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Topic: Grothendieck topos

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Grothendieck topology

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Introduction to topos theory

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	PlanetMath: topos
		First, there is the Grothendieck topos, which was developed by Grothendieck as part of his general reconstruction of algebraic geometry.
		Second, there is the elementary topos, which was introduced by Lawvere as a setting for work in categorical logic.
		This is version 11 of topos, born on 2007-01-19, modified 2007-02-10.
	planetmath.org /encyclopedia/GrothendieckTopos.html (372 words)

	Introduction to topos theory
		In the light of later work, 'descent' is part of the theory of comonads[?]; here we can see the way in which the Grothendieck school bifurcates in its approach from the 'pure' category theorists, a theme that is important for the understanding of how the topos concept was later treated.
		There was perhaps a more direct route available: the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules.
		The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.
	www.ebroadcast.com.au /lookup/encyclopedia/in/Introduction_to_topos_theory.html (1465 words)

	Mathematical topos
		A topos (plural: topoi or toposes - this is a contentious topic) in mathematics is a type of category which allows the formulation of all of mathematics inside it.
		For instance, constructivists may be interested in the topos of all "constructible" sets and functions in some sense.
		The historical origin of topos theory is algebraic geometry.
	www.ebroadcast.com.au /lookup/encyclopedia/to/Topos_theory.html (399 words)

	Geometry.Net - Scientists: Grothendieck Alexander
		Alexander Grothendieck, born in 1928 in Berlin, is one of the leading mathematicians of the twentieth century, with
		Alexander Grothendieck, born in in Berlin, is one of the leading mathematicians of the twentieth century, with monumental contributions to functional analysis and algebraic geometry Born to Jewish parents, he was a displaced person during much of his childhood due to the upheavals of World War II.
		A topos (plural: topoi) in mathematics is a type of category which allows the formulation of all of mathematics inside it.
	www.geometry.net /scientists/grothendieck_alexander.php (1556 words)

	Grothendieck topology - ExampleProblems.com
		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.
		Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory which he suspected would be the Weil cohomology.
	www.exampleproblems.com /wiki/index.php/Grothendieck_topology (1308 words)

	Categorical logic Information
		This can be traced in a number of stages, from 1960 onwards: the formulation of the Grothendieck topos, and then of the elementary topos, giving rise first to topos theory.
		Topos theory, as would now be understood, is the intuitionistic replacement for set theory.
		This was one consequence, certainly unanticipated, of Grothendieck's relative point of view; and not lost on Pierre Cartier, one of the broadest of the core group of French mathematicians around Bourbaki and IHES.
	www.bookrags.com /wiki/Categorical_logic (822 words)

	Topos
		Another important example of a topos (and historically the first) is the category of all sheaves of sets on a given topological space.
		It is also possible to encode a logical theory, such as the theory of all groupss, in a topos.
		For instance, the category of all directed graphs is a topos.
	www.guajara.com /wiki/en/wikipedia/t/to/topos.html (616 words)

	Topos Information
		In mathematics, a topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space.
		Topos theory is, in some sense, a generalization of classical point-set topology.
		A ringed topos is a pair (X,R), where X is a topos and R is a commutative ring object in X.
	www.bookrags.com /wiki/Topos (1600 words)

	Alexander Grothendieck: A biography - Helium (Site not responding. Last check: 2007-10-17)
		Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century.
		He is 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.
		Grothendieck's radical left-wing and pacifist politics were doubtless born by his family history and his wartime experiences.
	www.helium.com /tm/55009/alexander-grothendieck-biography (1627 words)

	The Ultimate Topos - American History Information Guide and Reference
		A topos exists in which the axiom of choice is invalid.
		Constructivists will be interested to work in a topos without the law of excluded middle (allowing some propositions to be neither true nor false).
		It is also possible to encode a logical theory, such as the theory of all groups, in a topos.
	www.historymania.com /american_history/Topos (963 words)

