1. Topos topos theory in a Nutshell. Learning more about all these concepts is probably thebest use of your time if you wants to learn a little bit of topos theory. http://math.ucr.edu/home/baez/topos.html

Topos Theory in a Nutshell John Baez January 27, 2004 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. 1. Hand-Wavy Vague Explanation 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:

2. Topos Theory Quantum Gravity and The Theory of Everything Butterfield (All Souls College, Oxford) Some Possible Roles for topos theory in Quantum Theory and Quantum Gravity ways in which topos theory (a http://www.mmsysgrp.com/QuantumGravity/topos.htm

Topos Theory and Quantum Theory Chris Isham (Imperial College, London): Quantum Theory and Reality 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. Chris Isham (Imperial College, London) and Jeremy Butterfield (All Souls College, Oxford): A Topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations 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.

3. Topos Theory wordtrade.com/science/mathematics. World Scientific Publishing Mathematics Book Sale. Sketches of an Elephant A topos theory Compendium by Peter T. Johnstone ( Oxford Logic Guides, 43 44 Oxford University Press) topos theory is an important branch of mathematical logic of http://www.wordtrade.com/science/mathematics/topostheory.htm

wordtrade.com/science/mathematics World Scientific Publishing Mathematics Book Sale Sketches of an Elephant : A Topos Theory Compendium by Peter T. Johnstone Oxford Oxford University Press) Topos theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject. Topos: A category modeled after the properties of the category of sets. Topos theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Now available in this two volume set, it contains all the important information both volumes provides. Considered to be a complete benefit for all researchers and academics in theoretical computer science, logicians and philosophers who study the foundations of mathematics, and those working in differential geometry and continuum physics. Citing the old Indian story about the blind men feeling different parts of an elephant and coming up with divergent descriptions of the animal Johnstone says that topos theory can be described in divergent ways depending on what part is examined. The original conception of toposes arose in the 1960s as a "generalized space" supporting cohomology theory, but has grown to be used by category theory and other branches of mathematics. Addressing those already familiar with topos theory, Johnstone offers these volumes (and a scheduled 3rd) as a complete treatment of all of the pieces of elementary topos theory together with fully worked-out results. Volume one discusses toposes as categories. Toposes as spaces and theories are reserved for the second. The final volume expected to discuss homotopy and cohomology, and toposes as mathematical universes.

4. Category Theory Bunge, M., 1974, "topos theory and Souslin's Hypothesis", Journal of Pure and Applied Algebra, 4, 159187 Johnstone, P.T., 1977, topos theory, New York Academic Press http://plato.stanford.edu/entries/category-theory

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z This document uses XHTML-1/Unicode to format the display. Older browsers and/or operating systems may not display the formatting correctly. last substantive content change FEB Category Theory 1. General Definitions, Examples and Applications 2. Brief Historical Sketch 3. Philosophical Significance Bibliography ... Related Entries 1. General Definitions, Examples and Applications defined as sets, category theory cannot provide a philosophically enlightening foundation for mathematics. In terms of collections, 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 Mor a b ), namely, the morphisms from a to b in C (when f is a morphism from a to b , we write f a b For every triple a b and c of objects, there is a partial operation from pairs of morphisms in Mor a b ) X Mor b c ) to morphisms in Mor a c ), called the composition of morphisms in C (when f a b and g b c g o f a c is their composition);

5. 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

Go to Pure Maths Department Home Page Steven Vickers Lecturer click here for personal details and interests Current Projects Maths interests Geometric logic, locales, toposes Quantales Applications to computer science including semantics and specification Home Page Pure Mathematics Department The Open University Walton Hall Milton Keynes United Kingdom Office Location: Office Phone: +44 Office Fax: +44 E-mail: s.j.vickers@open.ac.uk

6. 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

October 29, 1995 This Week's Finds in Mathematical Physics (Week 68) John Baez 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....

7. 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

John C. Baez (University of California, Riverside):Categorification '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 John C. Baez (University of California, Riverside):Higher-Dimensional Algebra and Planck-Scale Physics This is a nontechnical introduction to recent work on quantum gravity using ideas fromhigher-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'.

