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	Grothendieck universe - Wikipedia, the free encyclopedia
		A Grothendieck universe is meant to provide a set in which all of mathematics can be performed.
		Grothendieck universes are equivalent to strongly inaccessible cardinals.
		The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.
	en.wikipedia.org /wiki/Grothendieck_universe (576 words)

	Alexander Grothendieck - QuickSeek Encyclopedia (Site not responding. Last check: 2007-10-08)
		Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians active in the 20th century.
		Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.
		Initially, Grothendieck attended Élie Cartan's Seminar at École Normale Superieure, but lacking the neccessary background to follow the high powered seminar, he moved to University of Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950.
	encyclopedia.quickseek.com /index.php/Alexander_Grothendieck (1608 words)

	Universe (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use.
		Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis.
		This concept of a universe is reflected in the use of Venn diagrams.
	en.wikipedia.org /wiki/Universe_set (1858 words)

	PlanetMath: universe
		In order for uncountable universes to exist, it is necessary to adopt an extra axiom for set theory.
		Finally, one must be careful when using relations within universes; the details are too technical for Bourbaki to work out (!), but see the appendix to Exposé 1 of [SGA4] for more detail.
		This is version 5 of universe, born on 2003-03-14, modified 2004-04-07.
	planetmath.org /encyclopedia/Universe.html (278 words)

	ALEXANDER GROTHENDIECK : Encyclopedia Entry (Site not responding. Last check: 2007-10-08)
		Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century.
		Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Superieure, but lacking the necessary background to follow the high powered seminar, he moved to University of Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950.
		Grothendieck political views were considered to be radical left-wing and pacifist.
	bahairesearch.com /LookUpDefinitions/Alexander_Grothendieck (1599 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		We can drop the "replacement" condition on a Grothendieck universe U: for every x in U and every onto function h:x-->y with y a subset of U, y is also a member of U.
		Grothendieck's is equivalent to extending ZF by a proper class of inaccessible cardinals.
		I believe all the apparatus of Grothendieck's TOHOKU paper, and the topos theory and cohomology of the SGAs, is provable in Zermelo set theory with axiom of choice.
	www.mta.ca /~cat-dist/catlist/1999/small-universes (413 words)

	Edward Witten - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-08)
		Witten was born in Baltimore, Maryland, the son of Lorraine W. Witten and Louis Witten, a physicist specializing in gravitation and general relativity.
		Afterwards, he worked at Harvard University as a Junior Fellow and at Princeton as a professor.
		Likewise, in his bestseller The Elegant Universe, Columbia University physicist Brian Greene writes that Witten is "widely regarded as Einstein's successor in the role of the world's greatest living physicist."
	www.pole.ws /nph-proxy.pl/010110A/http/en.wikipedia.org/wiki/Edward_Witten (459 words)

	Home - Alexander Grothendieck (Site not responding. Last check: 2007-10-08)
		Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century.
		Grothendieck is one of the few mathematicians who matches the French concept of maître à penser; some go further and call him maître-penseur.) The bulk of Grothendieck's published work is collected in the monumental, and yet incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA).
		Today, most mathematicians are professors at a university or other research institution; however, a minority have a non-academic career and are often known as amateur mathematicians.
	alexander.grothendieck.en.infoax.org (9888 words)

	Universe (mathematics): Encyclopedia topic (Site not responding. Last check: 2007-10-08)
		This concept of a universe is reflected in the use of Venn diagram (Venn diagram: A diagram that uses circles to represent set theory; the position and overlap of the circles indicate the relations among the sets) s.
		We can give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model (model: A representative form or pattern) of Zermelo set theory (Zermelo set theory: zermelo set theory, as set out in an important paper in 1908 by ernst zermelo, is the...
		There is another approach to universes which is historically connected with category theory (category theory: category theory is a mathematical theory that deals in an abstract way with mathematical...
	www.absoluteastronomy.com /reference/universe_mathematics (2446 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		If C is endowed with a Grothendieck topology ø * (in the sense of [SGA4-I, M-M ]), one can define a notion of ø-local isomorphisms in P r(C) by r *equiring injectivity and surjectivity only up to a ø-covering1.
		Universes will be denoted by U 2 V 2 W.
		This universal property will* * not be used in the rest of the paper, but we believe it makes the meaning of the simplica* *il localization more transparent.
	www.math.purdue.edu /research/atopology/Toen-Vezzosi/agmod-I-fin-web.txt (17432 words)