	[No title]
		Y be a map of I-diagrams of simplicial objects in a topos E, and suppose that Y is a homotopy colimit diagram.
		A Grothendieck topos E is a category equivalent to some category ShC of sheaves of sets on a small Grothendieck site C.
		Among the many properties of a Grothendieck topos E, we note that E has all small limits * *and colimits, and that E is cartesian closed.
	www.math.purdue.edu /research/atopology/Rezk/rezk-sharp-maps.txt (5482 words)

	Bibliografia
		There is a good treatment of the construction of the latter topos, using Freyd's method of assemblies, and results from the last chapter which discusses the construction of universally adding quotients of equivalence relations to a regular category (via categories of relations) and provides a condition of when this construction yields a topos.
		Grothendieck's "double-plus" construction of the sheafification functor and its properties are given a carefully motivated exposition.
		Every Grothendieck topos is a classifying topos: the points of the topos (wherever they be taken) are the models for the theory classified.
	www.disi.unige.it /person/RosoliniG/ILM/bib01.html (4412 words)

	Background and genesis of topos theory at AllExperts
		The question of points was close to resolution by 1950; Grothendieck took a sweeping step (appealing to the Yoneda lemma) that disposed of it — naturally at a cost, that every variety or more general scheme should become a functor.
		An abelian category is supposed to be closed under certain category-theoretic operations — by using this kind of definition one can focus entirely on structure, saying nothing at all about the nature of the objects involved.
		Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier (see Verdier's Bourbaki seminar Analysis Situs).
	en.allexperts.com /e/b/ba/background_and_genesis_of_topos_theory.htm (1594 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		Grothendieck's proof that every AB5 category has enough injectives uses the axiom of choice (actually Zorn's lemma--which John Bell points out to me is significantly weaker than choice in toposes).
		And the proof in Johnstone's TOPOS THEORY that the category of Abelian groups over any Grothendieck topos has enough injectives uses Barr's theorem: Every Grothendieck topos is covered by one that satisfies the axiom of choice.
		And the proof in > Johnstone's TOPOS THEORY that the category of Abelian groups over any > Grothendieck topos has enough injectives uses Barr's theorem: Every > Grothendieck topos is covered by one that satisfies the axiom of choice.
	www.mta.ca /~cat-dist/catlist/1999/injchoice (340 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-17)
		A category equipped with a Grothendieck topology, that is, with a structure of "coverings" which makes it possible to define the notion of a sheaf on the category.
		is a topos, and is a reflective subcategory of
		The category of Abelian groups in a Grothendieck topos (equivalently, the category of sheaves of Abelian groups on a site) is a Grothendieck category, which makes it possible to define sheaf cohomology on a site; the cohomology groups
	eom.springer.de /s/s085660.htm (506 words)

	topos
		Then you might want to work in the topos of presheaves on X, or the topos of sheaves on X. Sheaves are important in twistor theory and other applications of algebraic geometry and topology to physics.
		Or, you might like to work in the topos of sheaves on a topological space - or even on a "site", which is a category equipped with something like a topology.
		These ideas were invented by Alexander Grothendieck as part of his strategy for proving the Weil conjectures.
	math.ucr.edu /home/baez/topos.html (1839 words)

	FOM: no foundations, but real analysis
		Dear all, this message tries to clarify some aspects of topos theory, related to the question whether or not one can develop real analysis inside a topos, and if yes, how far it can be done.
		They were invented by Grothendieck and his school to have the right set-up to do cohomology theory (the category of sheaves of abelian groups of a Grothendieck topos is an abelian category satisfying AB5*).
		In a topos with natural number object, you can develop intuitionistic analysis, with all its pro's and con's, for example, Dedekind reals and Cauchy reals may differ in such a set-up.
	www.cs.nyu.edu /pipermail/fom/1998-January/000862.html (1253 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-17)
		Blass [a1] showed that the presence of a natural numbers object is necessary as well as sufficient for the construction of classifying toposes.
		The existence of a natural numbers object, as a postulate, plays much the same role in topos theory as the axiom of infinity (see Infinity, axiom of) does in set theory.
		In classical set theory, this axiom is normally viewed as giving rise to the incompleteness phenomenon (see Gödel incompleteness theorem), via Gödel numbering of formulas; however, in the constructive logic of toposes, the picture is rather different.
	eom.springer.de /N/n120030.htm (746 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-10-17)
		Grothendieck's insight was that the definition of a sheaf depends only of the open sets of a topological space, not on the individual points.
		This allowed Grothendieck to define Ã©tale cohomology and l-adic cohomology, which were eventually used to prove the Weil conjectures.
		The notion of a topos was later abstracted by William Lawvere and Miles Tierney to define an elementary topos, which has connections to mathematical logic.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=sheaf_theory (5586 words)

