21. The Primitive Topology Of A Scheme, By Mark E. Walker We define a grothendieck topology on the category of schemes whose associatedsheaf theory coincides in many cases with that of the Zariski topology. http://www.math.uiuc.edu/K-theory/0214/

The primitive topology of a scheme, by Mark E. Walker We define a Grothendieck topology on the category of schemes whose associated sheaf theory coincides in many cases with that of the Zariski topology. We also give some indications of possible advantages this new topology has over the Zariski topology. 0214.bib (224 bytes) primitive.dvi (131488 bytes) [July 29, 1997] primitive.dvi.gz (53508 bytes) primitive.ps.gz (141420 bytes) Mark E. Walker

22. Charles Rezk's Papers And Preprints. Abstract We show that homotopy pullbacks of sheaves of simplicial sets over agrothendieck topology distribute over homotopy colimits; this generalizes a http://www.math.uiuc.edu/~rezk/papers.html

Charles Rezk's papers and preprints. Here is my cv Papers and Preprints. "Simplicial structures on model categories and functors" , with B. Shipley and S. Schwede. ( dvi .) This has appeared in American Journal of Mathematics , v.123 (2001). Abstract: We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or `continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence. "Every homotopy theory of simplicial algebras admits a proper model" dvi .) This has appeared in Topology and its Applications , v.119 (2002).

23. Sheaf - InformationBlast By precisely analyzing the properties of X needed to define sheaves, he definedthe notion of a grothendieck topology on a category (this came in a somewhat http://www.informationblast.com/Sheaf.html

Sheaf Alternate meanings: River Sheaf , King Sceaf 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 a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property , as applied to a function Sheaves are used in topology algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (i.e., depending on the open set). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as open sets, continuous, analytic, differentiable functions, and so on. For a typical example, consider a topological space X , and for every open set U in X , let F U ) be the set of all continuous functions U R . If V is an open subset of U , then the functions on U can be restricted to V , and we get a map F U F V ). "Gluing" describes the following process: suppose the

24. Sheaf :: Online Encyclopedia :: Information Genius of algebraic geometry, introducing schemes and general sheaves on them, local cohomology,the derived category (with Verdier), and the grothendieck topology. http://www.informationgenius.com/encyclopedia/s/sh/sheaf.html

Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Sheaf Online Encyclopedia Alternate meaning: River Sheaf 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 patching smaller open sets to obtain a bigger one. Table of contents 1 Introduction 2 Timeline of the history of sheaf theory 3 The formal definition 3.1 Definition of a presheaf ... 7 Generalization Introduction Sheaves are used in topology algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical object. For a typical example, consider a topological space X , and for every open set U in X , let F U ) be the set of all continuous functions U R . If V is an open subset of U , then the functions in F U ) can be restricted to V , and we get a map F U F V ). "Patching" describes the following process: suppose the U i are given open sets with union U , and for each i we are given an element f i F U i ), i.e. a continuous function

25. CS 59/93 It is proven that a class of finite automata defines a grothendieck topology andthe conditions are developed when a set of states of an automation determines http://www.cs.ioc.ee/~bibi/resrep/cs/cs59.html

Number: CS 59/93 Author(s): KALJULAID, Uno, MERISTE, Merik, PENJAM, Jaan. Title: Algebraic theory of tape-controlled attributed automata.28p. Language: ABSTRACT. Compositional theory of tape-controlled attributed automata is considered together with related developments in formal languages theory and algebra. It is proven that a class of finite automata defines a Grothendieck topology and the conditions are developed when a set of states of an automation determines a sheaf of sets of objects in the induced topological category. These two results are expected to be used in the proof that the induced fiber product of a Grothendieck topology is suitable for decomposition of tape- controlled attributed automata.

26. SmartPedia.com - Free Online Encyclopedia - Encyclopedia Books. Grothendieck, Alexander Grothendieck s Galois theory Grothendieck s Séminairede géométrie algébrique grothendieck topology Group Group http://www.smartpedia.com/smart/browse/List_of_mathematical_topics_(G-I)

Search: Math and Natural Sciences Applied Arts Social Sciences Culture ... Interdisciplinary Categories List of mathematical topics (G-I) List of mathematical topics A-C D-F G-I J-L M-O P-R S-U ... Mathematicians G G2 (mathematics) G-delta set Gabriel's horn Galilean transformation ... Gumbel, Eric H H infinity H-principle H-space Haar integral ... Hypothesis testing I IACR Icosagon Icosahedron Icosidodecahedron ... Caption This This document is licensed under the GNU Free Documentation License (GFDL), which means that you can copy and modify it as long as the entire work (including additions) remains under this license. GFDL SOURCE

