81. Untitled Document Johnstone, topos theory Johnstone, Sketches of an Elephant Lawvere and Schanuel, Conceptual Mathematics Mac Lane, Categories for the Working Mathematician http://dwispc10.vub.ac.be/topoi.html

Topos Theorie Een uiterst informeel seminarie met als doel topos theorie te begrijpen en te kunnen toepassen. De lezingen worden beurtelings verzorgd door de deelnemers. Neem in geval van vragen, opmerkingen of drang tot een lezing contact op met Tim Van der Linden In principe is het seminarie tweewekelijks en heeft het plaats op dinsdag van 17 tot 19 uur in lokaal 10F734. De volgende lezingen hebben plaatsgevonden of zijn reeds aangekondigd: 12/10 Tim: "Inleiding" 26/10 Tor: "Schoven over een topologische ruimte I" Tor: "Schoven over een topologische ruimte II" 3/12, 16u Tim: "Elementen I" 17/12, 16u Tor: " Schoven over een topologische ruimte III Tomas: "Elementaire topoi I" 5/1, 16u Frank: "Gentlemen! Is there really enough room in Hilbert space? An operational view on classical and quantum theories. 21/1, 16u Rudger: "Elementaire topoi II" 28/1, 16u Rudger: "Elementaire topoi III: exactheidseigenschappen" 1/2, 14u30 Isar: "Topoi van omega-verzamelingen" 4/2, 16u Rudger: "Logische aspecten I"

82. Historia Matematica Mailing List Archive: Re: [HM] Sets & Categories Uses and Abuses of the History of topos theory, Brit. J. Phil. Sci. 41 category theory, and in particular topos theory and its subsequent http://sunsite.utk.edu/math_archives/.http/hypermail/historia/jul99/0093.html

John Pais paisj@medicine.wustl.edu Mon, 19 Jul 1999 10:24:21 -0700 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Barry Cipra: "[HM] Abel's joke" Previous message: Prof. Lueneburg: "Re: [HM] Alternating group" Maybe in reply to: Bill Barton wrote: Dear Bill, I have included the complete quote from page 407 of Mac Lane's book below. Also, I have now had an opportunity to read Colin McLarty's, The Uses and Abuses of the History of Topos Theory, Brit. J. Phil. Sci. 41 (1990), 351-375. This is a fascinating description of the development of category theory, and in particular topos theory and its subsequent misconstrual by some as having orginally developed as an alternative to set theoretic foundations for mathematics. The paper gives one the feeling of a view into live history as the original sources include conversations with some of the developers of the subject (e.g. Mac Lane, Freyd, Lawvere), and since it illustrates how the psychological (intuitive) precedence of one concept, e.g. set, can tend to both mediate and obscure another, e.g. category.

83. Predmety - Predmety The basic notions of topos theory and their relation to the relevant notions The aim of topos theory concerning the foundations of mathematics and logic http://www.mff.cuni.cz/vnitro/is/sis/predmety/kod.php?kod=MAT044

84. PlanetMath: Structure Sheaf Also, is the footnote about topos theory correct? I m unclear what isomorphism of topos means an isomorphism of toposes, or an isomorphism within one http://planetmath.org/encyclopedia/StructureSheaf.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About structure sheaf (Definition) Let be an irreducible algebraic variety over a field , together with the Zariski topology Fix a point and let be any affine open subset of containing . Define where is the coordinate ring of and is the fraction field of . The ring is independent of the choice of affine open neighborhood of The structure sheaf on the variety is the sheaf of rings whose sections on any open subset are given by and where the restriction map for is the inclusion map There is an equivalence of categories under which an affine variety with its structure sheaf corresponds to the prime spectrum of the coordinate ring . In fact, the topological embedding gives rise to a lattice-preserving bijection between the open sets of and of , and the sections of the structure sheaf on are isomorphic to the sections of the sheaf Footnotes Those who are fans of topos theory will recognize this map as an isomorphism of topos.

