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 . A category together with a choice of Grothendieck topology is called a site 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 étale 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 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

2. Grothendieck Topology Grothendieck topology In mathematics, 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 http://www.fact-index.com/g/gr/grothendieck_topology.html

Main Page See live article Alphabetical index Grothendieck topology In mathematics , 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 site . This tool is used in algebraic number theory and algebraic geometry schemess , but also for flat cohomology and crystalline cohomology. Note that a Grothendieck topology is not a topology in the classical sense. History and idea At a time when cohomology for sheaves on topological spaces was well established, Alexander Grothendieck wanted to define cohomology theories for other structures, his schemess . He thought of a sheaf on a topological space as a "measuring rod" for that space, and the cohomology of such a measuring rod as a rough measure for the underlying space. His goal was thus to produce a structure which would allow the definition of more general sheaves or "measuring rods"; once that was done, the model of topological cohomology theories could be followed almost verbatim. Motivating example Start with a topological space X and consider the sheaf of all continuous real-valued functions defined on X . This associates to every open set U in X the set F U ) of real-valued continuous functions defined on U . Whenver U is a subset of V , we have a "restriction map" from F V ) to F U ). If we interpret the topological space

3. Wapedia - Wiki: Grothendieck Topology 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://wapedia.mobi/en/Grothendieck_topology

Wiki: Grothendieck topology 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 . A category together with a choice of Grothendieck topology is called a site 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 étale 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 rigid analytic geometry There is a natural way to associate a site to an ordinary topological space sobriety , this is completely accurate—it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies which do not come from topological spaces.

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5. Grothendieck Topology - Wikivisual 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. http://en.wikivisual.com/index.php/Grothendieck_topology

Francais English 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 Alexandre Grothendieck to define the étale cohomology of a scheme . It has been used to define many 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 Grothendieck topologies are not comparable to the classical notion of a topology on a space. While it is possible to interpret sober spaces in terms of Grothendieck topologies, more pathological spaces have no such representation. Conversely, not all Grothendieck topologies correspond to topological spaces.

6. Grothendieck Topology - Definition In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the http://www.wordiq.com/definition/Grothendieck_topology

Grothendieck topology - Definition In mathematics , 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 site This tool has been used in algebraic number theory and algebraic geometry , initially to define étale cohomology of schemes , but also for flat cohomology and crystalline cohomology, and in further ways. Note that a Grothendieck topology is a true generalisation. It is not a topology in the classical sense (and may not be equivalent to giving one). Contents showTocToggle("show","hide") 1 History and idea 2 Motivating example 3 Formal definition 4 Beyond cohomology History and idea See main article Background and genesis of topos theory At a time when cohomology for sheaves on topological spaces was well established, Alexander Grothendieck wanted to define cohomology theories for other structures, his schemes . He thought of a sheaf on a topological space as a "measuring rod" for that space, and the cohomology of such a measuring rod as a rough measure for the underlying space. His goal was thus to produce a structure which would allow the definition of more general sheaves or "measuring rods"; once that was done, the model of topological cohomology theories could be followed almost verbatim. Motivating example Start with a topological space X and consider the sheaf of all continuous real-valued functions defined on

7. Science Fair Projects - Grothendieck Topology The Ultimate Science Fair Projects Encyclopedia Grothendieck topology http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/Grot

All Science Fair Projects Science Fair Project Encyclopedia for Schools! Search Browse Forum Coach ... Dictionary Science Fair Project Encyclopedia For information on any area of science that interests you, enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.). Or else, you can start by choosing any of the categories below. Science Fair Project Encyclopedia Contents Page Grothendieck topology In mathematics , 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 site This tool has been used in algebraic number theory and algebraic geometry , initially to define étale cohomology of schemes , but also for flat cohomology and crystalline cohomology , and in further ways. Note that a Grothendieck topology is a true generalisation. It is not a topology in the classical sense (and may not be equivalent to giving one). Contents showTocToggle("show","hide")

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13. Grothendieck Topology In - Dictionary And Translation Grothendieck topology. Dictionary terms for Grothendieck topology, definition for Grothendieck topology, Thesaurus and Translations of Grothendieck topology to English, Spanish http://www.babylon.com/definition/Grothendieck_topology/

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Mul tili ngual Ar chi ve Po wer ed by Wor ldLi ngo Grothendieck topology Home Multilingual Archive Index Ch oo se your la ngua ge: English Italiano Deutsch Nederlands ... Svenska var addthis_pub="anacolta"; Grothendieck topology 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 étale 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

15. Grothendieck Topology - On Opentopia, Find Out More About Grothendieck Topology 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 set s of a topological space. http://encycl.opentopia.com/term/Grothendieck_topology

About Opentopia Opentopia Directory Encyclopedia ... Tools Grothendieck topology Encyclopedia G GR GRO : Grothendieck topology 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 set s 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 Alexandre Grothendieck to define the étale cohomology of a scheme . It has been used to define many 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 Grothendieck topologies are not comparable to the classical notion of a topology on a space. While it is possible to interpret

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Contents Grothendieck topology 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 site . This tool is mainly used in algebraic geometry , for instance to define . Note that a Grothendieck topology is not a topology in the classical sense. The motivating example is the following: start with a topological space X and consider the sheaf of all continuous real-valued functions defined on X . This associates to every open set U in X the set F U ) of real-valued continuous functions defined on U . Whenver U is a subset of V , we have a "restriction map" from F V ) to F U ). If we interpret the topological space X as a category, with the open sets being the objects and a morphism from U to V if and only if U is a subset of V , then F is revealed as a contravariant functor from this category into the category of sets. In general, every contravariant functor from a category C to the category of sets is therefore called a pre-sheaf of sets on C . Our functor F has a special property: if you have an open covering ( V i ) of the set U , and you are given mutually compatible elements of F V i ), then there exists precisely one element of

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19. Grothendieck Topology - VisWiki Grothendieck topology Scheme (mathematics), Alexander Grothendieck, Yoneda lemma, Cohomology, Springer Science+Business Media - VisWiki http://www.viswiki.com/en/Grothendieck_topology

20. Grothendieck Topology 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. http://pandapedia.com/wiki/Grothendieck_topology

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