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1. Grothendieck Topology - Wikipedia, The Free Encyclopedia
http://en.wikipedia.org/wiki/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 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 Ã©tale cohomology of schemes , but also for flat cohomology and crystalline cohomology . Note that a Grothendieck topology is not a topology in the classical sense. 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 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

 2. The Primitive Topology Of A Scheme The Primitive Topology of a Scheme We define a grothendieck topology on the category of schemes whose associated sheaf theory coincides in many cases with the Zariski topology. We also give somehttp://rdre1.inktomi.com/click?u=http://citeseer.ist.psu.edu/302227.html&y=0

3. List Of Algebraic Topology Topics - Wikipedia, The Free Encyclopedia
Splitting lemma; Extension problem. Abelian category; Group cohomology;Sheaf grothendieck topology. History. Combinatorial topology. Edit
http://en.wikipedia.org/wiki/List_of_algebraic_topology_topics
##### List of algebraic topology topics
This is a list of algebraic topology topics , by Wikipedia page. Table of contents 1 Homology (mathematics) 2 Homotopy theory 3 Further developments 4 Homological algebra ...
##### Homology (mathematics)
Main article: Homology theory

4. Grothendieck Topology
grothendieck topology. In mathematics, a theories. A category togetherwith a grothendieck topology on it is called a site. This tool
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

5. Background And Genesis Of Topos Theory
that a Grothendieck topos was a category of sheaves, where now the word sheaf hadacquired an extended meaning with respect to the idea of grothendieck topology
http://www.fact-index.com/b/ba/background_and_genesis_of_topos_theory.html
Main Page See live article Alphabetical index
##### Background and genesis of topos theory
This page gives some very general background to the mathematical idea of topos . This is an aspect of category theory , and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. Table of contents 1 In the school of Grothendieck
2 From pure category theory to categorical logic

3 Position of topos theory

4 Summary
##### In the school of Grothendieck
During the latter part of the , the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of etale cohomology With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems, for a long time. The first was to do with its points : back in the days of projective geometry it was clear that the absence of 'enough' points on an algebraic variety was a barrier to having a good geometric theory (in which it was somewhat like a compact manifold ). There was also the difficulty, that was clear as soon as

6. Grothendieck Topology - Encyclopedia Article About Grothendieck Topology. Free A
encyclopedia article about grothendieck topology. grothendieck topology in Freeonline English dictionary, thesaurus and encyclopedia. grothendieck topology.
http://encyclopedia.thefreedictionary.com/Grothendieck topology
Dictionaries: General Computing Medical Legal Encyclopedia
##### Grothendieck topology
Word: Word Starts with Ends with Definition In mathematics Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
Click the link for more information. , a Grothendieck topology is a structure defined on an arbitrary category Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "abstract nonsense". See list of category theory topics for a breakdown of the relevant Encyclopedia pages.
##### Background
A category attempts to capture the essence of a class of related mathematical objects

7. Notions Of Flatness Relative To A Grothendieck Topology
Notions of flatness relative to a grothendieck topology. PanagisKarazeris. Completions of (small) categories under certain kinds
http://www.tac.mta.ca/tac/volumes/12/5/12-05abs.html
##### Panagis Karazeris
Keywords: flat functor, postulated colimit, geometric logic, exact completion, pretopos completion, left exact Kan extension 2000 MSC: 18A35, 03G30, 18F10 Theory and Applications of Categories, Vol. 12, 2004, No. 5, pp 225-236.
http://www.tac.mta.ca/tac/volumes/12/5/12-05.dvi

