale invariant general relativity? Next by Date: Re: Self-learning quantum mechanics Prev by thread: Re: Topos theory for physicists Next by thread: Re: Hopf algebras, quantum groups Index(es): Date Thread
On Branched Covers In Topos Theory (ResearchIndex)
On Branched Covers In topos theory (2000) (Make Corrections)
http://citeseer.nj.nec.com/funk00branched.html

Extractions: Abstract: . We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. Our main result is a description of the covers of this subtopos as a category of fractions of branched covers, in the sense of Fox [10], of the including topos. We also have some new results concerning the general theory of KZ-doctrines, such as the... (Update)
PhilSci Archive: Topos Theory As A Framework For Partial Truth
PhilSci Archive, topos theory as a Framework for Partial Truth. Butterfield,Jeremy (2000) topos theory as a Framework for Partial Truth.
http://philsci-archive.pitt.edu/documents/disk0/00/00/01/92/
Theory And Applications Of Categories
1. On branched covers in topos theory. Jonathon Funk, 122
http://www.tac.mta.ca/tac

Extractions: A survey of definitions of n-category Tom Leinster, 1-70 abstract dvi dvi.gz ps ... A homotopy double groupoid of a Hausdorff space Ronald Brown, Keith A. Hardie, Klaus Heiner Kamps, Timothy Porter, 71-93 abstract dvi dvi.gz ps ... Entity-relationship-attribute designs and sketches Michael Johnson, Robert Rosebrugh and R.J. Wood, 94-112 abstract dvi dvi.gz ps ... pdf Emzar Khmaladze, 113-126 abstract dvi dvi.gz ps ... Exponentiability of perfect maps: four approaches Susan Niefield, 127-133 abstract dvi dvi.gz
Course In Topos Theory
topos theory, spring term 1999. synthetic differential geometry. This graduatecourse offers an introduction to topos theory and categorical logic.
http://www.math.uu.se/~palmgren/topos-eng.html

Extractions: A graduate course (6 course points) in mathematical logic. Topos theory grew out of the observation that the category of sheaves over a fixed topological space forms a universe of "continuously variable sets" which obeys the laws of intuitionistic logic. These sheaf models, or Grothendieck toposes, turn out to be generalisations of Kripke and Beth models (which are fundamental for various non-classical logics) as well as Cohen's forcing models for set theory. The notion of topos was subsequently extended and given an elementary axiomatisation by Lawvere and Tierney, and shown to correspond to a certain higher order intuitionistic logic. Various logics and type theories have been given categorical characterisations, which are of importance for the mathematical foundations for programming languages. One of the most interesting aspects of toposes is that they can provide natural models of certain theories that lack classical models, viz. synthetic differential geometry. This graduate course offers an introduction to topos theory and categorical logic. In particular the following topics will be covered: Categorical logic: relation between logics, type theories and categories. Generalised topologies, including formal topologies. Sheaves. Pretoposes and toposes. Beth-Kripke-Joyal semantics. Boolean toposes and Cohen forcing. Barr's theorem and Diaconescu covers. Geometric morphisms. Classifying toposes. Sheaf models of infinitesimal analysis.
Kurs I Toposteori
Springer 1992. Referenslitteratur PT Johnstone. topos theory. Academic Press 1977.J. Lambek and PJ Scott An introduction to Higher Order Categorical Logic.
http://www.math.uu.se/~palmgren/topos.html

Extractions: En forskarutbildningskurs (6p) i matematisk logik. (In English, Please.) S. Mac Lane and I. Moerdijk: Sheaves in Geometry and Logic. Springer 1992. P.T. Johnstone. Topos Theory. Academic Press 1977. J. Lambek and P.J. Scott: An introduction to Higher Order Categorical Logic. Cambridge University Press 1986 S. Mac Lane: Categories for the Working Mathematician. Springer 1971. A.M. Pitts. Categorical Logic. Chapter in the Handbook of Logic in Computer Science, vol. VI. Oxford University Press (under utgivning) Aktuell kursinformation : hittills behandlade moment etc. 1998-02-15, Erik Palmgren, E-post: palmgren@math.uu.se
Natures New Math( Topos Theory?)
Natures New Math( topos theory?). Follow Ups (Reload page to see mostrecent) Re Natures New Math( topos theory?) DickT 7/18/02 (4)
http://superstringtheory.com/forum/philboard/messages13/193.html

Extractions: String Theory Discussion Forum String Theory Home Forum Index Natures New Math( Topos Theory?) Follow Ups Post Followup Philosophy of Physics XIII FAQ Posted by sol on July 18, 2002 at 15:08:55: In Reply to: Re: u2) Theory of Everything (TOE) without the fantasy of Mind? You Bet! posted by DickT on July 18, 2002 at 14:09:57: Dickt, And we are still discovering new ways in which to see what nature has hidden Movement in a world, that we had never understood before. We understand now what energy can do, and helps, to shape matters, but we are in a space now, that requires dimensional thinking, and evidence of the force carriers, to have seen this intergration in a world, as we see in movement of those same matters. Just thought I would add my two bits:) Sol
A Link On Topos Theory (not Too Hard)
A link on topos theory (not too hard). In Reply to Re Natures New Math(topos theory?) posted by DickT on July 19, 2002 at 111618 sol,.
http://superstringtheory.com/forum/philboard/messages13/202.html

Extractions: String Theory Discussion Forum String Theory Home Forum Index A link on Topos theory (not too hard) Follow Ups Post Followup Philosophy of Physics XIII FAQ Posted by DickT on July 19, 2002 at 19:34:32: In Reply to: Re: Natures New Math( Topos Theory?) posted by DickT on July 19, 2002 at 11:16:18: sol, I found this Topos Paper . Chris Isham is a coauthor, he is a big name in this area. The first three sections are very clear (to me) and maybe give you an idea. I know you understand better from pictures, but there don't seem to be any. Regards,
Lars Birkedal / Teaching / Topos Theory Seminar --- Spring 2003
topos theory Seminar Spring 2003. This is a Ph.D. seminar in whichwe study aspects of topos theory relevant to computer science.
http://www.it-c.dk/people/birkedal/teaching/topos-theory-Spring-2003/

Extractions: Spring 2003 Organizers: Lars Birkedal birkedal@it-c.dk , Room 2.21, 3816 8868 Carsten Butz butz@it-c.dk , Room 1.17, 3816 8820 This is a Ph.D. seminar in which we study aspects of topos theory relevant to computer science. This semester we plan to continue reading and discussing material from Peter Johnstone's opus: Sketches of an Elephant: A Topos Theory Compendium . In the schedule below, readings refer to Johnstone's books. Meeting time: We meet on Fridays, 14:0015:30, Room 2.03 (except January 10: Room 2.55; February 7: Room 2.31, March 7: Room 2.55). Schedule: Date Speaker Reading Fri Jan CB A.4.2: Surjections and Inclusions Fri Jan LB A.4.3: Cartesian Reflectors and Sheaves Fri Jan LB A.4.3: Cartesian Reflectors and Sheaves Fri Jan VS A.4.4: Local Operators Fri Feb VS A.4.4: Local Operators Fri Feb REM A.4.5 (pages 204211, incl. 4.5.9): Examples of Local Operators Fri Feb REM A.4.5 (pages 211217, incl. 4.5.16): Examples of Local Operators Fri Feb LB A.4.5 (pages 217223): Examples of Local Operators

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