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1. Grothendieck Topology - Wikipedia, The Free Encyclopedia
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##### Grothendieck topology
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 accurateâit is possible to recover a sober space from its associated site. However simple examples such as the

2. Grothendieck Topology - Encyclopedia Article - Citizendium
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This is a draft article , under development. These unapproved articles are subject to edit intro The notion of a Grothendieck topology or site' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin.
##### Definition
A Grothendieck topology T consists of
• A category, denoted c a t T A set of coverings , denoted c o v T , such that
• for each object U of c a t T If , and is any morphism in c a t T , then the canonical morphisms of the fiber products determine a covering If and , then
• ##### Examples
• A standard topological space X becomes a category o p X when you regard the open subsets of X as objects, and morphisms are inclusions. An open covering of open subsets U clearly verify the axioms above for coverings in a site. Notice that a
• 3. Grothendieck Topology - Indopedia, The Indological Knowledgebase
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Category theory Sheaf theory
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... Wikipedia Article
##### Grothendieck topology
à¤à¥à¤à¤¾à¤¨à¤à¥à¤¶: - The Indological Knowledgebase 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 , mainly to define étale cohomology of schemes , but also for flat cohomology and crystalline cohomology . Note that a Grothendieck topology is not a topology in the classical sense. Contents showTocToggle("show","hide") 1 History and idea
2 Motivating example

3 Formal definition

4 Beyond cohomology
...
edit
##### 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 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. edit
##### Motivating example
Start with a topological space X and consider the sheaf of all continuous real-valued functions defined on

4. Grothendieck Topology - Wikipedia
http://nostalgia.wikipedia.org/wiki/Grothendieck_topology
##### Grothendieck topology
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

5. Grothendieck Topology - MathWiki
\$ /amsmath . If math is a sieve over math , and math , then math . Retrieved from http//mathwiki.gc.cuny.edu/index.php/grothendieck_topology
http://mathwiki.gc.cuny.edu/index.php/Grothendieck_topology
edit
##### Formal Definition
A Grothendieck topology on a category is a function from the objects of to sets of sieves of such that
• The maximal sieve for all If , and , then , where is defined as the pullback
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• If is a sieve over , and , then

6. Grothendieck Topology - ááªáªáµ á¦á§á¾á³ á¦
##### Grothendieck topology

7. Grothendieck Topology Articles And Information
Grothendieck topology. In mathematics,Grothendieck topology isstructure defined on an arbitrary category C which allowsdefinitionsheaves on C,with
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Current Article
##### Grothendieck topology
In mathematics Grothendieck topology isstructure defined on an arbitrary category C which allowsdefinition sheaves on C ,with thatdefinitiongeneral cohomology theories. A category together withGrothendieck topology on itcalled site . This toolused algebraic number theory algebraic geometry schemess , but alsoflat cohomologycrystalline cohomology. Note thatGrothendieck topologynot topology inclassical sense.
##### Historyidea
Attime when cohomologysheaves on topological spaces was well established, Alexander Grothendieck wanteddefine cohomology theoriesother structures, his schemess . He thought ofsheaf ontopological space as"measuring rod"that space, andcohomologysuchmeasuring rod asrough measure forunderlying space. His goal was thusproducestructure which would allowdefinitionmore general sheaves or "measuring rods"; once that was done,modeltopological cohomology theories could be followed almost verbatim.
##### Motivating example
Start withtopological space X considersheafall continuous real-valued functions defined on X . This associatesevery open set U X set F U )real-valued continuous functions defined on U . Whenver U issubset V , we have"restriction map" from F V F U ). If we interprettopological space

8. Grothendieck Topology
A selection of articles related to Grothendieck topology.
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##### A Wisdom Archive on Grothendieck topology
Grothendieck topology A selection of articles related to Grothendieck topology More material related to Grothendieck Topology can be found here: Index of Articles
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Grothendieck Topology
Grothendieck topology Page 2
##### ARTICLES RELATED TO Grothendieck topology
Grothendieck topology: Encyclopedia II - Grothendieck topology - Definition
Grothendieck topology Grothendieck topology - Introduction Grothendieck topology - Definition Grothendieck topology - Sites and sheaves Read more here: Grothendieck topology: Encyclopedia - Alexander Grothendieck Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of its most extreme scientific personalities, with achievements over a short span of years that are still scarcely credible in their broad scope and sheer bulk, and an approach that antagonised even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and co-awarded the Crafoord Prize with Pierre Delig ...
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9. Grothendieck Topology
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##### Grothendieck Topology
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##### References
Mark Hovey. Morita theory for Hopf algebroids and presheaves of groupoids.

10. 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://pedia-site.com/Grothendieck_topology
 Site Go to The Main Page Add Site to favorite! 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 accurateâit is possible to recover a sober space from its associated site. However simple examples such as the

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12. Grothendieck Topology - Mathematics Dictionary And Research Guide
Wikipedia and Wikis. Grothendieck topology Wikipedia http//en.wikipedia.org/wiki/grothendieck_topology. Keywords and Synonyms
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The Language of Mathematics - Dictionary and Research Guide Provided by
##### 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.
##### Keywords and Synonyms
• Grothendieck topology, Grothendieck topos, Grothendieck site, Grothendieck topologies

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13. Tiny Wikipedia Dump Test
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15. Good Math, Bad Math : Big To Small, Small To Big: Topological Properties Through
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##### Profile
Mark Chu-Carroll (aka MarkCC) is a PhD Computer Scientist, who works for Google as a Software Engineer. My professional interests center on programming languages and tools, and how to improve the languages and tools that are used for building complex software systems.
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17. Grothendieck Topology - Biocrawler
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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 (although it can be used to faithfully model sober spaces Table of contents showTocToggle("show","hide") 1 History and idea
2 Motivating example

3 Formal definition

4 Presheaves and sheaves
...
edit
##### 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

18. All Articles: Page 59
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19. Grothendieck Topology
Contents. Grothendieck topology. A Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C,
 Make eBroadcast my Homepage Contact Us It's Web Guide Encyclopedia 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

20. Grothendieck Topologie
Grothendieck topologie. V matematice, Grothendieck topologie je struktura definovaná na libovolný kategorie C který dovolí definici svazk na C,
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##### Grothendieck topologie
V matematice Grothendieck topologie je struktura definovan¡ na libovoln½ kategorie C kter½ dovol­ definici svazkÅ¯ na C , a s t­m definice gener¡la cohomology teorie. Kategorie spolu s Grothendieck topologie na tom je naz½v¡na m­stem . Tento n¡stroj je pouÅ¾­v¡n v teorii algebraick©ho Ä­sla a algebraick© geometrii , hlavnÄ vymezit ©tale cohomology schemess , ale tak© pro ploch© cohomology a krystalick© cohomology. Poznamenejte, Å¾e Grothendieck topologie nen­ topologie v klasick©m smyslu.