1. Grothendieck Topology - Wikipedia, The Free Encyclopedia In category theory, a branch of mathematics, a grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a http://en.wikipedia.org/wiki/Grothendieck_topology

Grothendieck topology From Wikipedia, the free encyclopedia Jump to: navigation search In category theory , a branch of mathematics , a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space . Grothendieck topologies axiomatize the notion of an open cover . Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology . This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the ©tale cohomology of a scheme . It has been used to define other cohomology theories since then, such as l-adic cohomology flat cohomology , and crystalline cohomology . While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate 's theory of rigid analytic geometry There is a natural way to associate a category with a Grothendieck topology (a site ) to an ordinary topological space , and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety , this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the

2. Grothendieck Topologies « Rigorous Trivialities A grothendieck topology (note, the wikipedia article defines Grothendieck Topologies differently than I do) on a category is a collection of sets of http://rigtriv.wordpress.com/2007/09/17/grothendieck-topologies/

var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www."); document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E")); Rigorous Trivialities Home About Grothendieck Topologies ... metric topology (usually on ), followed by point set topology . Then in graduate school, pretty much everyone gets an introduction to algebraic topology . However, it all really just studies sets with distinguished collections of subsets. There is, however, a more general version of topology, in fact, a way to put a topology onto a category due to, of course, Grothendieck So the first question is: how should topologies be defined? The traditional way is that, given a set , a topology is a collection of subsets satisfying the following properties: 2. If with an arbitrary index set, then 3. If then algebraic geometry When algebraic geometers work over the complex numbers, they get two topologies to work with, the classical topology, which has the small open sets that analysts and differential geometers love so much. Also, though, there is the Zariski topology Unfortunately, when algebraic geometers decide to work in positive characteristic, say in the algebraic closure of

3. IngentaConnect Grothendieck Topology, Coherent Sheaves And Serre's Theorem For S We construct a generalised grothendieck topology for the free monoid on all Oresets of a schematic algebra R. This allows us to develop a sheaf theory http://www.ingentaconnect.com/content/els/00224049/1995/00000104/00000001/art001

Home About Ingenta Ingenta Labs Ingenta Blog ... Check our FAQs Or contact us to report problems with: Subscription access Article delivery Registration Library administrator tasks ... Other problems For Publishers Why go online? Why choose IngentaConnect? Beyond Print: enhancing your service Access and authentication ... Contact us For Researchers For Authors About IngentaConnect Search and browse Publications available ... Register For Librarians Resource Zone Why choose IngentaConnect? Why choose IngentaConnect Complete? Activating Subscriptions ... Register CM8ShowAd("HorizontalBanner"); Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras Authors: Van Oystaeyen F. ; Willaert L. Source: Journal of Pure and Applied Algebra , Volume 104, Number 1, 13 October 1995 , pp. 109-122(14) Publisher: Elsevier view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper"); Language: English Document Type: Research article DOI: Affiliations: Department of Mathematics and Computer Science, University of Antwerp U.I.A., Universiteitsplein 1, 2610 Antwerpen, Belgium

4. Derived Categories For Dummies, Part I | The String Coffee Table The answer is called grothendieck topology (def. 2.24 in Vistoli’s review). A category is said to have a grothendieck topology when its objects and http://golem.ph.utexas.edu/string/archives/000528.html

@import url("/string/styles-site.css"); The String Coffee Table A Group Blog on Physics Skip to the Main Content Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. Main March 13, 2005 Derived categories for dummies, Part I Posted by urs I am still trying to learn about derived categories, mostly using P. Aspinwall: D-Branes on Calabi-Yau Manifolds hep-th/0403166 First of all: what is a derived category anyway? The short answer is: The derived category D of any abelian category is the category of complexes in An abelian category is a category equipped with the necessary structure so as to make it possible to have sequences of morphisms in that category which form a complex . So in an abelian category we can have sequences d n n d n n d n n d n with all the n being objects and all the d n n being morphisms in is zero Furthermore, we can speak of the n-th

5. Grothendieck Topology - Wikipedia A grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of http://nostalgia.wikipedia.org/wiki/Grothendieck_topology

