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  1. The Grothendieck Festschrift Volume I, II + III Set: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck (Progress ... V. 86-88.) (English and French Edition)
  2. Frobenius Categories versus Brauer Blocks: The Grothendieck Group of the Frobenius Category of a Brauer Block (Progress in Mathematics) by Lluís Puig, 2009-05-04
  3. Produits Tensoriels Topologiques Et Espaces Nucleaires (Memoirs : No.16) by Alexander Grothendieck, 1979-06
  4. A general theory of fibre spaces with structure sheaf by A Grothendieck, 1958
  5. Classifying Spaces and Classifying Topoi (Lecture Notes in Mathematics) by Izak Moerdijk, 1995-11-10
  6. Local Cohomology: A Seminar Given by A. Groethendieck, Harvard University. Fall, 1961 (Lecture Notes in Mathematics) by Robin Hartshorne, 1967-01-01
  7. Fundamental Algebraic Geometry (Mathematical Surveys and Monographs) by Barbara Fantechi; Lothar Göttsche; Luc Illusie; Steven L. Kleiman; Nitin Nitsure; and Angelo Vistoli, 2005-12-08
  8. Algebraic Geometry for Associative Algebras (Pure and Applied Mathematics)

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

22. Kids.Net.Au - Encyclopedia > Grothendieck Topology
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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

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.
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Grothendieck Topology
, Grothendieck topology scheme Zariski topology poset small category Grothendieck topology small category site
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Mac Lane Moerdijk Kashiwara Schapira Vistoli Vis Site sheaf site sheaf category category Grothendieck 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
[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. topology
Grothendieck topology
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Internal Logic

26. PlanetMath: Alexander Grothendieck
Some concepts named after him include the Grothendieck group, the Grothendieck topology, the Grothendieck category and the Grothendieck universe.
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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 math
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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

29. Coarsest Encyclopedia Topics |
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.

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

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
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.
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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

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 topology
Lawvere-Tierney topology
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Lawvere–Tierney topologies
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.”
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

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 $\mathcal{S}h(\mathcal{C})$,
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Induced Grothendieck topology on a presheaf or sheaf category of a site?
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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
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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
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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
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  • 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
    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
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    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

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