27. SmartPedia.com - Free Online Encyclopedia - Encyclopedia Books. Grothendieck s Galois theory, Grothendieck s Séminaire de géométriealgébrique, grothendieck topology. Grothendieck topos, Grotius, Groton. http://www.smartpedia.com/smart/browse/Special:Allpages&from=Gresham_College

Search: Math and Natural Sciences Applied Arts Social Sciences Culture ... Interdisciplinary Categories All pages Gresham College Gresham Park, Georgia Greshams Law Greshams law ... Caption This This document is licensed under the GNU Free Documentation License (GFDL), which means that you can copy and modify it as long as the entire work (including additions) remains under this license. GFDL SOURCE

28. Research Topics Fujiwara, et al, gives a remedy for such a difficulty, changing virtually the topologicaltexture of spaces by means of grothendieck topology (in Fujiwara s http://www.math.kyoto-u.ac.jp/~kato/Research/topics.html

Research Topics Fake Projective Plane This is a compact complex algebraic surface of general type having the same betti numbers as the projective plane. Such a surface was first imagined in connection with Severi's conjecture, which expects that the projective plane would be characterized only by its topological type, and, at first, it was completely unknown as to whether such a surface really exists or not. Since then, although the first knowledge of it was anything but substantial, the fake projective plane has attracted mathematician's curiosity even after S.-T. Yau's affirmative solution to Severi's conjecture because of the fact that, as well as being looking like a mysterious of the projective plane, it satisfies the equality in the famous Miyaoka-Yau inequality. The first example of such surfaces was discovered by Mumford in 1979, well-known nowadays as Mumford's fake projective plane . Amazing is not only his discovery itself but also the way of construction by means of dyadic uniformization , which realizes the surface in question as a discrete fixed-point free quotient of a certain symmetric domain in rigid analysis. The last fact can be seen in an interesting parallelism with the fact that any possible fake projective plane, being on the Miyaoka-Yau critical line, should be realized as a discrete fixed-point free quotient of the complex unit-ball by the procedure usually referred to as a uniformization in complex analysis.

29. Morita Theory For Hopf Algebroids And Presheaves Of Groupoids Mark We prove the general theorem that internal equivalences of presheaves of groupoidswith respect to a grothendieck topology on Aff give rise to equivalences of http://claude.math.wesleyan.edu/~mhovey/papers/hopfalgebroids.abstract

Morita theory for Hopf algebroids and presheaves of groupoids Mark Hovey Wesleyan University Middletown, CT mhovey@wesleyan.edu 5/17/01 AMS classification nos: 14L05, 14L15, 16W30, 18F20, 18G15, 55N22 Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel change of rings theorems in algebraic topology.

30. PSSL 77 17.00 John Kennison (Clark) Integral domain representation and related topics Saturday10.00 Rob Goldblatt (Wellington) grothendieck topology as geometric http://www.wraith.u-net.com/PSSL/1977.html

THE PERIPATETIC SEMINAR ON SHEAVES AND LOGIC Fifth Meeting: Cambridge, 2223 January 1977 Friday Robert Seely (Cambridge) Exact functors and measurable cardinals Saturday Martin Hyland (Cambridge) Elementary constructive analysis Peter Johnstone (Cambridge) On a topological topos Chris Mulvey (Sussex) Banach spaces I Gavin Wraith (Sussex) Local equivalence of toposes Charles Burden (Sussex) Banach spaces II Sunday Martin Hyland (Cambridge) Continuity in spatial topoi Jean-Roger Roisin (Louvain) A categorical approach to model theory Sixth Meeting: Sussex, 1920 March 1977 Friday Dana Scott (Oxford) The strange story of continuous lattices Saturday Robin Grayson (Oxford) Well-orderings Jon Zangwill (Bristol) Local set theory Peter Johnstone (Cambridge) How to prove Barr's theorem Jack Duskin (Buffalo) Towards a non-abelian Kan-Dold-Puppe theorem Sunday Dana Scott (Oxford) J-operators Chris Mulvey (Sussex) The spectrum of a commutative C*-algebra Richard Lewis (Sussex) Sometimes additive functions Seventh Meeting: Lille, 45 June 1977 Saturday Rudolphe Bkouche Mike Fourman Peter Johnstone (Cambridge) A topos for topologists Michel Coste (Paris-Nord) A negative result on lim-theories Sabah Fakir (Lille) On differentially closed fields Sunday Yves Diers (Valenciennes) Locally algebraic categories Eighth Meeting: Cambridge, 1213 November 1977