8. Topos - Wikipedia, The Free Encyclopedia Main article Background and genesis of topos theory The historical origin of topostheory is algebraic geometry. to topos theory, Springer, New York, 1992. http://en.wikipedia.org/wiki/Topos

Topos From Wikipedia, the free encyclopedia. For discussion of topoi in literary theory , see literary topos In mathematics , a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. Table of contents 1 Introduction 2 History 3 Formal definition 4 Further examples ... edit Introduction Traditionally, mathematics is built on set theory , and all objects studied in mathematics are ultimately sets and functions . It has been argued that category theory could provide a better foundation for mathematics. By analyzing precisely which properties of the category of sets and functions are needed to express mathematics, one arrives at the definition of topoi, and one can then formulate mathematics inside any topos. Of course, the category of sets forms a topos, but that is boring. In more interesting topoi, the axiom of choice may no longer be valid, or the law of excluded middle (every proposition is either true or false) may break down. It is thus of some interest to collect those theorems which are valid in all topoi, not just in the topos of sets. One may also work in a particular topos in order to concentrate only on certain objects. For instance

9. Topos - Wikipedia, The Free Encyclopedia Topos. (Redirected from topos theory). History. Main article Background and genesisof topos theory The historical origin of topos theory is algebraic geometry. http://en.wikipedia.org/wiki/Topos_theory

Topos From Wikipedia, the free encyclopedia. (Redirected from Topos theory For discussion of topoi in literary theory , see literary topos In mathematics , a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. Table of contents 1 Introduction 2 History 3 Formal definition 4 Further examples ... edit Introduction Traditionally, mathematics is built on set theory , and all objects studied in mathematics are ultimately sets and functions . It has been argued that category theory could provide a better foundation for mathematics. By analyzing precisely which properties of the category of sets and functions are needed to express mathematics, one arrives at the definition of topoi, and one can then formulate mathematics inside any topos. Of course, the category of sets forms a topos, but that is boring. In more interesting topoi, the axiom of choice may no longer be valid, or the law of excluded middle (every proposition is either true or false) may break down. It is thus of some interest to collect those theorems which are valid in all topoi, not just in the topos of sets. One may also work in a particular topos in order to concentrate only on certain objects. For instance

10. Topos Theory As A Framework For Partial Truth topos theory as a Framework for Partial Truth This paper develops some ideas from previous work (coauthored, mostly with C.J.Isham). In that work, the main proposal is to assign as the value of a http://rdre1.inktomi.com/click?u=http://philsci-archive.pitt.edu/archive/0000019

11. 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

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)

12. On Branched Covers In Topos Theory On Branched Covers In topos theory . 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 http://rdre1.inktomi.com/click?u=http://citeseer.ist.psu.edu/321707.html&y=0

13. Topos structure. History. Main article Background and genesis of topos theoryThe historical origin of topos theory is algebraic geometry. http://www.fact-index.com/t/to/topos.html

Main Page See live article Alphabetical index Topos In mathematics , a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. (For discussion of topoi in literary theory , see literary topos Table of contents 1 Introduction 2 History 3 Formal definition 4 Further examples ... 5 References Introduction Traditionally, mathematics is built on set theory , and all objects studied in mathematics are ultimately sets and functions . It has been argued that category theory could provide a better foundation for mathematics. By analyzing precisely which properties of the category of sets and functions are needed to express mathematics, one arrives at the definition of topoi, and one can then formulate mathematics inside any topos. Of course, the category of sets forms a topos, but that is boring. In more interesting topoi, the axiom of choice may no longer be valid, or the law of excluded middle (every proposition is either true or false) may break down. It is thus of some interest to collect those theorems which are valid in all topoi, not just in the topos of sets. One may also work in a particular topos in order to concentrate only on certain objects. For instance

14. Some Possible Roles For Topos Theory In Quantum Theory And Quantum 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 http://rdre1.inktomi.com/click?u=http://citebase.eprints.org/cgi-bin/citations?i

15. Background And Genesis Of Topos Theory Background and genesis of topos theory. This page gives some very generalbackground to the mathematical idea of topos. Position of topos theory. http://www.fact-index.com/b/ba/background_and_genesis_of_topos_theory.html

