21. Grothendieck Topology: Encyclopedia II - Grothendieck Topology - Definition The classical definition of a sheaf begins with a topological space X. A sheaf associates information to the open sets of X. This information can be phrased abstractly by letting O http://www.experiencefestival.com/a/Grothendieck_topology_-_Definition/id/144379

22. Kids.Net.Au - Encyclopedia > Grothendieck Topology Kids.Net.Au is a search engine / portal for kids, children, parents, and teachers. The site offers a directory of child / kids safe websites, encyclopedia, dictionary, thesaurus http://encyclopedia.kids.net.au/page/gr/Grothendieck_topology

Search the Internet with Kids.Net.Au Encyclopedia > Grothendieck topology Article Content 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

23. Grothendieck Topology : Quiz (The Full Wiki) Question 1 In , 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 http://quiz.thefullwiki.org/Grothendieck_topology

24. Grothendieck Topology In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Definition. Let C be a category, and let c be an object of C. http://pantodon.shinshu-u.ac.jp/topology/literature/Grothendieck_topology.html

Your language? Japanese English Oct, 2010 Sun Mon Tue Wed Thu Fri Sat Updated Pages date page Oct 31 Oct 30 topological_pi1.html Oct 30 smooth_manifold.html Oct 30 equivariant_de_Rham.html Oct 30 equivariant_manifold.html Oct 30 equivariant_homology.html Oct 30 pi1.html Oct 30 equivariant_homotopy.html Oct 29 Oct 29 configurations.html Oct 29 Oct 29 configuration_space.html Oct 28 numerical_analysis.html Oct 28 Hopf_algebra_basics.html Oct 28 Oct 28 Hopf_algebra_basics.html Oct 27 vector_bundle.html Oct 27 vector_space.html Oct 27 Oct 27 von_Neumann_algebra.html objectives precautions index ... list of pages Grothendieck Topology , Grothendieck topology scheme Zariski topology poset small category Grothendieck topology small category site site Mac Lane Moerdijk Kashiwara Schapira Vistoli Vis Site sheaf site sheaf category category Grothendieck topos topos Grothendieck topology Mark Johnson spectrum Hopf algebroid affine groupoid scheme , Hopf algebroid comodule groupoid scheme quasi-coherent sheaf local pospace model category BW Wor Grothendieck topology ionad , Garner Gar Zariski topology Grothendieck topology , unique factorization system Anel Ane References [Ane] Mathieu Anel. Grothendieck topologies from unique factorisation systems

25. Grothendieck Topology In NLab Idea. A Grothendieck topology on a category is a choice of morphisms in that category which are regarded as covers. A category equipped with a Grothendieck topology is a site. http://ncatlab.org/nlab/show/Grothendieck topology

nLab Grothendieck topology Skip the Navigation Links Home Page All Pages Recently Revised ... Toposes Background category theory category functor Toposes (0,1)-topos Heyting algebra locale pretopos ... quasitopos Internal Logic internal logic type theory subobject classifier natural numbers object Topos morphisms geometric morphism direct image inverse image global sections ... logical morphism Extra stuff, structure, properties topological locale localic topos petit topos/gros topos locally connected topos ... smooth topos Cohomology and homotopy cohomology homotopy abelian sheaf cohomology model structure on simplicial presheaves In higher category theory higher topos theory (0,1)-topos (0,1)-site 2-topos ... Edit this sidebar Contents Idea Definition Saturation Examples ... Terminology Idea A Grothendieck topology on a category is a choice of morphisms in that category which are regarded as covers A category equipped with a Grothendieck topology is a site . Sometimes all sites are required to be small Probably the main point of having a site is so that one can define sheaves , or more generally stacks , on it. In particular, the category of sheaves on a (small) site is a

26. PlanetMath: Alexander Grothendieck Some concepts named after him include the Grothendieck group, the Grothendieck topology, the Grothendieck category and the Grothendieck universe. http://planetmath.org/encyclopedia/AlexanderGrothendieck.html