31. 2 The Model Category Of D-stacks In turns out that the data of a model pretopology on a model category M is moreor less equivalent to the data of a grothendieck topology on its homotopy http://www.mimuw.edu.pl/~jacho/test/HagWord/HagV3se2.html

next prev prev-tail tail ... up The model category of D -stacks In this Section we will present the construction of a model category of D -stacks . It will be our derived version of the category of stacks that is commonly used in moduli theory, and all our examples of derived moduli stacks will be objects of this category. The main idea of the construction is the one used in HAG-I , and consists of adopting systematically the functorial point of view. Schemes, or stacks, are sheaves over the category of commutative algebras. In the same way, D -stacks will be sheaves-like objects on the category of commutative differential graded algebras. This point of view may probably be justified if one convinces himself that commutative differential graded algebras have to be the affine derived moduli spaces, and that therefore they are the elementary pieces of the theory that one would like to glue to obtain global geometric objects. Another, more down to earth, justification would just be to notice that all of the wanted derived moduli spaces we are aware of, have a reasonable model as an object in our category of

32. Intuitionistic Modal Logic Publications R. Goldblatt. grothendieck topology as Geometric Modality. Zeitschrift fuer MathematischeLogik und Grundlagen der Mathematik, 27495529, 1981. R. Goldblatt. http://www.cs.bham.ac.uk/~vdp/publications/IMLA-papers.htm

Intuitionistic Modal Logic Publications P. Aczel, The Russel-Prawitz Modality, N. Alechina and D. Shkatov, On decidability of intuitionistic modal logics , Proceedings of Methods for Modalities Workshop, 2003, Loria, Nancy, France. N. Alechina, M. Mendler, V. de Paiva and E. Ritter, Categorical and Kripke Semantics for Constructive S4 Modal Logic . In Proc. of Computer Science Logic (CSL'01), LNCS 2142, ed L. Fribourg. 2001. Sergei Artemov, Deep isomorphism of modal derivations and lambda-terms S. Awodey, L. Birkedal, D.S. Scott, Local realizability toposes and a modal logic for computability. Mathematical Structures in Computer Science, vol. 12 (2002), pp. 319-334. G. Bellin, V. de Paiva and E. Ritter, Extended Curry-Howard Correspondence for a Basic Constructive Modal Logic In Methods for Modalities 2001, Amsterdam, November 2001. Z. Benaissa, E. Moggi, W. Taha, T. Sheard A Constructive presentation for the modal connective of necessity , Journal of Logic and Computation, 2(1):31-50, 1992. M. Benevides, T. Maibaum

33. Promenade 11 The concept of a locale or of a grothendieck topology ( a preliminary formof the topos) can clearly be discerned in the wake of the scheme. http://www.fermentmagazine.org/rands/promenade11.html

Promenade 11 The Magical Spectrum - or Innocence The two powerful ideas that had the most to contribute to the initiation and development of the new geometry are schemes and toposes . Having made their appearance in a somewhat symbiotic fashion at more or less the same time. The concept of a locale or of a "Grothendieck topology" ( a preliminary form of the topos) can clearly be discerned in the wake of the scheme. This, in its turn, supplies the needed new language for ideas such as "descent" and "localisation", which are employed at every stage in the development of this theme and of the schematic tools. The more inherently geometric notion of the topos The concept of the scheme is the natural one to start with. As "self-evident" as one could imagine, it comprises in a single concept an infinite series of versions of the idea of an (algebraic) variety, that were previously used( one version for each prime number(*). (*)It is convenient to include as well the case p = "infinity", corresponding to algebraic varieties of "nul characteristic". In addition, one and the same "scheme" ( or "variety" in the new sense) can give birth, for each prime number p, to a well-defined "algebraic variety of characteristic p". The collection of these different varieties with different characteristics can thereby be seen as a kind of" (infinite) spectrum of varieties", (one for each characteristic). The "scheme" is in fact this magical spectrum, which connects between them, as so many different "branches", its "avatars", or "incarnations" in all possible characteristics. By virtue of this it furnishes an effective "principle of transition" for tying together these "varieties", arising out of geometries which, up until that point, seemed more or less isolated, cut off from each other. For the present they are all ensconced within a common "geometry" that establishes the connections between them. One might call it