	Boolean localization, in practice, by J.F. Jardine (Site not responding. Last check: 2007-10-17)
		The Boolean localization theorem asserts that every Grothendieck topos can be faithfully imbedded into a topos E that satisfies the axiom of choice.
		This theorem is a combination of results of Diaconescu and Barr, and has been known to have homotopy theoretic consequences since Van Osdol's proof of the Illusie conjecture in the late 1970's, but it has only appeared in the literature in fragmentary form until the recent appearance of the Mac Lane-Moerdijk text.
		These results have been known by other means since about 1984; the proof given here for simplicial sheaves roughly approximates that given by Joyal, but it does not involve sheaves of homotopy groups.
	www.math.uiuc.edu /K-theory/0129 (176 words)

	Citations: Simplicial objects in a Grothendieck topos - Jardine (ResearchIndex)
		Jardine, Simplicial objects in a Grothendieck topos, Contemp.
		Jardine, Simplicial objects in a Grothendieck topos, in "Applications of algebraic Ktheory to algebraic geometry and number theory, part I", vol.
		J.F. Jardine, Simplicial objects in a Grothendieck topos, Comtemp.
	citeseer.ist.psu.edu /context/178243/0 (1948 words)

	[No title]
		Although the latter offers a simpler approach and is meaningful for arbitrary elementary toposes, it is tied up with the question of the representability of stack completions for the category objects (``axiom of stack completions'').
		The stack completion (for the regular epis) of a (localic) groupoid G in a topos S is the S-indexed category Tors^1(G) of G-torsors.
		An alternative description, for an etale complete (localic) groupoid G, has been shown [Bunge 1979, 1990] to be describable in terms of the groupoid of (essential) points of the classifying topos BG (``classification theorem'').
	www.math.mcgill.ca /rags/seminar/seminar.listings.02 (783 words)

	Akademika
		Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets.
		Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced.
		Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.
	www.akademika.no /vare.php?ean=9780521441803 (297 words)

	CMS/CAIMS Summer 2004 Meeting
		We recall also that the category of small categories in a Grothendieck topos admits a Quillen model structure whose fibrant objects are stacks [TandJ].
		Any given object in a smooth topos will induce a function presheaf on finite-dimensional varieties; since continuous functions are not usually smooth, it is unlikely that the Dedekind reals (even two-sided) will be included in R.
		Motivated by this, Andy Pitts began the study of Grothendieck toposes as certain objects in the 2-category of cocomplete categories.
	www.cms.math.ca /Events/summer04/abs/tt.html (1220 words)

	[No title]
		For an elemenatry topos E satisfying 1) and 2) the condition 3) is equivalent to the requirement that any object X of E is a SUBQUOTIENT of a Delta(I).
		This yields a topos and the forgetful functor to the cat of sets is logical (which tells one how to construct power-sets).
		Formulas of set theory in which all quantifiers are bounded by V{sub} alpha (A)'s have a well-known interpretation in {script}E, since they are formulas of the internal logic of {script}E. The completeness and local smallness of E enable the author to interpret unbounded quantifiers as well.
	www.mta.ca /~cat-dist/catlist/1999/loc-sm-coc-top (981 words)

	Abstracts of Papers by Carsten Butz
		In fact, a topos with a natural numbers object is an adequate universe in which to develop intuitionistic mathematics, and such a topos may be seen as a categorical analogue of a model of intuitionistic Zermelo-Fraenkel set-theory.
		Abstract: Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces, so-called `topological semantics'.
		Abstract: By a classifying topos for a first-order theory T, we mean a topos E such that, for any topos F, models of T in F correspond exactly to open geometric morphisms F->E.
	www.itu.dk /people/butz/research/abstracts.html (1387 words)

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