	Not Even Wrong » Blog Archive » Grothendieck Biographical Article (Site not responding. Last check: 2007-10-08)
		Evidently Winfried Scharlau is writing a biography of Grothendieck, and Jackson’s article is partially based on materials he has gathered.
		Illusie was a student of Grothendieck’s, and Jackson’s article has some of his reminiscences about what that experience was like.
		At a fundamental level of this association was quantified in terms of quantum grvaity, then I saw computerization deeply valued in relation to discriptors of that early uiverse.
	www.math.columbia.edu /~woit/blog/archives/000078.html (1414 words)

	Grothendieck universe: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-08)
		A Grothendieck universe is meant to provide a set in which all of mathematics can be performed, Exception Handler: No article summary found.
		Grothendieck universes are equivalent to strongly inaccessible cardinal (In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular...)
		The idea of universes is due to Alexander Grothendieck (Alexander grothendieck (born march 28, 1928, berlin) was one of the most important mathematicians...)
	www.absoluteastronomy.com /ref/grothendieck_universe (939 words)

	PlanetMath: site
		The reference to universes and small sets in the definition may be safely ignored for most purposes; they exist to deal with set-theoretic difficulties one can encounter when dealing with certain sites (such as the crystalline site or the big étale site).
		Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3, available on the web at http://www.math.mcgill.ca/˜archibal/SGA/SGA.html
		Cross-references: étale site, sheaves, presheaves, fibre product, factor, projection, products, isomorphism, morphisms, maps, universe, complex, Weil conjectures, étale, fundamental group, theory, cohomology, open sets, Zariski topology, category, algebraic, topology
	planetmath.org /encyclopedia/GrothendieckTopology.html (340 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		We show that it gives an interesti* ng naturally occurring situation in which all but one of the hypotheses of the for mal Grothendieck* isomorphism of [2] hold, but the conclusion fails because the rele* *vant left adjoint fails to preserve compact objects.
		Since G-spectra in C are indexed on an N-trivial universe, SG =N ~= SJ and [2, 4.3] holds in the trivial form D(SJ) ~= SG =N. Thus all of the formal hypotheses of the formal Wirthmüller isomorphism theorem, [2, 6.1], are satisfied.
		The G-fixed point functor from D to the stable homotopy category of spectra is the composite of i* ** and the G-fixed point spectrum functor from C to spectra.
	hopf.math.purdue.edu /May/WirthRev.txt (3972 words)

	Surreal number Definition / Surreal number Research (Site not responding. Last check: 2007-10-08)
		By limiting the construction to a Grothendieck universeIn mathematics, if κ is a strongly inaccessible cardinal, the corresponding Grothendieck universe is the set of all sets with rank less than κ.
		By a theorem of Mirimanoff, the Grothendieck universe is a set, not a proper class.
		The idea is due to Alexander Grothendieck, who used it as...
	www.elresearch.com /Surreal_number (320 words)

	Some metaphysical questions about physics
		4) Although the realm of physics in the strict sense is only the physical universe, there is also the universe of mental states, sense qualia etc. The physical universe is for us as conscious observers only the determinator of experience but the experiences themselves transcend - in my opinion - their material preconditions.
		This invokes a system theoretic model of the universe as a hierarchy of nested conscious systems, many of which transcend human consciousness - for example towns, countrys, political, economical or other organizations, planets, etc. 6a) An interesting idea is that a given physical universe might have a multitude of experiential interpretations.
		Of course in such a universe the "binding laws" between matter and experiential state must be different.
	www.lns.cornell.edu /spr/2001-03/msg0031873.html (1036 words)

	Re: being inside a universe
		On the subject of "being inside a universe," there are some exciting papers by Fotini Markopoulou and others on a "category theory" (more precisely, "topos theory") outlook on this.
		One of her papers is "The internal description of a causal set; What the universe looks like from the inside," 1999.
		A popular treatment of the "what it means to be inside a universe" point of view is in the cosmologist Lee Smolin's book, "Three Roads to Quantum Gravity," less than a couple of years old.
	www.mail-archive.com /everything-list@eskimo.com/msg03682.html (842 words)

	[No title]
		These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function.
		Usually one takes as logical axioms at least some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.
		Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicans can actually prove all they need in systems weaker than ZFC, such as second order arithmetic.
	www.homestayfinder.com /Dictionary.aspx?q=axiom (2903 words)