34. The Lax Logic Project Page Journal of Symbolic Logic 17249265, 1952. RI Goldblatt grothendieck topologyas geometric modality. Z. Math. Logick Grundlag. Math. 27495-529, 1981. http://www.dcs.shef.ac.uk/~mattw/ll_project/main.html

The Lax Logic Project Page Personnel Matt Fairtlough (Dept. Computer Science, University of Sheffield) Michael Mendler (Dept. Mathematics and Computer Science, University of Passau) Matt Walton (Dept. Computer Science, University of Sheffield) Mike Holcombe (Dept. Computer Science, University of Sheffield) Project Description This is a collaborative project with Dr. Michael Mendler of the Department of Mathematics and Computer Science at the University of Passau. The "Lax" in "Lax Logic" indicates the looseness associated with the notion of correctness-up-to constraints. We are investigating the proof theory and model theory of both Propositional Logic (PLL) and its application to the formal verification/synthesis of combinational circuits with delays. This investigation has provided the construction of an automatic theorem prover for PLL for use in generating correctness proofs for combinational circuits. Because PLL proofs are constructive, it is possible to extract data-dependent timing constraints from them. Lax Logic also promises to be more widely applicable to the formal verification of software and of other types of hardware. Recent work has extended Lax Logic from its propositional roots to a full first-order version (QLL). This has enabled us to examine the use of QLL as a framework for capturing more general notions of constraints and also abstractions, as widely found in artificial intelligence. One application of such a framework is to the paradigm of Constraint Logic Programming (CLP). Not only can "concrete" CLP programs be expressed in QLL, but we can factor them into two parts: an abstract program (also expressed in QLL) and an associated set of constraints. QLL can then be used to recover concrete answer constraints for CLP programs using proofs of the abstract program. Thus we provide new denotational and operational semantics for CLP in a way which differs from other approaches, in that, proof-theoretically, we are not just interested with

35. Practical Foundations Of Mathematics (c) L also includes some stable colimits, encoded by a grothendieck topology J.The category S = Set C op of presheaves must be replaced by the category Shv(C,J http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/s77.html

Practical Foundations of Mathematics Paul Taylor Gluing and Completeness To complete the equivalence between syntax and semantics, it remains to prove confluence, strong normalisation for the l -calculus and the disjunction property for intuitionistic logic. The conceptual content of these results, when proved syntactically, is drowned in a swamp of symbolic detail which cannot be transferred to new situations. The remarkable construction which we use illustrates how much can be discovered simply by playing with adjoints and pullbacks. The origin of the name gluing is that this is how to recover a topological space from an open set and its complementary closed set (Exercise ). The construction for Grothendieck toposes was first set out in [ p p S U S x A or p S U A are called surjections and open inclusions respectively (geometrically, S x A is the disjoint union of S and A The gluing construction Recall from Example 7.3.10(i) that, for any functor U A S , the gluing construction is the category S U whose objects consist of I ob S G ob A and f I U G in S , and whose morphisms are illustrated by the diagram below. We shall say that (

36. Categories: Re: Effective Topos recursive sets and recursive maps (or equivalently just the monoid of recursiveendomaps of N). Its canonical grothendieck topology turns out to be finitary. http://north.ecc.edu/alsani/ct02(1-2)/msg00016.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Re: effective topos To categories@mta.ca Subject : categories: Re: effective topos From wlawvere@acsu.buffalo.edu Date : Thu, 10 Jan 2002 13:13:51 -0500 (EST) In-reply-to Pine.GSO.3.96.1020109153019.7372L-100000@lisa Reply-to wlawvere@acsu.buffalo.edu Sender cat-dist@mta.ca http://www.acsu.buffalo.edu/~wlawvere References categories: effective topos From: Prev by Date: categories: Re: lluf subcategory Next by Date: categories: effective topos Previous by thread: categories: Re: effective topos Next by thread: categories: effective topos Index(es): Date Thread

37. Ingenta: Article Summary -- Whole And Part In Mathematics Keywords Gelfand duality; grothendieck topology; mereosis; Stone duality;part; whole Document Type Research article ISSN 11221151 http://www.ingenta.com/isis/searching/ExpandTOC/ingenta?issue=pubinfobike://klu/

