21. Topos Theory Seminar This is an advanced Ph.D......Home Forskning PhD studies Courses topos theory Seminar, E2002,topos theory Seminar. http://www.it-c.dk/Internet/research/phd/courses/TTS/

Hjem Nyhedsbrev Søg Find person ... Courses Topos Theory Seminar, E2002 Topos Theory Seminar Description: This is an advanced Ph.D. course, in which we read and present material from Peter Johnstone's new book Sketches of an Elephant: A topos theory compendium. This semester we cover the chapters on Allegories and exact completions, and on Geometric morphisms of toposes. Meetings: Fridays 14-15:30, Room 2.03. Credits: 7,5 ECTS. Ph.D. students who wish to take the course for credit should prepare lectures and present the prepared material at least twice during the semester. Opdateret 20/02-2003

22. AMCA: Covering Morphisms In Topos Theory Presented By Marta Bunge Covering morphisms in topos theory by Marta Bunge Department of Mathematicsand Statistics, McGill University, Montreal, Canada http://at.yorku.ca/cgi-bin/amca/cajf-21

AMCA Document # cajf-21 Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories September 23-28, 2002 Fields Institute Toronto, Ontario, Canada Organizers George Janelidze, Georgian Academy of Sciences, Bodo Pareigis, University of Munich, Walter Tholen, York University View Abstracts Conference Homepage Covering morphisms in topos theory by Marta Bunge Department of Mathematics and Statistics, McGill University, Montreal, Canada In the work of Janelidze in 1990, a formal notion of covering morphism arises from an abstract categorical framework given by a pair of adjoint functors. Associated with any such class of covering morphisms, there is a corresponding (pure) Galois theory, of which there are examples in different areas of mathematics. In topos theory, the class of all covering projections (local homeomorphisms determined by a locally constant object of the topos) that appears explicitly in the work of Barr and Diaconescu in 1981, has been shown already by Janelidze to be an instance of the above notion of covering morphism, but only under the conditions that either the base topos be Set (that is, for Grothendieck toposes), or else where the splitting cover in the topos is assumed connected. U (E) is defined by means of the pushout, in the 2-category

23. AMCA: Spectral Decomposition Of Ultrametric Spaces And Topos Theory Presented By Homepage. Spectral Decomposition of Ultrametric Spaces and topos theoryby Alex J. Lemin Moscow State University of Civil Engineering http://at.yorku.ca/cgi-bin/amca/cacl-81

AMCA Document # cacl-81 1999 Summer Conference on Topology and its Applications August 4-7, 1999 C.W. Post Campus of Long Island University Brookville, NY, USA Organizers Sheldon Rothman, Ralph Kopperman View Abstracts Conference Homepage Spectral Decomposition of Ultrametric Spaces and Topos Theory by Alex J. Lemin Moscow State University of Civil Engineering We consider categories M E T R and M E T R c U L T R A M E T R and U L T R A M E T R c of ultrametric spaces and the same maps. Given a family of ultrametric spaces, we prove that sums and products, equalizer and co-equalizer, pull-back and push-out, limits of direct and inverse spectra, if exist, are ultrametric. A product and a limit of inverse spectrum of complete metric spaces are complete. A space (X,d) is uniformly discrete 'for all' x, y X. This is necessarily complete. Theorem Every complete ultrametric space is isometric to a limit of a countable inverse spectrum of uniformly discrete ultrametric spaces (and vise versa) (see [1]). Corollary 1 Every compact ultrametric space is isometric to a limit of inverse sequence of skeletons of finite dimensional isosceles simplexes lying in Euclidean spaces (see [2]).

24. Homotopical Algebraic Geometry I Topos Theory, By Bertrand Toen And Gabriele Vez Homotopical Algebraic Geometry I topos theory, by Bertrand Toen andGabriele Vezzosi. This is the first of a series of papers devoted http://www.math.uiuc.edu/K-theory/0579/

Homotopical Algebraic Geometry I Topos theory, by Bertrand Toen and Gabriele Vezzosi 0579.bib (254 bytes) agmod-I-fin-web.dvi (516636 bytes) [July 2, 2002] agmod-I-fin-web.dvi.gz (185402 bytes) agmod-I-fin-web.ps.gz (345410 bytes) Bertrand Toen Gabriele Vezzosi

25. Atlas: Covering Morphisms In Topos Theory By Marta Bunge Covering morphisms in topos theory presented by Marta Bunge Departmentof Mathematics and Statistics, McGill University, Montreal, Canada http://atlas-conferences.com/c/a/j/f/21.htm