Grothendieck topology From Wikipedia HomePage Recent changes View source Page history ... Log in Special pages Double redirects Broken redirects Disambiguation pages Log in Preferences My watchlist Recent changes Upload file File list Gallery of new files User list Statistics Random page Orphaned pages Uncategorized pages Uncategorized categories Uncategorized files Uncategorized templates Unused categories Unused files Wanted pages Wanted categories Most linked to pages Most linked to categories Most linked-to templates Pages with the most categories Most linked to files Pages with the most revisions Pages with the fewest revisions Short pages Long pages New pages Oldest pages Dead-end pages Protected pages Protected titles All pages Prefix index List of blocked IP addresses and usernames User contributions What links here Book sources Categories Export pages Version System messages Logs MIME search List redirects Unused templates Random redirect Pages without language links File path List of Wikimedia wikis Expand templates CategoryTree Gadgets Parser diff test Search Cross-namespace links Wikimedia Board of Trustees election Search web links Printable version A Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C , and with that the definition of general cohomology theories. A category together with a Grothendieck topology on it is called a

6. Atlas: Grothendieck Topologies And Ideal Closure Operations - The Integral Closu These closures are induced by sheafification in a suitable grothendieck topology. The viewpoint of Grothendieck topologies does also yield concepts of http://atlas-conferences.com/c/a/r/v/21.htm

Atlas home Conferences Abstracts about Atlas Conference on Valuation Theory and Integral Closures in Commutative Algebra July 16-22, 2006 Fields Institute and University of Ottawa Ottawa, Ontario, Canada Organizers Steven Dale Cutkosky, Sara Faridi, Franz-Viktor Kuhlmann, Irena Swanson View Abstracts Conference Homepage Grothendieck topologies and ideal closure operations - The integral closure and the submersive topology by Holger Brenner University of Sheffield Coauthors: Manuel Blickle This is a report on ongoing work relating closure operations for ideals and submodules like the radical, tight closure, integral closure, plus closure etc. to non-pure (in particular non-flat) Grothendieck theories. These closures are induced by sheafification in a suitable Grothendieck topology. The viewpoint of Grothendieck topologies does also yield concepts of stalks, of exactness, of module of global sections and of a cohomology theory. Date received: May 15, 2006 Atlas Conferences Inc. Document # carv-21.

7. Springer Online Reference Works grothendieck topology. A structure of coverings on a category, making it possible to define the notion of sheaf. For details, see Site. http://eom.springer.de/g/g045180.htm

Encyclopaedia of Mathematics G Article referred from Article refers to Grothendieck topology category , making it possible to define the notion of sheaf . For details, see Site This text originally appeared in Encyclopaedia of Mathematics - ISBN 1402006098

8. Wiley InterScience :: Session Cookies Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://doi.wiley.com/10.1002/malq.19810273104

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9. Uspekhi Matematicheskikh Nauk Geronimus, Lie groups and grothendieck topology, UMN, 1971, 261(157), 219–220. Linking options; http//mi.mathnet.ru/eng/umn5172 http://www.mathnet.ru/php/journal.phtml?wshow=paper&jrnid=rm&paperid=5172&year=1

10. Noncommutative Topology (1) At Neverendingbooks He then went on to properly define what a noncommutative topology (and even more generally a noncommutative grothendieck topology) should be by using this http://www.neverendingbooks.org/index.php/noncommutative-topology-1.html

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11. Grothendieck Topology Translate this page grothendieck topology. Grothendieck Grothendieck http://homepage.mac.com/yenlung/WebWiki/GrothendieckTopology.html