85. Eduardo Ochs - Academic Research - Categories, NSA, The "Typical Point Notation" what I hope to be able to claim is this a significant part of topos theory is easy . Corollary a topos semantics for the language with infinitesimals http://angg.twu.net/math.html

Eduardo Ochs - Academic Research - Categories, NSA, the "Typical Point Notation", and a language for skeletons of proofs (...quick notes added in 2001dec26; I still haven't had time to glue the pieces together and make a coherent whole. The rest of the page is older.) Some important online texts: On Category Theory, plus some unclassified things: Barr/Wells: Toposes, Triples, and Theories (1985): http://www.cwru.edu/artsci/math/wells/pub/ttt.html Phil Scott: Some Aspects of Categories in Computer Science (1998/2000; a chapter of the "Handbook of Algebra") Pitts : Categorical Logic (2001): http://citeseer.nj.nec.com/pitts01categorical.html http://www.cl.cam.ac.uk/~amp12/papers/ ftp://ftp.cl.cam.ac.uk/papers/amp12/catl.ps.gz Thomas Streicher 's notes on Fibered Categories à là Jean Bénabou Wesley Phoa: An Introduction to Fibrations, Topos Theory, the Effective Topos and Modest Sets Caccamo/Hyland/ Winskel Lecture Notes in Category Theory [Draft] A Higher-Order Calculus for Categories Milly Maietti's home page ... Martin Hofmann : Syntax and Semantics of Dependent Types (1997): http://citeseer.nj.nec.com/hofmann97syntax.html

86. Encyclopedia: Topos Main article Background and genesis of topos theory This page gives some very general The historical origin of topos theory is algebraic geometry. http://www.nationmaster.com/encyclopedia/Topos

Supporter Benefits Signup Login Sources ... Pies Related Articles People who viewed "Topos" also viewed: Elementary topos List of category theory topics Sheaf theory Functor category ... Group action What's new? Our next offering Latest newsletter Student area Lesson plans Recent Updates Paul Laffoley Palestine Liberation Organization Oude wetering Orange (word) ... More Recent Articles Top Graphs Richest Most Murderous Most Taxed Most Populous ... More Stats Encyclopedia: Topos Updated 60 days 1 hour 54 minutes ago. Other descriptions of Topos 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 In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

87. Information Systems And The Theory Of Categories modern Kantian judgments and categories of pure and applied reasoning both analytic and synthetic on to a postmodern formalism of topos theory 3. http://computing.unn.ac.uk/staff/CGNR1/liege heather abstract 2003.htm

Information Systems and the Theory of Categories. Is Every Model an Anticipatory System? Michael. Heather, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, m.heather@unn.ac.uk B. Nick Rossiter, Informatics, Northumbria University, nick.rossiter@unn.ac.uk Keywords: information systems, theory of categories, anticipatory strength, formal models, analytic and synthetic reasoning, monads and sketches. Abstract Knowledge advances by using the known to understand the unknown. Information systems are important repositories and sources of the known but also contain the unknown by reason of yet unrealised connections between what is known. The philosophy of idealism and categories, the limits and colimits of knowledge, have developed from classical Platonic idealism and Aristotolean categories through modern Kantian judgments and categories of pure and applied reasoning both analytic and synthetic on to a postmodern formalism of topos theory [3]. Subcategory j jj j To deal with organisms as complex, not just simple mechanisms [4], modern information systems have to cope with the dynamic, open and non-local nature of knowledge beyond set theory [1,2]. This is exemplified in the current interest with sketches. The figure represents the possible unknown behaviour of a reactive system whether physical, biological or social. The change may not be fully understood but may be modelled in an information system by a corresponding behavioural change. The systems in this figure may be formalised as 2-cell categories in a topos. In the upper limiting case the universe is a reactive system and the information system belongs to it as a subcategory. Any other existing reactive system is a subcategory of the universe as a topos. If the information system is predictive it may be termed anticipatory. Where the anticipatory is a subcategory of the reactive system it is often referred to as strong anticipation. Otherwise it is weak. However the strength of an anticipatory system is not just Boolean because the internal topos logic is Heyting. There are quantitative and qualitative degrees of sameness.

