21. Br.crashed.net/~akrowne/math/sheaves.txt nerdy2 (and because of (3), we require our category to have pullbacks to applythis defn) 022557 nerdy2 so in particular, a grothendieck topology on a http://br.crashed.net/~akrowne/math/sheaves.txt

and if the elements of F(U_i) which are chosen restrict to the same elements of F(U_i and U_j) for all intersections (meaning x_i in F(U_i) and x_j in F(U_j) restrict to the same thing in F(U_i intersect U_j)), then there exists an element of F(U) which simultaneously restricts to the elements of F(U_i) [02:13:30] that completes the defn :) [02:14:05] examples, basically the F(U) often represent 'regular' functions on U, for whatever defn of regular makes sense [02:14:38] glasnost: given some topological space the idea is to give local bits of algebra. like on a manifold, one gives the ring of continuous functions to R defined on open subsets of the manifold [02:14:47] for example, let X be a topological space, and F(U) be continuous functions U -> R, with the obvious restriction maps [02:15:05] let X be a smooth manifold, and F(U) be smooth functions U -> R, also with the function restriction maps [02:15:20] glasnost: the first part says that the bits of algebra agree if you look at smaller parts. the second part says that knowing everything locally gives you global information [02:16:33] the two sheaf conditions (i.e. the two requirements involving the open cover) mean precisely that the sheaves are determined by local data [02:16:49]

22. Felix.unife.it/Root/d-Mathematics/d-Algebraic-geometry/t-Categories-in-algebraic can be applied to define a canonical functor from an analytic category to the categoryof locales, which is a special type of grothendieck topology (ie framed http://felix.unife.it/Root/d-Mathematics/d-Algebraic-geometry/t-Categories-in-al

From e-prints@e-math.ams.org Fri Jun 5 12:48:55 1998 Date: Fri, 5 Jun 1998 01:00:33 -0400 From: AMSPPS

23. Re: Category Theory the characteristic p case and other such theories For example, instead of justhaving a set theoretic topology, one uses a grothendieck topology defined in http://www.mail-archive.com/haskell@haskell.org/msg02882.html

haskell Chronological Find Thread Re: category theory From: Hans Aberg Subject: Re: category theory Date: Thu, 15 Oct 1998 16:42:08 +0200 mailto:haberg@member.ams.org http://www.matematik.su.se/~haberg/ http://www.ams.org/cml/ Prev by Date: Re: category theory Next by Date: Re: category theory Prev by thread: Re: category theory Next by thread: Re: category theory Index(es): Main Thread Reply via email to

24. Regarding "realizability, Covers And Sheaves, ..." For this purpose, we believe that the notion of a cover algebra (a specialcase of a grothendieck topology) is particularly well suited. http://www.cis.upenn.edu/~bcpierce/types/archives/1993/msg00077.html

[Prev] [Next] [Index] [Thread] Regarding "realizability, covers and sheaves, ..." To types@dcs.gla.ac.uk Subject : Regarding "realizability, covers and sheaves, ..." From jean@saul.cis.upenn.edu Date : Mon, 24 May 1993 13:36:13 EDT Approved : types@dcs.gla.ac.uk Prev: linear topology, hopf algebras and *-autonomous categories Next: What's new? Index(es): Main Thread

25. 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

26. 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

27. [algebra Seminar] Freddy Van Oystaeyen the noncommutative geometry in terms of C*algebras, we develop noncommutative topologyboth in the analytic sense or in the sense of grothendieck topology. http://listas.math.ist.utl.pt/pipermail/algebraist/2002/000042.html

[algebra seminar] Freddy van Oystaeyen Pedro Resende pmr@math.ist.utl.pt Wed, 23 Oct 2002 06:01:18 +0100 Previous message: [algebra seminar] proxima 6a feira Next message: [algebra seminar] this wednesday Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Algebra Seminar Thursday, October 31, 14h, Room P3.10 Noncommutative topology and geometry Freddy van Oystaeyen (University of Antwerp) Abstract: Trying to discover a unifying theory connecting the noncommutative geometry of schematic algebras to the classical basis for quantum mechanics and the noncommutative geometry in terms of C*-algebras, we develop noncommutative topology both in the analytic sense or in the sense of Grothendieck topology. The relation with the lattice of linear closed subspaces of a Hilbert space H is indicated. Function theory in a quaternion variable viewed as two noncommuting complex variables is developed and leads to noncommutative Riemann manifolds and other examples of noncommutative manifolds. Previous message: [algebra seminar] proxima 6a feira Next message: [algebra seminar] this wednesday Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