Grothendieck Topology ³o¸Ì§Ú­Ì­n¤¶²Ð¥N¼Æ´X¦ó±`¥Îªº¡u©Ý¾ë¡v¡ßGrothendieck©Ý¾ë¡C¨Æ¹ê¤W§Ú»¡±`¥Î¦³ÂI´ÛÄFªº¶ûº¡A¦]¬°«Ü¦h¥N¼Æ´X¦ó¨ä¹ê¤]¤£¥ÎGrothendieck©Ý¾ë¡C¤j·§ªº¨Ó»¡¡AGrothendieck©Ý¾ë¦³¨â¤j¯S©Ê¡G «Ü®e©ö©w¸q¥Xsheaf¡A¥i¥H¨Ï¥Îcohomology²z½×¡]¦pªGª¾¹Dmotives²z½×ªº¤H¡A´Nª¾¹DGrothendieck¤ß²z·Q¤°»ò¤F¡^ ¹ï©ó¬Y¤@­ÓGrothendieck©Ý¾ë¡A©Ò¦³¦b¨ä¤Wªºsheavesªº¶°¦X¡A¤]¦³Grothendieck©Ý¾ë ¨Æ¹ê¤W©O¡AGrothendieck©Ý¾ë©M§Ú­Ì­ì¨Ó·Qªº©Ý¾ë¤]¨S¦³®t¤Ó¦h¡A¦ý¬O¤£¤Ó¤@¼Ë¡C°ò¥»¤W§Ú­Ì¦³ ¥H«e©Ò¦³ªº©Ý¾ë¡A³£¬OGrothendieck©Ý¾ë Grothendieck©Ý¾ë¨¤£¤@©w¬O´¶³qªº©Ý¾ë ´«¥y¸Ü»¡©O¡A§Ú­Ì¥i¥H·Q¦¨§â¥H«eªº©Ý¾ë§ó¤@¯ë¤Æ¤F¡C ²³æªº»¡¡AGrothendieck©Ý¾ë´N¬O¥Î¥Î¡u½d¥¡v(category¡^ªº²z½×¨Ó¬Ý©Ý¾ë¡C §Ú­Ì«ç»ò¼Ë¥Î½d¥¨Ó©w¸q¤@¼ËªºªF¦è©O¡H§Ú­Ìª¾¹D¡A¤@­Ó½d¥¥]¬A¨â¤jþªºªF¦è¡G ¬M«¬¡]morphisms¡^ ¦n¡AC(X)³o­Ó½d¥¬O¦³¤F¡C¤£¹L¡A§Ú­ÌÁÙ­n¤@ÂI§V¤O¡A¤~¯à»¡¥¦¬O¤@­Ó©Ý¾ë¡C §Ú­Ì¦^¨ý¤@¤U¥H«eªº©Ý¾ë©w¸q¡C§Ú­Ì¦³ ¥©§®ªº¦a¤è´N¬O³o¸Ì°Õ¡C Grothendieck©Ý¾ë ±q¥t¤@­Ó¨¤«×¨Ó¬Ý¬°¤°»ò§Ú­Ì­nGrothendieck©Ý¾ë «e­±¬Ý¨Ó³£¬O«Ü¾÷¥©ªº©w¸q¡C¦³½ì¬O®¼¦³½ìªº¡A¦³·N«ä¤]¥i¯à¦³·N«ä¡C¦ý¬O¡A¬°¤°»ò§Ú­Ì­n³o¼Ë¥h©w¸q©O¡A«o¤£¬O¤Q¤Àªº©ú¥Õ¡C Updated: 2003-04-18 Home Index

12. Seminars And Talks 27July-05 (MFG) What is a grothendieck topology? I may write up a set of notes covering this and the previous talk (below). http://www.cs.man.ac.uk/~hsimmons/newslides.html

Seminars and talks Over the years I must have given hundreds of seminars, talks, and short research courses. Most of these are lost forever. In January 03 I started to keep a record of these. Here is a table of recent seminars where you can find links to the slides and relevant notes where available. Here is a brief description of the content these talks. List of seminars given in reverse chronological order 27-may-07 (Manchester) Galois connections done properly and Classical rings of fractions (two separate topics) Slides available for first part. 15-May-07 and 22-may-07 (Manchester) From rings of fractions to localizations A wander around these topics as part of a longish teaching seminar. Slides available 02-May-07 (Leeds) Is the Ackermann function optimal? A standard construction, originally due to Ackermann, produces a recursive function that is not primitive recursive. This can be relativized to produce a jump on the poset of degrees up to primitive recursive equivalence. What is there between a degree and its jump? Quite a lot. Slides and a write-up available.