http://www.tac.mta.ca/tac/volumes/12/5/12-05.ps

http://www.tac.mta.ca/tac/volumes/12/5/12-05.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/12/5/12-05.dvi
...
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8. PlanetMath: Site
cohomology, covering space Other names grothendieck topology. Also definescover, covering, morphism of sites. Keywords etale cohomology
http://planetmath.org/encyclopedia/GrothendieckTopology.html
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##### Definition
A site is a generalization of a topology , designed to address the problem that in the algebraic category , the only reasonable topology is the Zariski topology , in which the open sets are much too large. In order to obtain a well-behaved cohomology theory (and an algebraic version of the fundamental group Using the machinery of sites, one can construct (or -adic) cohomology, and one can construct crystalline cohomology, both of which can be used to prove the Weil conjectures, and both of which serve as generalizations of the familiar cohomology from topology and complex analysis. Fix a universe Definition A site is a -category of collections of maps is a small set of morphisms in . These objects must satisfy the following:
• If is an isomorphism , then is a covering; if
• 9. Grothendieck Topology - Information
An online Encyclopedia with information and facts grothendieck topology Information,and a wide range of other subjects. grothendieck topology - Information.
http://www.book-spot.co.uk/index.php/Grothendieck_topology
Grothendieck topology - Information Home
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adsonar_pid=2712;adsonar_ps=1199;adsonar_zw=120;adsonar_zh=600;adsonar_jv='ads.adsonar.com'; 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 of schemes , 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 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 X . This associates to every open set U in X the set F U ) of real-valued continuous functions defined on

10. List Of Algebraic Topology Topics - Information
Splitting lemma; Extension problem. Abelian category; Group cohomology;Sheaf grothendieck topology. History. Combinatorial topology. All
http://www.book-spot.co.uk/index.php/List_of_algebraic_topology_topics
List of algebraic topology topics - Information Home
Mathematical and natural sciences

Applied arts and sciences

Social sciences and philosophy
...
Interdisciplinary categories

2 Homotopy theory

3 Further developments

4 Homological algebra
...
##### Homology (mathematics)
Main article: Homology theory

11. Grothendieck Topology
The summary for this Chinese (Traditional) page contains characters that cannot be correctly displayed in this language/character set.
http://homepage.mac.com/yenlung/WebWiki/GrothendieckTopology.html
##### Grothendieck Topology
• Â¬MÂ«Â¬Â¡]morphismsÂ¡^

12. Grothendieck Topology Definition Meaning Information Explanation
COMMENTSA noncommutative grothendieck topology? We have seen that a non-commutativel-point is an algebra P=S1 Sk with each Si a simple
http://www.free-definition.com/Grothendieck-topology.html
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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 , mainly to define of scheme s, 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 space s was well established, Alexander Grothendieck wanted to define cohomology theories for other structures, his scheme s. 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

13. Grothendieck Topology
grothendieck topology. In mathematics, a theories. A category togetherwith a grothendieck topology on it is called a site. This tool
http://www.sciencedaily.com/encyclopedia/grothendieck_topology
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##### Encyclopedia
Main Page See live article
##### 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.

14. Week68
symmetries. Then there are *really* highpowered things like topoi ofsheaves on a category equipped with a grothendieck topology .
http://math.ucr.edu/home/baez/week68.html
##### 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 *really* high-powered things like topoi of sheaves on a category equipped with a Grothendieck topology....

 15. Also Available At Http//math.ucr.edu/home/baez/week68.html symmetries. Then there are *really* highpowered things like topoi ofsheaves on a category equipped with a grothendieck topology . Andhttp://math.ucr.edu/home/baez/twf.ascii/week68

16. Seminar On Cohomology
Langlands conjectures, October 15, Eyal Goren; Gil Alon, Some remarkson etale morphism; Descent and a grothendieck topology. October 22,
http://www.math.mcgill.ca/goren/SeminarOnCohomology.html
 Seminar on Cohomology Theories Schedule: September 17 Gabriel Chenevert and Payman Kassaei Review of homological algebra, sheaf cohomology via derived functors and via Cech cohomology, some key theorems, examples of cohomology groups (in particular, Zariski is no good for constant coefficients). September 24 Gabriel Chenevert and Payman Kassaei Continued. notes Andrew Archibald and Alex Ghitza Grothendieck topologies. The etale topology: flat and unramified morphisms. Etale morphisms and criteria. Lots of examples. Sheaves in the etale topology. The stalk of a sheaf. October 1 Andrew Archibald and Alex Ghitza Continued. October 8 Andrew Archibald and Alex Ghitza Continued notes October 10 Elena Mantovan (BURN 1205, 10:30-11:00 and 1:30-2:30) The role of the geometry of Shimura varieties in the Langlands' conjectures October 15 Eyal Goren; Gil Alon Some remarks on etale morphism; Descent and a Grothendieck topology. October 22 Gil Alon Descent and a Grothendieck topology (Cont'd). notes October 29 Pete Clark and Marc-Hubert Nicole de Rham cohomology and the Hodge to de Rham spectral sequence. Algebraic de Rham cohomology. The crystalline topology. Divided powers. The crystalline site. Lots of examples.