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Alexander Grothendieck (Biography) Alexander Grothendieck (1928 - ) German-born, French mathematician, one of the pioneers of topos theory , and the `new algebraic geometry and number theory '. In 1966, he was awarded the Fields Medal for fundamental contributions to mathematics (an algebraic proof of one of the Riemann-Roch theorems , previously conjectured), but boycotted the ceremony held in Moscow (USSR), (as further explained in the text). Two decades later, he declined the Crafoord Prize that was awarded to him and his student Pierre Deligne, because he didn't want the money and because the award was in recognition of work he had done much earlier in his career. Some concepts named after him include the Grothendieck group , the Grothendieck topology , the Grothendieck category and the Grothendieck universe In 1949, Grothendieck worked on

27. [math/0612471] Grothendieck Topologies And Ideal Closure Operations Abstract We relate closure operations for ideals and for submodules to nonflat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure http://arxiv.org/abs/math/0612471

arXiv.org math Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PDF PostScript Other formats Current browse context: math new recent NASA ADS Bookmark what is this? Mathematics > Algebraic Geometry Title: Grothendieck topologies and ideal closure operations Authors: Holger Brenner (Submitted on 16 Dec 2006) Abstract: We relate closure operations for ideals and for submodules to non-flat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendieck topology which induces this operation. In this way we relate the radical to the surjective topology and the constructible topology, the integral closure to the submersive topology, to the proper topology and to Voevodsky's h-topology, the Frobenius closure to the Frobenius topology and the plus closure to the finite topology. The topologies which are induced by a Zariski filter yield the closure operations which are studied under the name of hereditary torsion theories. The Grothendieck topologies enrich the corresponding closure operation by providing cohomology theories, rings of global sections, concepts of exactness and of stalks. Subjects: Algebraic Geometry (math.AG)

28. Springer Online Reference Works Encyclopaedia of Mathematics G Grothendieck topology This text originally appeared in Encyclopaedia of Mathematics http://eom.springer.de/g/g045180.htm

29. Coarsest Encyclopedia Topics | Reference.com 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://www.reference.com/browse/Coarsest

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

31. Notions Of Flatness Relative To A Grothendieck Topology Notions of flatness relative to a Grothendieck topology Panagis Karazeris Completions of (small) categories under certain kinds of colimits and exactness conditions have been http://emis.library.cornell.edu/journals/TAC/volumes/12/5/12-05abs.html

Notions of flatness relative to a Grothendieck topology 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 ... TAC Home

32. Cohomology In Grothendieck Topologies And Lower Boundsin Boolean On the other hand, as shown inSGA4.I{VI, any Grothendieck topology has analogues of sheaves, cohomology, and related concepts that are strikingly similar to what one is accustomed http://www.math.ubc.ca/~jf/pubs/web_stuff/groth1.pdf

33. Lawvere-Tierney Topology In NLab In fact, it is a generalisation of Grothendieck topology in this sense If C is a small category, then choosing a Grothendieck topology on C is equivalent to choosing a Lawvere http://ncatlab.org/nlab/show/Lawvere-Tierney topology

nLab Lawvere-Tierney topology Skip the Navigation Links Home Page All Pages Recently Revised ... Export Lawvere–Tierney topologies Idea Definition Equivalence with Grothendieck topologies Sheaves ... References Idea A Lawvere–Tierney topology (or operator, or modality, also called geometric modality ) is a way of saying that something is ‘locally’ true. Unlike a Grothendieck topology , this is done directly at the stage of logic , defining a geometric logic . In fact, it is a generalisation of Grothendieck topology in this sense: If C is a small category, then choosing a Grothendieck topology on C is equivalent to choosing a Lawvere–Tierney topology in the topos Set C op of presheaves on C The use of “topology” for this and the related Grothendieck concept is regarded by some people as unfortunate; see Grothendieck topology for some reasons why. A proposed replacement for “Grothendieck topology” is (Grothendieck) coverage ; see Grothendieck topology for some possible replacements for “Lawvere–Tierney topology.” Definition Let E be a topos , with subobject classifier . A Lawvere–Tierney topology in E is a map j that satisfies certain axioms.