Atlas Document # cajf-21 Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories September 23-28, 2002 Fields Institute Toronto, Ontario, Canada Conference Organizers George Janelidze, Georgian Academy of Sciences, Bodo Pareigis, University of Munich and Walter Tholen, York University View Abstracts Conference Homepage Covering morphisms in topos theory presented by Marta Bunge Department of Mathematics and Statistics, McGill University, Montreal, Canada In the work of Janelidze in 1990, a formal notion of covering morphism arises from an abstract categorical framework given by a pair of adjoint functors. Associated with any such class of covering morphisms, there is a corresponding (pure) Galois theory, of which there are examples in different areas of mathematics. In topos theory, the class of all covering projections (local homeomorphisms determined by a locally constant object of the topos) that appears explicitly in the work of Barr and Diaconescu in 1981, has been shown already by Janelidze to be an instance of the above notion of covering morphism, but only under the conditions that either the base topos be Set (that is, for Grothendieck toposes), or else where the splitting cover in the topos is assumed connected. U (E) is defined by means of the pushout, in the 2-category

26. Atlas: Spectral Decomposition Of Ultrametric Spaces And Topos Theory By Alex J. Spectral Decomposition of Ultrametric Spaces and topos theory presentedby Alex J. Lemin Moscow State University of Civil Engineering http://atlas-conferences.com/c/a/c/l/81.htm

Atlas Document # cacl-81 1999 Summer Conference on Topology and its Applications August 4-7, 1999 C.W. Post Campus of Long Island University Brookville, NY 11548, USA Conference Organizers Sheldon Rothman and Ralph Kopperman View Abstracts Conference Homepage Spectral Decomposition of Ultrametric Spaces and Topos Theory presented by Alex J. Lemin Moscow State University of Civil Engineering We consider categories M E T R and M E T R c U L T R A M E T R and U L T R A M E T R c of ultrametric spaces and the same maps. Given a family of ultrametric spaces, we prove that sums and products, equalizer and co-equalizer, pull-back and push-out, limits of direct and inverse spectra, if exist, are ultrametric. A product and a limit of inverse spectrum of complete metric spaces are complete. A space (X,d) is uniformly discrete 'for all' x, y X. This is necessarily complete. Theorem Every complete ultrametric space is isometric to a limit of a countable inverse spectrum of uniformly discrete ultrametric spaces (and vise versa) (see [1]). Corollary 1 Every compact ultrametric space is isometric to a limit of inverse sequence of skeletons of finite dimensional isosceles simplexes lying in Euclidean spaces (see [2]).

27. On Branched Covers In Topos Theory On Branched Covers in topos theory. Jonathon Funk. We present somenew findings concerning branched covers in topos theory. Our 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

28. Lars Birkedal / Teaching Current Suggestions for Master's Theses topos theory Seminar Ph.D. seminar courseon topos theory. topos theory Seminar Ph.D. seminar course on topos theory. http://www.itu.dk/people/birkedal/teaching/

Teaching Current: Suggestions for Master's Theses Topos Theory Seminar Ph.D. seminar course on Topos Theory. Spring 2002. Past: Advanced XML / Data on the Web Fall 2002. Topos Theory Seminar Ph.D. seminar course on Topos Theory. Fall 2002. Models and Languages for Concurrency and Mobility Spring 2002. Distributed Systems Spring 2002. Topos Theory Seminar Ph.D. seminar course on Topos Theory. Spring 2002. Reasoning about Resources (RaR) Seminar: Reasoning about Low-level Languages Ph.D. seminar course on logics for reasoning about resources with a focus on reasoning about low-level languages, in particular languages with pointers and mutable data structures. Fall 2001. Topos Theory Seminar Ph.D. seminar course on Topos Theory. Fall 2001. Category Theory Project Introductory graduate project course on category theory. Fall 2001. Introductory Programming Introductory programming course using Java. Spring 2001. Constructive Logic Project Introductory graduate project on constructive logic. Spring 2001.