17. Abstract:001108bm
Suppose J is a grothendieck topology on C which is generated by the subcanonicalpretopology J for which a family (C i D) is in J if and only if the
http://www.maths.usyd.edu.au:8000/u/stevel/auscat/abstracts/001108bm.html
##### Bachuki Mesablishvili (8/11/00 and 22/11/00)
Let C be a category with pullbacks and let E be a pullback-stable class of morphisms of C which is closed under composition with the isomorphisms. E defines a pseudofunctor from C op to Cat , also denoted by E , which sends an object C to E C , the full subcategory of the slice category C C consisting of arrows in E with codomain C . We may then consider the category of E -descent morphisms in C as defined in G. Janelidze and W. Tholen, Facets of Descent I, Applied Categorical Structures 2, 1994:1-37 Suppose J is a Grothendieck topology on C which is generated by the subcanonical pretopology J' for which a family (C i ->D) is in J' if and only if: the coproduct C of the C i exists, it is universal and disjoint, and the induced morphism C>D is both a universal regular epimorphism and an E -descent morphism. Using the Yoneda embedding Y: C op ->Sh( C ,J) we prove several results related to E -effective descent morphisms, Galois objects, and torsors in C . As an application, we get the following two theorems.

18. Sheaves And Espaces Etal\'es By Allan Adler
I think that if one works with Grothendieck topological spaces (ie sets equippedwith a grothendieck topology of subsets, one can also get an espace etal\ e
http://mathforum.org/epigone/sci.math.research/yimproibroo
 reply to this message post a message on a new topic Back to sci.math.research Subject: Author: ara@zurich.ai.mit.edu Date: The Math Forum

19. Cohomology Of Abelian Group Stacks By Angelo Vistoli
Harvard University Date Thu, 30 Jan 1997 174737 0500 Let X be a topologicalspace, or, more generally, a category with a grothendieck topology.
http://mathforum.org/epigone/sci.math.research/zendthermsex
##### Cohomology of abelian group stacks by Angelo Vistoli
post a message on a new topic

Back to sci.math.research
Subject: Cohomology of abelian group stacks Author: vistoli@abel.math.harvard.edu Organization: Department of Mathematics - Harvard University Date: The Math Forum

20. Topology
Hints Resolve sieve_prop. Sites. A site is a category with a grothendieck topology.Record Site Type = { site_cat Cat; site_top (Topology site_cat) }.
http://math1.unice.fr/~maggesi/coq/zariski/html/topology.html
##### Module topology
Require Export cat
Require Export sieve Implicit Arguments On.
##### Grothendieck Topologies.
Module Topology. Export Sieve. Section Topology. Variable A:Cat. Section Datum of a topology.
Variable Axioms for a topology.
Record
ax_top_maximal : (u:A; s:(Sieve u))
ax_top_larger : (u:A; s,t:(Sieve u))
ax_top_transport : (u,v:A; f:(mor u v); s:(Sieve v))
(u:A; s:(Sieve u); Hs:(J s))
(t:(x:A; f:(mor x u); Hf:(sieve s f))(Sieve x))
(Ht:(x:A; f:(mor x u); Hf:(sieve s f))(J (t x f Hf)))
(J (sieve_comp t)) End Type of a topology. Record Section Variable J:Topology. The maximal sieve is a covering sieve. Lemma top_maximal : (u:A; s:(Sieve u)) Proof Elim J; Simpl. NewDestruct 1; Auto. Qed Any larger sieve of a covering sieve is also a covering sieve. Lemma top_larger : (u:A; s,t:(Sieve u)) Proof Elim J; Simpl. NewDestruct 1; Auto. Qed Transport property. Lemma top_transport : (u,v:A; f:(mor u v); s:(Sieve v)) Proof Elim J; Simpl. NewDestruct 1; Auto. Qed Composition property. Lemma (u:A; s:(Sieve u); Hs:(cov_obj J s)) (t:(x:A; f:(mor x u); Hf:(sieve s f))(Sieve (src f)))

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