28. Lambda Definability With Sums Via Grothendieck Logical Relations - Fiore, Simpso 0 1 ; w 0 l ; wj 1 ; wkg 2 S This definition appears to be related tothe notion of a grothendieck topology (Fiore and Simpson, 1999). http://citeseer.nj.nec.com/55838.html

Lambda Definability with Sums via Grothendieck Logical Relations (1999) (Make Corrections) (2 citations) Marcelo Fiore, Alex Simpson Typed Lambda Calculus and Applications Home/Search Context Related View or download: dcs.ed.ac.uk/home/als/Resea glr.ps.gz dcs.ed.ac.uk/~als/Research/glr.ps.gz Cached: PS.gz PS PDF DjVu ... Help From: dcs.ed.ac.uk/home/als/Research (more) From: dcs.ed.ac.uk/~als/Research/ Homepages: M.Fiore A.Simpson HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: . We introduce a notion of Grothendieck logical relation and use it to characterise the definability of morphisms in stable bicartesian closed categories by terms of the simply-typed lambda calculus with finite products and finite sums. Our techniques are based on concepts from topos theory, however our exposition is elementary. Introduction The use of logical relations as a tool for characterising the -definable elements in a model of the simply-typed -calculus originated in the work of... (Update) Context of citations to this paper: More 0 1 ; w l ; w j 1 ; w k g 2 S: This definition appears to be related to the notion of a Grothendieck topology

29. Www.infomag.ru8082/dbase/B003E/951108-004.txt symmetries. Then there are *really* highpowered things like topoi ofsheaves on a category equipped with a grothendieck topology . And http://www.infomag.ru:8082/dbase/B003E/951108-004.txt

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. Colloquium Of The Mathematical Sciences Group, Department Of Computer Science, S Abstract In this talk we will show that the class of maps generated by densesubobjects for a given grothendieck topology on a partially ordered set P http://math.usask.ca/fvk/coll01.htm

COLLOQUIUM of the Mathematical Sciences Group at the Department of Computer Science University of Saskatchewan Colloquium chair: Salma Kuhlmann Colloquium Talks 2001/2002 Friday, November 30, 2001, 3:30 p.m. Professor Gordon Sarty Department of Psychology , Saskatoon gave a talk on BOLDfold for fMRI: Statistics or Pattern Recognition? This event was organized jointly with the Cognitive and Neurosciences Seminar Department of Psychology. Friday, January 4, 2002, 3:30 p.m. Professor Eric Neufeld Department of Computer Science , Saskatoon gave a talk on A variation on the Puzzle of the Two Envelopes Friday, January 11, 2002, 4:00 p.m. Professor Patrick Browne Saskatoon gave a talk on Sturm-Liouville problems with eigenparameter dependent boundary conditions Abstract: This talk gave an overview of results concerning existence and asymptotes of eigenvalues, and oscillation theory for Sturm-Liouville problems of the type -(py')' + qy = l ry with boundary conditions at x=0: (py'/y)(0) = cot a at x=1: (py'/y)(1) = g( l for various functions g( l Friday, January 25, 2002, 3:30 p.m.

32. Goldblatt, Robert: Mathematics Of Modality Arithmetical Necessity, Provability and Intuitionistic Logic 5 Diodorean Modalityin Minkowski Spacetime 6 grothendieck topology as Geometric Modality 7 The http://www.press.uchicago.edu/cgi-bin/hfs.cgi/00/12591.ctl

Go to ... Full text search Excerpts Subject catalogs Subject index ... Shopping cart contents or Print an order form Goldblatt, Robert Mathematics of Modality . Distributed for the Center for the Study of Language and Information. 274 p. 1993 Series: (CSLI-LN) Center for the Study of Language and Information-Lecture Notes Cloth $54.95tx 1-881526-24-0 Spring 1994 Paper $24.95tx 1-881526-23-2 Spring 1994 Modal logic is the study of modalitiesexpressions that qualify assertions about the truth of statementslike some ordinary language phrases and mathematically motivated expressions. The study of modalities dates from antiquity, but has been most actively pursued in the last three decades. This volume collects together a number of Golblatt's papers on modal logic, beginning with his work on the duality between algebraic and set-theoretic models, and including two new articles, one on infinitary rules of inference, and the other about recent results on the relationship between modal logic and first-order logic. Table of Contents Introduction 1: Metamathematics of Modal Logic 2: Semantic Analysis of Orthologic 3: Orthomodularity is not Elementary 4: Arithmetical Necessity, Provability and Intuitionistic Logic