29. HallPhilosophy.com Sketches Of An Elephant A Topos Theory HallPhilosophy.com Sketches of an Elephant A topos theory Compendium(Oxford Logic Guides, 43 44). HallPhilosophy.com. the most http://hallphilosophy.com/index.php/Mode/product/AsinSearch/019852496X/name/Sket

30. [gr-qc/9607069] Topos Theory And Consistent Histories: The Internal Logic Of The 34 +0100 (28kb) topos theory and Consistent Histories The InternalLogic of the Set of all Consistent Sets. Authors CJ Isham Comments http://arxiv.org/abs/gr-qc/9607069

General Relativity and Quantum Cosmology, abstract gr-qc/9607069 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) Links to: arXiv gr-qc find abs

31. Categories: Topos Theory And Large Cardinals categories topos theory and large cardinals. Andrej Bauer asked whether large cardinalsother than inaccessible ones have a natural definition in topos theory. http://north.ecc.edu/alsani/ct99-00(8-12)/msg00128.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Topos theory and large cardinals To categories@mta.ca Subject : categories: Topos theory and large cardinals From wlawvere@acsu.buffalo.edu Date : Tue, 21 Mar 2000 16:32:09 -0500 (EST) Reply-To wlawvere@acsu.buffalo.edu Sender cat-dist@mta.ca http://www.acsu.buffalo.edu/~wlawvere Prev by Date: categories: Preprint: a paper on injective hulls Next by Date: categories: Functorial injective hulls Prev by thread: categories: Topos theory and large cardinals Next by thread: categories: dom fibration Index(es): Date Thread

32. Categories: Topos Theory And Large Cardinals categories topos theory and large cardinals. To categories@mta.ca; Subjectcategories topos theory and large cardinals; From Andrej.Bauer@cs.cmu.edu; http://north.ecc.edu/alsani/ct99-00(8-12)/msg00117.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Topos theory and large cardinals To categories@mta.ca Subject : categories: Topos theory and large cardinals From Andrej.Bauer@cs.cmu.edu Date : 01 Mar 2000 22:29:58 -0500 Sender cat-dist@mta.ca Source-Info : Sender is really andrej+@gs2.sp.cs.cmu.edu User-Agent : Gnus/5.0803 (Gnus v5.8.3) XEmacs/20.4 (Emerald) Can you complete this analogy? ``Large cardinals are to ZFC, as are to topos theory.'' One answer is "Grothendieck universes", but they correspond to rather small large cardinals. Can we go further than that? Andrej Bauer School of Computer Science Carnegie Mellon University http://andrej.com Prev by Date: categories: Johnstone review of Clark/Davey Next by Date: categories: dom fibration Prev by thread: categories: Johnstone review of Clark/Davey Next by thread: categories: Topos theory and large cardinals Index(es): Date Thread

33. Citations: Topos Theory - Johnstone (ResearchIndex) Similar pages Partial morphisms in categories of effective objects Moggi ( Correct) 0.0 A Guided Tour in the Topos of Graphs - Sebastiano Vig Na (Correct)0.0 On Branched Covers In topos theory - Funk (2000) (Correct) Related http://citeseer.nj.nec.com/context/13760/0

72 citations found. Retrieving documents... Johnstone, P.T., Topos Theory , L.M.S. Monographs, vol. 10, Academic Press, 1977. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: First 50 documents Next 50 Solving Recursive Domain Equations with Enriched Categories - Wagner (1994) (14 citations) (Correct) ....However, the exposition should serve more as a reminder and a statement about notation than a first introduction to the concepts just listed. For that a much longer text is needed, and we refer the reader to the following list of core references. Fourman Scott 77] Mac Lane Moerdijk 92] Johnstone 77 ] Troelstra van Dalen 88] Wyler 91] Fourman 74] and also [Fourman 77] Rosolini 80] Ambler 92] and [Nawaz 85] The intention behind the notion of an m set is to model sets in a constructive universe with truth values in . Thus operations like equality (between members of sets) and .... ....on or internally, using logical connectives instead. We use the logic of forcing over , that is, the logic in h( For a good reference to forcing over a cHa, see [Troelstra van Dalen 88] volume II, and also in general for internal logic in a topos, for instance [Mac Lane Moerdijk 92] Johnstone 77 ] or [Fourman 74] 69 Definition 4.