34. Stacks For Everybody Stacks for Everybody Barbara Fantechi Abstract. Let Sbeacategorywitha Grothendieck topology. A stack over Sis a category fibered in groupoids over S, such that isomorphisms forma sheaf http://www.cgtp.duke.edu/~drm/PCMI2001/fantechi-stacks.pdf

35. Induced Grothendieck Topology On A Presheaf Or Sheaf Category Of A Site? - MathO In general, is there a natural induced Grothendieck topology on $\mathcal{P}sh(\ mathcal{C})$ or $\mathcal{S}h(\mathcal{C})$, http://mathoverflow.net/questions/11069/induced-grothendieck-topology-on-a-presh

login faq how to ask meta ... Induced Grothendieck topology on a presheaf or sheaf category of a site? Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points Question, then: Edit: I removed the other parts of the question regarding t-Schemes and algebraic spaces as functors of points to ask them at some other time. ag.algebraic-geometry ct.category-theory flag edited Jan 9 at 0:07 Harry Gindi community wiki 1 Answer oldest newest votes The answer to your first question is yes. Suppose $C$ a site. covering morphism covering family link flag answered Jan 8 at 2:06 Clark Barwick David Brown Jan 8 at 21:16 You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you. Your Answer OpenID Login Get an OpenID or Name Email never shown Home Page Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry ct.category-theory or ask your own question Welcome to Math Overflow A place for mathematicians to ask and answer questions.

36. A Noncommutative Grothendieck Topology | Neverendingbooks Feb 15, 2004 We have seen that a noncommutative $l$-point is an algebra$P=S_1 \oplus \ oplus S_k$with each $S_i$ a simple finite dimensional http://www.neverendingbooks.org/index.php/a-noncommutative-grothendieck-topology

@import "http://www.neverendingbooks.org/wp-content/themes/mystique/style.css"; @import "http://www.neverendingbooks.org/index.php/a-noncommutative-grothendieck-topology.html?mystique=css"; neverendingbooks Follow me on Twitter RSS Feeds Home ... Van Eck phreaking a noncommutative Grothendieck topology non-commutative geometry connected component coalgebra Galois and the Brauer group a noncommutative Grothendieck topology noncommutative geometry noncommutative geometry 2 projects in noncommutative geometry points and lines ... noncommutative geometry : a medieval science? We have seen that a non-commutative $l$-point is an algebra$P=S k$with each $S i$ a simple finite dimensional $l$-algebra with center $L i$ which is a separable extension of $l$. The centers of these non-commutative points (that is the algebras $L k$) are the open sets of a Grothendieck -topology on $l$. To define it properly, let $L$ be the separable closure of $l$ and let $G=Gal(L/l)$ be the so called absolute Galois group. Consider the category with objects the finite cover of $V$ if the images of the finite number of $Vi$ is all of $V$. Let $Cov$ be the set of all covers of finite $G$-sets, then this is an example of a

37. Cats In The Jungle Just as in point set topology, we can define a basis to generate a topology on a given space, a Grothendieck pretopology induces a Grothendieck topology, and it is usualy easier http://catsinthejungle.wordpress.com/

Cats in the Jungle 1 Comment satisfies the sheaf axiom for a covering family , it will also satisfy it for any covering family that contains , and the converse is true if every morphism of factors through one of . In order to avoid this we turn to coverings that are downward saturated, in other words, sieves. Definition: Let be a category. A Grothendieck topology on is given by associating to each a collection of of -sieves such that (i) The maximal sieve is in (ii) If and is a morphism of , then is in (iii) If and is a -sieve such that for any in , then A category with a Grothendieck topology is called a site Proposition 1: Let be a category with pullbacks. Every Grothendieck pretopology on induces a Grothendieck topology on by setting In this case, we say that is a basis for Proposition 2: Let be a category with pullbacks and a Grothendieck topology on with basis . A presheaf is a -sheaf if, and only if, it is a -sheaf. Proofs can be found in SGL Given two Grothendieck topologies on a category we say that is finer than if for every Thus the Grothendieck topologies on are partially ordered. The topology in which every sieve is a covering is called the

38. Grothendieck Topology And The Theory Of Representations Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.springerlink.com/index/g464h62017047461.pdf

39. NOTIONS OF FLATNESS RELATIVE TO A GROTHENDIECK TOPOLOGY 1 File Format Adobe PostScript View as HTML http://www.emis.de/journals/TAC/volumes/12/5/12-05.ps

40. Background And Genesis Of Topos Theory - VisWiki Crystalline cohomology JeanLouis Verdier Profinite group William Lawvere Ramification Pointless topology Yoneda lemma Grothendieck topology Monad (category http://www.viswiki.com/en/Background_and_genesis_of_topos_theory

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