38. Algebra-Number Theory Seminar Abstracts (March 20) Stephen Lichtenbaum The Weilétale topology - We definea new grothendieck topology on varieties over finite fields. http://www.math.umd.edu/research/seminars/algebra/abstracts01-02.html

University of Maryland Algebra-Number Theory Seminar Abstracts (September 12) Lorenzo Ramero Almost purity and p-adic analytic geometry - I'll discuss a joint work with O.Gabber, on the foundations of "Almost ring theory". Almost rings provide a natural setup for Faltings' theory of almost etale extensions. First I'll explain what almost rings are and how one can develop "almost commutative algebra". Then I'll talk of our new approach to Faltings' "almost purity" (which is at the heart of his version of p-adic Hodge theory). Our method relies on p-adic analytic geometry and the analysis of ramification in higher rank valuation rings (while Faltings used local cohomology on higher dimensional noetherian rings). (September 19) Mohamed Saidi Galois action on tame covers - We describe the galois action of a p-adic field on tame quotients of the fundamental group of algebraic curves over p-adic fields. (October 15) Jordan Ellenberg Q -curves, modular forms, and solutions to A + B = C p - We discuss the problem of classifying all solutions to the generalized Fermat equation A + B = C p with A, B, C coprime. We show that such solutions give rise to certain elliptic curves over number fields, which are modular by a result of a author and C. Skinner, and whose mod-p Galois representations have small image. We prove the non-existence of such elliptic curves along the line of Mazur's bounds for the degree of a rational isogeny between elliptic curves over

39. Modalhistory.html Milner logic, temporal logic of concurrency, the modal mucalculus, Solovay on provabilityin arithmetic as a modality, grothendieck topology as intuitionistic http://www.mcs.vuw.ac.nz/~rob/papers/modalhistory.html

Mathematical Modal Logic: a View of its Evolution This is a survey of the origins of mathematical semantics for modal logics, and its development over the last century or so. It focuses on the algebraic semantics using Boolean algebras with operators and the relational semantics using structures often called Kripke models. Chapter 1: Introduction Chapter 2: Beginnings MacColl's algebraic analysis of modalised statements. C. I. Lewis's systems S1 - S5. Godel's work on provability as a modality. Chapter 3: Modal Algebras McKinsey's algebraic construction for the finite model property. Topological models of S4. Jonsson and Tarski's theory of Boolean algebras with operators. Could Tarski have invented Kripke semantics? Chapter 4: Relational Semantics Kripke's relatively possible worlds. So who invented relational models? The ideas of Carnap, Bayart, Meredith, Prior, Geach, Kanger, Montague and Hintikka. The place of Kripke. Chapter 5: The Post-Kripkean Boom of theSixties The Lemmon-Scott collaboration. Bull's tense algebra, Segerberg's Essay. Chapter 6: Metatheory of the Seventies and Beyond Incompleteness, decidability and complexity, first-order-definability, reduction of second-order logic to propositional modal logic, duality, canonicity.

40. Seminar Information presented. This formulation uses the notion of a sheaf over a categorywith a grothendieck topology (which is called a site). This http://math.shinshu-u.ac.jp/seminars.php.en

Seminar Information Japanese The following is our Colloquium and seminar schedule in the academic year 2004 (from April to March). Date Contents Apr 9, 2004 From 15:00 to 16:00 Speaker: Janusz Zielinski (Nicholas Copernicus University, Poland) Title: Local derivations in the Kadison sense Abstract: We give a description of all local derivations for some k-algebras. In particularly we determine which of these k-algebras are Kadison algebras. Apr 12, 2004 From 13:15 to 14:15 Speaker: Goro Kato (California Polytechnic State University) Title: Sheaves and Physics (elements of t-Topos and Quantum Gravity) Abstract: A mathematical model (or formulation) for solving fundamental problems in quantum mechanics will be presented. This formulation uses the notion of a sheaf over a category with a Grothendieck topology (which is called a site). This model explain the notions of quantum entanglement (non-locality), particle - wave duality, discrete notions of space and time in a micro (or ultra micro) level obtaining a ?"smooth?" space-time using the concept of a sheaf. It was mentioned as a possible quantum gravity at the 2nd International Conference on Topos and Theoretical Physics at Imperial College, London, July, 2003. Apr 12, 2004

Page 2     21-40 of 72    Back | 1  | 2  | 3  | 4  | Next 20