34. Mathematical Topos - Wikipedia The historical origin of topos theory is algebraic geometry. References John Baeztopos theory in a nutshell, http//math.ucr.edu/home/baez/topos.html. http://www.wikipedia.org/wiki/Mathematical_topos

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Mathematical topos From Wikipedia, the free encyclopedia. A topos (plural: topoi) in mathematics 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 . 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 the

35. Untitled Research Areas Category theory, topos theory, topology. It has natural connectionswith topos theory. See the Seminar of the Rough Set Technology Lab. http://www.math.uregina.ca/~funk/research.html

Research Areas Category theory, topos theory, topology Current Research (newest to oldest) Toposes and rough set theory: The idea of a rough set comes from computer science. It has natural connections with topos theory. See the Seminar of the Rough Set Technology Lab Inverse semigroups and etendue: Joint with David Cowan. Braid group orderings: Patrick Dehornoy has discovered that an Artin braid group carries a left-invariant linear ordering. In my article ``The Hurwitz action and braid group orderings,'' Theory and Applications of Categories , Vol. 9 (2001), No. 7, pp 121-150, a ramified covering space is used to find a linear ordering of a countably generated free group, which is not invariant under the product in the free group, and an action of the countably generated Artin braid group in the free group that preserves the ordering. Cosheaf spaces and ramified covers: A theory of branched covers in topos theory, which is based on the ideas of Ralph Fox, is developed in ``On branched covers in topos theory,'' Theory and Applications of Categories , Vol. 7 (2000), pp 1-22. This approach uses complete spread geometric morphisms.

36. Casual Category Theory - Fall 2000 Tuesday 1215 24/10/2000 (Luigi Santocanale), Introduction to topos theoryI'll define elementary toposes and introduce their basic properties. 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.

37. HallMathematics.com Sketches Of An Elephant A Topos Theory HallMathematics.com Sketches of an Elephant A topos theory Compendium(Oxford Logic Guides, 43 44). HallMathematics.com. the http://hallmathematics.com/index.php/Mode/product/AsinSearch/019852496X/name/Ske

38. Categorical Logic Using sheaves, topos theory also subsumes Kripke semantics for intuitionisticlogic. Prerequisites. 80413/713 Category Theory, or equivalent. http://www.andrew.cmu.edu/user/awodey/catlog/

Categorical Logic Fall 2002 Course Information Instructor: Steve Awodey Office: Baker 152 (mail: Baker 135) Office Hour: Thursday 1-2, or by appointment Phone: 8947 Email: awodey@andrew Secretary: Baker 135 Overview This course focuses on applications of category theory in logic and computer science. A leading idea is functorial semantics, according to which a model of a logical theory is a set-valued functor on a structured category determined by the theory. This gives rise to a syntax-invariant notion of a theory and introduces many algebraic methods into logic, leading naturally to the universal and other general models that distinguish functorial from classical semantics. Such categorical models occur, for example, in denotational semantics. In this connection the lambda-calculus is treated via the theory of cartesian closed categories. Similarly higher-order logic is modelled by the categorical notion of a topos. Using sheaves, topos theory also subsumes Kripke semantics for intuitionistic logic. Prerequisites 80-413/713 Category Theory, or equivalent.

39. Abstract:010801bunge In this lecture I intend to introduce the notion of a (Fox complete)spread in topology and then in topos theory. This requires http://www.maths.usyd.edu.au:8000/u/stevel/auscat/abstracts/010808bunge.html

Spreads and their completions Marta Bunge (1/8/01) The notion of a (complete) spread was introduced by R.H. Fox [6] in topology in order to give a common generalization of two different types of coverings with singularities (branched and folded). A different notion of a (proper) spread was given by E. Michael [10] in connection with topological cuts. In both cases, the basic ideas is that of a spread , meaning a continuous map p:Y>X , with Y locally connected, satisfying the property that the connected components (or more generally, the clopen subsets) of the p (U) , for U the opens of X , form a base for the topology of Y A topos-theoretic version of the notion of a spread was given by J. Funk and myself [3] as that of a geometric morphism p:F>E between toposes bounded over a base topos S , with F locally connected, for which there is a generating family a:f>p (E) of F over E which is an S-definable The two types of completions (Fox [6] and Micheal [10]) have a topos-theoretic counterpart, which, unlike the notion of a spread, are far from obvious. We deal with the Fox-like completion in [3] and with the Michael-type completion in [5]. An instance of complete spreads are the branched coverings, introduced in this context in [4] and (with minor variations) in [8]. On the other hand, the notion of folded cover has not been defined for toposes.

40. Subject Classification 61, SpringerVerlag (1968), 41-61. 19. Review of PM Cohn's Universal Algebra,2nd Edition, American Scientist (May-June 1982), 329. topos theory. 11. 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), 329. 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, Springer-Verlag (1972), 1-12 Continuously Variable Sets: Algebraic Geometry = Geometric Logic Proceedings of the Logic Colloquium , Bristol (1973), North Holland (1975), 135-157. Introduction to Part I of Model Theory and Topoi Springer Lecture Notes in Mathematics No.445

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