13. Research Topics For instance, the topology of such fields, being totally disconnected, prevents one of spaces by means of grothendieck topology (in Fujiwara s approach, 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.

14. Elemental Principles Of T-topos Let S be a site, namely, a category with a grothendieck topology and let $\hat{S}$ i) See 3 or 5 for Grothendieck topologies which is sufficient for http://stacks.iop.org/0295-5075/68/i=4/a=467/html

15. CS 59/93 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 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.

16. Week68 Then there are really highpowered things like topoi of sheaves on a category equipped with a grothendieck topology . And so on not an attempt to pick 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

17. Grothendieck のアイデアから発展した� Translate this page grothendieck topology topos Mac Lane Moerdijk MLM96, MLM94 Borceux Handbook Bor94 http://pantodon.shinshu-u.ac.jp/topology/literature/Grothendieck.html

Your language? Japanese English Mar, 2008 Sun Mon Tue Wed Thu Fri Sat Updated Pages date page Mar 14 stable_homotopy.html Mar 13 poset_topology.html Mar 13 poset_topology.html Mar 13 Mar 13 Mar 13 Mar 13 Mar 13 Mar 12 A-infinity.html Mar 11 Mar 11 Mar 08 combinatorial_model.html Mar 08 mapping_space_models.html Mar 08 mapping_space.html Mar 08 CO.html Mar 08 mapping_space.html Mar 07 unstable_chromatic.html Mar 07 unstable_chromatic.html Mar 07 Dror_localization.html Mar 07 objectives precautions index search Grothendieck のアイデアから発展した分野 を近代化したのは、もちろん Grothendieck の業績である。代数幾何にとどま らず、 1970 年に IHES を辞めてからも、 Grothendieck の数学は様〠な分野に影響を与え 続けている。 Grothendieck については、山下純一氏による伝記 Grothendieck Circle という site もあ る。やはり Grothendieck の数学のファンは多いようである。 Cartier による Bulletin of A.M.S. の記事 ã“ã®ãƒšãƒ¼ã‚¸ã«ã‚ã‚‹å† å®¹ã®è§£èª¬ã¨ã—ã¦ã€ Vistoli の Vis 基礎的な概念で、 Grothendieck が提示したものとしては、まずは topos が挙げられ る。 Grothendieck topology Topos 代数幾何を higher topos や model category TVa TVb Mah K -theory I Fibered category の定義では、 cotangent complex が用いられている。 Cotangent complex といえば Illusie の Cotangent complex このように、 Grothendieck ã®ã‚¢ã‚¤ãƒ‡ã‚¢ã«ã¯ãƒ›ãƒ¢ãƒˆãƒ”ãƒ¼è«–ã§ç™ºè¦‹ã•ã‚ŒãŸæ¦‚å¿µã¨å ±é€šã™ã‚‹ ものも多い。実際、 Grothendieck は、

18. Grothendieck Topologies (ResearchIndex) M, we can go to the etale topology, where nfg serves as an open neighborhood by way of the map xx allowing this as a neighborhood clearly makes the map into http://citeseer.ist.psu.edu/719258.html

19. Van Der Put: Rigid Analytic Spaces Tuo get something interesting, we have to consider on X a grothendiecktopology instead of the ordinary topology. For this purpose, we have introduced open http://www.numdam.org/numdam-bin/fitem?id=GAU_1975-1976__3_2_A6_0

20. Grothendieck Biography During this period grothendieck s work provided unifying themes in geometry, number theory, topology and complex analysis. He introduced the theory of http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Grothendieck.html

Alexander Grothendieck Born: 28 March 1928 in Berlin, Germany Click the picture above to see two larger pictures Show birthplace location Previous (Chronologically) Next Main Index Previous (Alphabetically) Next Biographies index Version for printing 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. ] 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 Fields Medal in . In looking back at this period, one marvels at the generosity with which Grothendieck shared his ideas with colleagues and students, the energy he and his collaborators devoted to meticulous redaction, the excitement with which they set out to explore a new land. During this period Grothendieck's work provided unifying themes in geometry

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