	[No title]
		Argues that the epistemological questions of the preceeding two decades are misdirected, not fitting well with the practice of mathematics.
		Cole, K. The Universe and the Teacup: The Mathematics of Truth and Beauty.
		Mathematics is the loom upon which God weaves the fabric of the universe.
	canyoninstitute.org /resources/URBibliography/061_foundations_math_b.htm (398 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		It is my responsibility as a conscientious reporter, to let you know that I have studied the old, classical Algebraic Geometry, (notably the texts of Hodge and Pedoe), and various branches of classical Number Theory, such as Analytic Number Theory, Additive Number Theory, the Geometry of Numbers, Diophantine Approximation, and all that.
		In all fairness to Thom, it is known that Alexandre Grothendieck didn't treat him very well when they were both at the Institute des Hautes Etudes Scientifiques (the French Institute for Advanced Study), in the 70's.
		To Grothendieck, Thom and anyone else associated with the Institute was, prima facie, a capitalistic tool, stoodge, running dog, and unregenerable militarist.
	rendezvous.com /tangledweb/conferences/fermat/aug11.html (249 words)

	Citebase - Tensor products of idempotent semimodules. An algebraic approach (Site not responding. Last check: 2007-10-08)
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
		Grothendieck, Produits tensoriels topologiques et espaces nucléairs, Mem.
		We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0101153 (957 words)

	Publications, Lecture Notes etc. - Thomas Streicher
		In this note we show how a Grothendieck universe can be "lifted" to a type theoretic universe within every topos of presheaves.
		In this note we discuss a notion of universes for elementary toposes which allows one to define sequence of sets by recursion, i.e.
		In this talk we investigate what happens if logic is interpreted in a proof-irrelevant universe Prop of propositional types in which case AC does not holds anymore in general.
	www.mathematik.tu-darmstadt.de /~streicher (2364 words)

	Inaccessible cardinal - (Site not responding. Last check: 2007-10-08)
		If κ is the smallest ordinal which is either a weak inaccessible in the universe or weakly inaccessible relative to any standard set model, then L
		If the Generalized Continuum Hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.
		The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.
	psychcentral.com /psypsych/Weakly_inaccessible_cardinal (489 words)

	[No title]
		In fact, these authors note that Grothendieck did something similar back in 1971: he classified all groupoids fibered over a groupoid B in terms of weak 2-functors from B to Gpd, which is the 2-groupoid of groupoids!
		It's too large to be a space unless you pass to a larger universe of sets, but otherwise it's perfectly fine.
		I prefer to use a new Grothendieck universe for each level of the n-categorical hierarchy, to handle such expressions.
	math.ucr.edu /home/baez/twf_ascii/week223 (6565 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		ABSTRACTS First Talk : DECISION PROBLEMS, CURVATURE, AND THE UNIVERSE OF FINITELY PRESENTED GROUPS One may reasonably take the view that the most basic of finitely presented groups are the finite groups and that the next class worthy of mention is formed by the virtually cyclic groups.
		In this talk I shall sketch a map of this "universe of finitely presented groups", letting it emerge from Dehn's original formulation of the decision problems at the heart of infinite group theory.
		Curvature enters the discussion through Gromov's theorem that the groups that admit "the most efficient" solution to the word problem are precisely the hyperbolic groups.
	www.math.technion.ac.il /~techm/20050110153020050110bri (436 words)

	week223
		Grothendieck also studied this kind of thing with categories replacing groupoids, so there should also be an n-category version, I think...
		Grothendieck invented the notion of a "Grothendieck universe" for precisely this purpose:
		I prefer to use a new Grothendieck universe for each level of the n-categorical hierarchy, to justify such expressions.
	math.ucr.edu /home/baez/week223.html (6940 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		If U is a Grothendieck universe inspection of the construction in 2.6 shows * *that the functor S preserves U-smallness.
		One fixes an E1 operad E in a Grothendieck universe U and considers the cat* *egory of U-small E-spaces.
		John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.math.mcgill.ca Lawrence Breen, Universite de Paris 13: breen@dmi.ens.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, University of Genoa: carboni@vmimat.mat.unimi.it P.
	www.math.purdue.edu /research/atopology/Thomason/thomason_SymMon_equals_Spectra.txt (7868 words)

	[No title]
		We can imagine their nonexistence (their existence is independent of ZFC) but so far we haven't been able to imagine their existence (a proof in ZFC that they don't exist is still on the cards).
		The first Grothendieck universe is rather large, so it is a fairly formidable kit of constructions.
		Pass to a second Grothendieck universe, and you used at least one miracle, to produce an individual from the first-universe construction.
	www.mta.ca /~cat-dist/catlist/1999/choice-etc (3692 words)

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