33. Publications with by a small extension of a result of Raynaud and Gruson showing that the imageof a flat map of affinoid varieties is open in the grothendieck topology. http://www.math.ohio-state.edu/~schoutens/publications.html

Publications This page has been accessed 1093 times. Downloads Click on the icon to obtain a postscript file (requires a laser printer for printing or a Ghostview program for viewing). Click on the icon to obtain a dvi file (requires a TeX program to view and print). Click on the icon to see the file online (via Acrobate Reader). Abstracts The more recent articles have an abtsract written in a TeX-patois (with some AMS-LaTeX words thrown in). Some papers are listed on ArXiv and a link is provided to that site, from which the paper can be downloaded. Books Rigid subanalytic sets (under constructionthe plan is to eventually publish this book in the Asterisque series) Table of contents: Journals and Proceedings In print Number of generators of a Cohen-Macaulay ideal , J. Algebra Computing the minimal number of equations defining an affine curve ideal-theoretically , J. Pure Applied Alg. There is an algorithm which computes the minimal number of generators of the ideal of a reduced curve $C$ in affine $n$-space over an algebraically closed field $K$, provided $C$ is not a local complete intersection. A non-standard proof of the Briancon-Skoda Theorem , Proc. Amer. Math. Soc.

34. 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.qmul.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 (

35. GROTHENDIECK'S RECONSTRUCTION PRINCIPLE AND 2-DIMENSIONAL TOPOLOGY AND GEOMETRY doi10.1142/S0219199799000079. grothendieck'S RECONSTRUCTION PRINCIPLEAND 2DIMENSIONAL topology AND GEOMETRY. FENG LUO Department http://www.worldscinet.com/ccm/01/0102/S0219199799000079.html

What's New New Journals Browse Journals Search ... Advertising Enquiries Communications in Contemporary Mathematics, Vol. 1, No. 2 (1999) 125-153 doi:10.1142/S0219199799000079 GROTHENDIECK'S RECONSTRUCTION PRINCIPLE AND 2-DIMENSIONAL TOPOLOGY AND GEOMETRY FENG LUO Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA No abstract received. PDF SOURCE (1,750 k) Back to Contents of Vol. 1, No. 2

36. Grothendieck During this period grothendieck's work provided unifying themes in geometry,number theory, topology and complex analysis. He introduced http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html

Alexander Grothendieck Born: 28 March 1928 in Berlin, Germany Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Alexander Grothendieck In 1949 Grothendieck moved to the University of Nancy where he worked on functional analysis with . He became one of the Bourbaki group of mathematicians which included Weil Henri Cartan and . He presented his doctoral thesis Grothendieck spent the years 1953-55 at the University of Sao Paulo and then he spent the following year at the University of Kansas. However it was during this period that his research interests changed and they moved towards topology and geometry. In fact during this period Grothendieck had been supported by the Centre National de la Recherche Scientifique, the support beginning in 1950. After leaving Kansas in 1956 he therefore returned to the Centre National de la Recherche Scientifique. However in 1959 he was offered a chair in the newly formed Institut des Hautes Etudes Scientifiques which he accepted. In [2] the next period in Grothendieck's career is described as follows:- It is no exaggeration to speak of Grothendieck's years algebraic geometry , and him as its driving force. He received the

37. Steve Vickers's WWW Home Page pour les vraiment nuls is a further introduction to grothendieck's idea of toposesas generalized topological spaces, using geometric logic to hide topology. http://mcs.open.ac.uk/sjv22/