88. Abstracts However, Johnstone goes on to say that the true value of topos theory is as a tool for understanding concepts in several areas of mathematics, and providing http://math.arizona.edu/~gradcolloq/fll01abs.html

August 29 Speaker - Alan Von Hermann Title - Von Neumann Regular Rings In this talk, I will give some classical examples of Von Neumann Regular Rings (VNRR) and discuss some interesting properties that such rings have. In particular, I will give a sufficient condition for a VNRR to be a division ring. Undergraduates with some background in Algebra are encouraged to attend. September 5 Speaker - Katrina Jimenez,Jennifer Christian Smith, and William Y. Velez Title - Outreach Opportunities for Graduate Students Outreach activities are a fun, rewarding way to add some spark to your resume during graduate school. We will present a variety of ways that you can get involved in outreach activities with local high school students through a developing program coordinated by graduate students in the department. Two types of outreach activities, the development and presentation of weekday mathematics workshops for high school students and "special guest" visits to local calculus classes, will be discussed. Students with VIGRE support are especially encouraged to get involved this year. Come and learn how you can get involved! September 12 Speaker - David Gay Title - How to Draw Pictures of 4-Manifolds This is partly designed as an introduction to some upcoming talks I will give in the geometry seminar, but it should also be entertaining all by itself. I will discuss "Kirby calculus", a technique for drawing pictures of 4-manifolds using knots and links; this is old-fashioned topology (no PDE's, no moduli spaces, no connections on weird bundles) but it continues to be a useful tool in modern work on 4-manifolds. I will also talk about how to add a little extra structure and use Kirby calculus to think about symplectic 4-manifolds.

89. Past Category Theory Seminars Information on current Category theory seminars in Cambridge is here. 15 October Peter Johnstone The nonclassifying topos of a first-order theory http://www.maths.gla.ac.uk/~tl/pastsems.html

Past Category Theory Seminars This is roughly speaking a list of the Category Theory seminars held at the Department of Pure Mathematics and Mathematical Statistics in Cambridge when I was there (October 1996 to September 2002). More exactly, it's a list of the ones at which I took notes and where these notes ended up in a file rather than a pile. Jump down to: My home page is here . Information on current Category Theory seminars in Cambridge is here Autumn term 1996 15 October Peter Johnstone The non-classifying topos of a first-order theory 22 October Martin Hyland Girard's completeness `theorem' for linear logic 29 October Jeff Egger Duality 5 November D Cubric Normalization via the Yoneda lemma 19 November Audrey Tan Some full completeness results for models of multiplicative linear logic 26 November Kim Wagner Enriched categories, adjoint bimodules, Cauchy completeness Spring term 1997 21 January Peter Johnstone Some remarks on Scott's category of equilogical spaces 28 January Martin Hyland What is the cyclic category? 4 February Jeff Egger Plonka's theorem 25 February Peter Johnstone The non-classifying topos, continued

90. MaMuX 2002-2003 http://recherche.ircam.fr/equipes/repmus/mamux/documents/ResumesMars.html

Samedi 20 mars 2004 Ircam, Salle I. Stravinsky Moreno Andreatta Guerino Mazzola Franck Jedrzejewski Thomas Noll : Transformational Logics in Harmonic and Metric Analysis Charles Alunni Peer Bundgaard Table ronde (avec la participation de Peter Johnstone Guerino Mazzola Following a series of sometimes very technical talks at IRCAM, I shall this time rather concentrate on the adequacy of topos theory for music, dealing with the questions of (1) formalization of musical ontology (which concept framework? which topoi? which addresses for denotators?), (2) the synthesis of aesthetics and logic (is there a common ground for beauty and truth?), and (3) languages for creativity (in the mental and technological realm). This will include comments on prominent reactions (is music THAT complicated?) relating to the mathematics used in "The Topos of Music". Franck Jedrzejewski (CEA Saclay - INSTN/UERTI) Z/12Z Thomas Noll The paper presents investigations following the strands of ideas that Guerino Mazzola and I presented in a joint talk "Extending Set Theory to Harmonic Topology and Topos Logic" at the Set Theory Conference at Ircam in october 2003. "Transformational Logics" in the title of my paper refers to the topos logics of functor categories. Functors F: C -> Sets from a small category C generalize the concept of group- or monoid actions, i.e in my applications I depart from small categories C and interpret functors F: C -> Sets music-theoretically as families of transformations on musical objects.

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