1. Topos I ll warn you despite Chris Isham s work applying topos theory to the So topos theory can be thought of as a merger of ideas from geometry and logic 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 topos theory in a Nutshell http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

3. Week68 One can develop topos theory within set theory if one wishes, but one can also To get deeper into topos theory, try. 3) Michael Barr and Charles Wells, 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....

4. Theory And Applications Of Categories 1. On branched covers in topos theory. Jonathon Funk, 122 abstract dvi dvi.gz ps ps.gz pdf http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Topos - Wikipedia, The Free Encyclopedia Main article Background and genesis of topos theory. The historical origin of topos theory is algebraic geometry. Alexander Grothendieck generalized the http://en.wikipedia.org/wiki/Topos

Topos From Wikipedia, the free encyclopedia. For discussion of topoi in literary theory , see literary topos For discussion of topoi in rhetorical invention , see Inventio In the mathematical field of category theory , a topos is a type of category that behaves like the category of sheaves of sets on a topological space Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on that space. This idea was expounded by Alexander Grothendieck by introducing the notion of a topos (plural: topoi or toposes - this is a contentious topic). The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest success of this yoga to date has been the introduction of the ©tale topos of a scheme Contents Introduction History Formal definition Further examples ... edit Introduction A traditional axiomatic foundation of mathematics is set theory , in which all mathematical objects are ultimately represented by sets (even functions which map between sets.) Recent work in

6. Topos Theory Seminar - Spring 2004 topos theory Seminar Spring 2004 Organizers Lars Birkedal, birkedal@itu.dk, Room 2.21, 3816 8868 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Background And Genesis Of Topos Theory - Wikipedia, The Free Encyclopedia Lawvere s definition picks out the central role in topos theory of the subobject Perhaps this is why topos theory has been treated as an oddity; http://en.wikipedia.org/wiki/Introduction_to_topos_theory

Background and genesis of topos theory From Wikipedia, the free encyclopedia. (Redirected from Introduction to 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. Contents In the school of Grothendieck From pure category theory to categorical logic Position of topos theory Summary ... edit 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 ©tale 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

8. Topos Theory Seminar - Fall 2003 topos theory Seminar Fall 2003 Organizers Lars Birkedal, birkedal@itu.dk, Room 2.21, 3816 8868 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. 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 http://www.tac.mta.ca/tac/volumes/7/n1/7-01abs.html

On Branched Covers in Topos Theory Jonathon Funk 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, of the including topos. We also have some new results concerning the general theory of KZ-doctrines, such as the closure under composition of discrete fibrations for a KZ-doctrine, in the sense of Bunge and Funk. Keywords: 1991 MSC: 18B25. Theory and Applications of Categories , Vol. 7, 2000, No. 1, pp 1-22. http://www.tac.mta.ca/tac/volumes/7/n1/n1.dvi http://www.tac.mta.ca/tac/volumes/7/n1/n1.ps http://www.tac.mta.ca/tac/volumes/7/n1/n1.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/7/n1/n1.dvi ... TAC Home

10. Categories Home Page CMS Special Session A special session of the Canadian Mathematical Society Summer 2004 Meeting on topos theory, June 1315, 2004 at http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Summer School On Topos Theory In Haute-Bodeux Summer School on topos theory. HauteBodeux, May 29 to June 5, 2005. Lectures Participants Pictures Credits http://www.math.ucl.ac.be/membres/topos/

Summer School on Topos Theory Haute-Bodeux, May 29 to June 5, 2005 Lectures Participants Pictures Credits This web site is a collective effort of the participants of the Summer School on Topos Theory that was held from May 29 to June 5, 2005, in Haute-Bodeux, Belgium. Its aim is in particular to publish lecture notes, written up by teachers and/or students. There is also a collection of photos taken during that week. If you have a document that you'd like to publish on this web site, contact Isar Stubbe. Lectures up Courses: Steve Awodey: Algebraic Set Theory [ pdf Peter Freyd: Realizability, independence proofs Peter Johnstone: Sites [ pdf Bill Lawvere: A topos framework for space in general and micromechanics in particular [ pdf Ieke Moerdijk: Classifying topos, descent, groupoid representation Myles Tierney: Homotopy in a topos [ pdf Contributed papers: Thomas Noll: The topos of triads Toby Kenney: Copower objects Matias Menni: Cocomplete toposes whose exact completions are toposes Jonathan Funk: Grothendieck topologies and comprehensive factorization for geometric morphisms [ pdf Kohei Kishida: Topological semantics for first-order modal logic [ pdf Benno van den Berg: Sheaves for predicative toposes Participants up Name Surname Affiliation E-Mail (write @ instead of [at]) Moreno Andreatta IRCAM, Paris

12. OUP Sketches Of An Elephant A Topos Theory Compendium Johnstone Sketches of an Elephant A topos theory Compendium 2 Volume Set http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Topos Theory Seminar --- Fall 2004 This is a Ph.D. seminar in which we study aspects of topos theory relevant to computer science. After having read material from Peter Johnstone s opus http://www.itu.dk/people/birkedal/teaching/topos-theory-Fall-2004/