Steven Vickers Department of Pure Maths The Open University , Walton Hall, Milton Keynes, MK7 6AA, England. s.j.vickers@open.ac.uk Tel: 01908 653144 I work in the area of Theoretical Computer Science, with particular interests in geometric logic, topology and topos theory. I have recently moved from the Department of Computing at Imperial College , where I worked in the Theory and Formal Methods group. You can download many of my papers , and also theses of some of my PhD students You can view the talk " Schemas as Toposes " that I gave at the OU on 2nd July 2002. Geometric Logic Most of my work is in investigating "geometric logic" (so-called from its origins in algebraic geometry). This has some unusual properties. First, it makes a hard distinction between formulae and axioms: a logical formula is restricted in the connectives it can use, to conjunction, disjunction, equality and existential quantification, and an axiom (for a geometric theory) expresses relationships between formulae in the form "for all x, y, z, ... (formula1 - formula2)". Hence the missing connectives can be introduced in just one layer - no nesting - in axioms. Second, it allows infinite disjunctions in formulae. Though these features may look weird, they have some conceptual justification in the idea of observation: formulae correspond to observable facts (and infinite disjunction is not a problem observationally) while the axioms correspond to background assumptions or scientific hypotheses.

38. 2 The Grothendieck-Riemann-Roch Theorem If K 0 (X) is the grothendieck ring of vector bundles on X, the Chern character(defined using Chern classes by the same formula as in topology) gives a ring http://www.imsc.ernet.in/~kapil/papers/harishconf/node3.html

Next: 3 Divisors on varieties Up: Algebraic Cycles Previous: 1 Model case of 2 The Grothendieck-Riemann-Roch theorem Let X be a non-singular variety over k . An algebraic cycle of codimension p is an element of the free Abelian group on irreducible subvarieties of X of codimension p ; the group of these cycles is denoted Z p X ). As in the case of curves one can introduce the effective cycles Z p X which is the sub-semigroup of Z p X ) consisting of non-negative linear combinations. There is a subgroup R p X Z p X ), defined to be the subgroup generated by all the cycles div f W where W ranges over irreducible subvarieties of codimension p - 1 in X , and f k W . The quotient X Z p X R p X ) is called the Chow group of codimension p cycles on X modulo rational equivalence; if n = dim X then we use the notation X X ). For p = 1 and X a smooth projective curve the Chow group X ) is precisely the class group X ) introduced above. The generalisation of Schubert calculus on the Grassmannians is the intersection product X X X making X X ) into an associative, commutative, graded ring, where X Z , and X ) = for p X . The Chow ring is thus an algebraic analogue for the even cohomology ring X Z ) in topology. A refined version of this analogy is examined in Section 6. In any case we note the following `cohomology-like' properties.

39. 1.4 Historical Notes spectral sequences of (grothendieck) topology are derived from the same theoremof Homological Algebra (which is , if maps injectives to -acyclic objects). http://home.t-online.de/home/berndt.schwerdtfeger/flat/flatnode7.html

40. Partners: Functional Analysis And Topology By Lawrence Narici topology to inspire functional analysis and it was progress in general topology throughout1930 in the works of Mackey (14, 15), and grothendieck (10, 11 http://at.yorku.ca/t/a/i/c/42.htm

Topology Atlas Document # taic-42 Topology Atlas Invited Contributions vol. 6 issue 1 (2001) p. 4-6 Partners: Functional Analysis and Topology Lawrence Narici Department of Mathematics and Computer Science, St. John's University, Jamaica, NY 11439, USA NARICIL@STJOHNS.EDU Amazon.com page for Functional Analysis by G. Bachman and L. Narici, Dover, Mineola, New York, 2000, a reprint of the the 1966 Academic Press book of the same title. See also the invited contribution What is functional analysis? by the same author. Introduction Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space-the space of continuous linear functionals-of a normed space played an especially important role in the formative years of functional analysis. To further this endeavor, many new kinds (weak, strong, etc.) of convergence and compactness were introduced . Metric and general topological spaces evolved in order to provide a framework in which to treat these types of convergence. As general topology gestated, many concepts were greatly clarified and simplified. (For example, "continuous" meant transforming convergent sequences into convergent sequences until about 1935.) These clarifications led to the development of general topological vector spaces in the 1930's. Beginnings As set theory developed at the end of the nineteenth century, its paradoxes revealed that mathematics had a disturbingly shaky foundation. With the aim of placing set theory in particular and mathematics generally on a firmer logical pedestal, Hilbert and others looked to Euclidean geometry for a model [

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