Main Page See live article Alphabetical index Background and genesis of topos theory This page gives some very general background to the mathematical idea of topos . This is an aspect of category theory , and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. Table of contents 1 In the school of Grothendieck 2 From pure category theory to categorical logic 3 Position of topos theory 4 Summary In the school of Grothendieck During the latter part of the , the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of etale cohomology With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems, for a long time. The first was to do with its points : back in the days of projective geometry it was clear that the absence of 'enough' points on an algebraic variety was a barrier to having a good geometric theory (in which it was somewhat like a compact manifold ). There was also the difficulty, that was clear as soon as

16. Homotopical Algebraic Geometry I Topos Theory Homotopical Algebraic Geometry I topos theory This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this http://rdre1.inktomi.com/click?u=http://citebase.eprints.org/cgi-bin/citations?i

17. Background And Genesis Of Topos Theory - Encyclopedia Article About Background A encyclopedia article about Background and genesis of topos theory. Backgroundand genesis of topos theory. Word Word. http://encyclopedia.thefreedictionary.com/Background and genesis of topos theory

Dictionaries: General Computing Medical Legal Encyclopedia Background and genesis of topos theory Word: Word Starts with Ends with Definition This page gives some very general background to the mathematical idea of topos For discussion of topoi in literary theory, see literary topos. In mathematics, a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. Introduction Traditionally, mathematics is built on set theory, and all objects studied in mathematics are ultimately sets and functions. Click the link for more information. . This is an aspect of category theory Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "abstract nonsense". See list of category theory topics for a breakdown of the relevant Encyclopedia pages. Background A category attempts to capture the essence of a class of related mathematical objects, Click the link for more information.

18. Topos Theory - Encyclopedia Article About Topos Theory. Free Access, No Registra encyclopedia article about topos theory. topos theory in Free onlineEnglish dictionary, thesaurus and encyclopedia. topos theory. http://encyclopedia.thefreedictionary.com/Topos theory

Dictionaries: General Computing Medical Legal Encyclopedia Topos theory Word: Word Starts with Ends with Definition For discussion of topoi in literary theory Literary theory is the theory (or the philosophy) of the interpretation of literature and literary criticism. Its history begins with classical Greek poetics and rhetoric and includes, since the 18th century, aesthetics and hermeneutics. In the 20th century, "theory" has become an umbrella term for a variety of scholarly approaches to reading texts, most of which are informed by various strands of Continental philosophy. (In much academic discussion, the terms "literary theory" and "Continental philosophy" are nearly synonymous, though some scholars would argue that a clear distinction can be drawn between the two.) Click the link for more information. , see literary topos In the context of classical Greek rhetoric a topos (literally "a place"; plural: topoi ) referred to a standardised method of constructing or treating an argument. Ernst Robert Curtius expanded this concept in studying topoi as commonplaces : reworkings of traditional material, particularly the descriptions of standardised settings, but extended to almost any literary

19. 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.itu.dk/people/birkedal/teaching/topos-theory-Spring-2003/

Topos Theory Seminar 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

20. IT-Universitetet I København - Topos Theory Seminar topos theory Seminar Time and place Thursdays at 1 to 3 pm, autumn2001 in room 1.05. Course description We go through Part A http://www.itu.dk/Internet/sw2951.asp

Hjem Nyhedsbrev Søg Find person ... Fall 2001 Topos Theory Seminar Topos Theory Seminar Time and place Thursdays at 1 to 3 p.m., autumn 2001 in room 1.05. Course description: We go through Part A of "A Topos Theory Compendium - Sketches of an Elephant" which gives a very thorough exposition of topos as categories. Prerequisites: A course in category theory, mathematical maturity. The course is given at PhD level. Lecturer: Lars Birkedal References: P. T. Johnstone: "A Topos Theory Compendium - Sketches of an Elephant" Course credit: To document understanding, participating PhD Students must give presentations incl. amplify proofs given in the text. Furthermore, the PhD Student must participate actively in the seminar. Rev. 18-03-2004 www_info@itu.dk

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