Topos Theory Seminar Fall 2004 Organizers: Lars Birkedal birkedal@itu.dk , Room 4C 15, 7218 5280 Carsten Butz butz@itu.dk , Room 4C 14, 7218 5274 This is a Ph.D. seminar in which we study aspects of topos theory relevant to computer science. After having read material from Peter Johnstone's opus Sketches of an Elephant: A Topos Theory Compendium , Lambek/Scott's Introduction to higher order categorical logic , Mac Lane/Moerdijk's Sheaves in Geometry and Logic we plan to continue with a semester on realizability toposes (without assuming too much knowledge of the material covered in previous semesters to allow new students to participate). Additional material can be found at Jaap van Oosten's page with links to recent articles on realizability and at Lars Birkedal's (somewhat dated) online bibliography on realizability The main goal of the seminar this Fall is to get an understanding of various realizability toposes and how theses are constructed from so-called triposes. In particular, we'll address recent attempts to create realizability toposes for Godel's dialectica interpretation. Further off-line topics could include how realizability toposes may be constructed and analyzed using exact completions; a deeper analysis of subcategories of realizablity toposes; how one may understand Krivine's notion of realizabilty as a tripos / topos; application of realizability models in semantics of programming languages; etc.

14. Citations Sheaves In Geometry And Logic A First Introduction To Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic a first introduction to topos theory. New York SpringerVerlag, 1992. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Topos Theory Seminar --- Fall 2003 This is a Ph.D. seminar in which we study aspects of topos theory relevant to computer science. After having read material from Peter Johnstone s opus http://www.itu.dk/people/butz/courses/TT-E2003.html

Topos Theory Seminar Fall 2003 Organizers: Lars Birkedal birkedal@itu.dk , Room 2.21, 3816 8868 Carsten Butz butz@itu.dk , Room 1.17, 3816 8820 This is a Ph.D. seminar in which we study aspects of topos theory relevant to computer science. After having read material from Peter Johnstone's opus: Sketches of an Elephant: A Topos Theory Compendium we plan to pause and read parts of Lambek/Scott: Introduction to higher order categorical logic Cambridge University Press 1986; and Mac Lane/Moerdijk: Sheaves in Geometry and Logic Springer-Verlag 1992. Additional material can be found in Jaap van Oosten's notes on Basic Category Theory . Focus will be on Lambek/Scott. To gain credit for this 7.5 ECTS seminar you have to participate actively in the discussions. Moreover, you have to take responsibility for at least one topic, stretching usually over more than a week. Meeting time: We meet on Fridays, 14:0015:30, Room 2.55. Participants: Lars Birkedal (LB) Carsten Butz (CB) Volodya Shavrukov (VS) Thomas Hildebrandt (TH) (RM) Noah Torp-Smith (NTS) Nina Bohr (NB) Rasmus Lerchedahl Petersen (RLP) Bodil Biering (BB) Marie Bjerrum (MB) (birkedal@itu.dk, butz@itu.dk, volodya@itu.dk, hilde@itu.dk, mogel@itu.dk, noah@itu.dk, nina@itu.dk, m98rlp@math.ku.dk, bodil@math.ku.dk, m98mb@math.ku.dk)

16. Sheaves In Geometry And Logic A First Introduction To Topos Sheaves in Geometry and Logic A First Introduction to topos theory (Universitext) Pop Culture Lists http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

17. Background And Genesis Of Topos Theory: Information From Answers.com Background and genesis of topos theory This page gives some very general background to the mathematical idea of topos. http://www.answers.com/topic/background-and-genesis-of-topos-theory

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Background and genesis of topos theory Wikipedia 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. 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 ©tale 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

18. Topos Theory Definition / Topos Theory Research topos theory Definition / topos theory Research Related Links Please wait Researching subject http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

19. Topos: Definition And Much More From Answers.com For discussion of topoi in literary theory, see literary topos. John Baez topos theory in a nutshell, http//math.ucr.edu/home/baez/topos.html. http://www.answers.com/topic/topos

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Dictionary Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping topos Dictionary to·pos tō pōs, -pŏs n. pl. -poi -poi A traditional theme or motif; a literary convention. [Greek, short for (koinos) topos , (common)place.] Wikipedia topos For discussion of topoi in literary theory , see literary topos For discussion of topoi in rhetorical invention , see Inventio Sheaves were introduced into mathematics in the 1940s, and a major theme since then has been to study a space by studying sheaves on that space. Grothendieck expounded on this idea by introducing the notion of a topos (plural: topoi or toposes - this is a contentious topic). A topos is a type of category that behaves like the category of sheaves of sets on a topological space . The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest success of this yoga to date has been the introduction of the ©tale topos of a scheme References John Baez Topos theory in a nutshell http://math.ucr.edu/home/baez/topos.html

20. Topos Theory (from Mathematics, Foundations Of) --  Encyclopædia Britannica topos theory (from mathematics, foundations of) The original purpose of category theory had been to make precise certain technical notions of algebra and http://www.britannica.com/eb/article?tocId=35466

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