41. Grothendieck Topology - Enyclopaedia Article About Grothendieck Topology Grothendieck topology encyclopaedia article In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the http://www.pro-researcher.co.uk/encyclopaedia/english/grothendieck_topology

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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 ©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

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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

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46. Grothendieck Topology - Free Encyclopedia Grothendieck topology. From Wacklepedia The Free Encyclopedia. In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C http://www.wacklepedia.com/g/gr/grothendieck_topology.html

Wacklepedia Home Page Index Grothendieck topology From Wacklepedia - The Free Encyclopedia 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

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Technology Computers Phones Cameras ... Links ABCworld >> Grothendieck topology 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 Alexandre Grothendieck to define the ©tale 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.

48. Everything About John Sheaf A sheaf associates information to the open sets A sheaf on a site, however, should allow en.wikipedia.org/wiki/grothendieck_topology http://www.exalead.com/wikipedia/results/John Sheaf/en

Customize your homepage Use Exalead in your browser Web Images Wikipedia Video Advanced search Wikipedia Results of about for John Sheaf View: Teddington times as Todyngton and Tutington. John Sheaf , Ken Howe: Hampton and Teddington Past, Historical Publications, Literature John Sheaf , Ken Howe: Hampton and Teddington en .wikipedia .org /wiki /Teddington Preview Categories: Districts of London Neighbourhoods of Richmond upon Thames See also: People: Ken Howe Stanley Juniors John Bodkin Adams ... Rudyard Kipling See also: Organizations: Carnegie Thames Tv and Haymarket Group Tearfund Hampton Court See also: Locations: Bushy Park London Twickenham England ... Fulwell, London Fulwell is located in the London Borough of Richmond upon Thames John Sheaf and Ken Howe country en .wikipedia .org /wiki /Fulwell,_London Preview Categories: Districts of London Neighbourhoods of Richmond upon Thames See also: People: Manuel II Ken Howe See also: Organization: Royal Mail See also: Locations: London Twickenham Hampton Hill Richmond upon Thames ... Grothendieck topology The classical definition of a sheaf begins with a topological space X. A

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Tiny Wiki : Fast loading, text only version of Wikipedia. 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 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

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Search the Archive Home Welcome to Station Information Mathematical and Natural Sciences Astronomy Biology Chemistry Computer science ... Statistics Applied Arts and Sciences Agriculture Architecture Business Communication ... Transport Social Sciences and Philosophy Archaeology Economics Geography History ... Sociology Culture and Fine Arts Classics Cooking Dance Entertainment ... Visual arts and design 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.

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