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Topic: Elementary topos

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Topos

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	Topos - Wikipedia, the free encyclopedia
		A topos exists in which the axiom of choice is invalid.
		Another important example of a topos (and historically the first) is the category of all sheaves of sets on a given topological space.
		The historical origin of topos theory is algebraic geometry.
	en.wikipedia.org /wiki/Topos (1097 words)

	Background and genesis of topos theory - Wikipedia, the free encyclopedia
		General problems of so-called 'descent' in algebraic geometry were considered, at the same period when the fundamental group was generalised to the algebraic geometry setting (as a pro-finite group).
		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.
		The topos concept arose in algebraic geometry, as a consequence of combining the concept of sheaf and closure under categorical operations.
	www.wikipedia.org /wiki/Background+and+genesis+of+topos+theory (1532 words)

	Topos Theory
		Topos theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas.
		The original notion of topos, as a `generalized space' suitable for supporting the exotic cohomology theories required in algebraic geometry, sprang from the fertile brain of Alexandre Grothendieck in the early 1960s, and was developed in his `Seminaire de Geometrie Algebrique du Bois-Marie' particularly during the academic year 1963-64.
		It is not addressed to those who are trying to learn about topos theory for the first time, but rather to those who already have some acquaintance with the subject and who wish to deepen their understanding, or to learn about aspects of it which they have not previously encountered.
	www.wordtrade.com /science/mathematics/topostheory.htm (3716 words)

	Sheaves in Geometry and Logic : A First Introduction to Topos Theory (Universitext), Springer, Saunders MacLane, Ieke ...
		This introduction to topos theory begins with a number of illustrative examples that explain the origin of these ideas and then describes the sheafification process and the properties of an elementary topos.
		The use of this book to learn topos theory certainly puts this view to rest, as the authors have given the readers an introduction to topos theory that is crystal clear and nicely motivated from an historical point of view.
		A "topos" is essentially a category that allows the construction of pullbacks, products, and so on, with the philosophy being that objects are to be viewed not only as things but as also having maps (functors) between them.
	allentech.net /bookstore/item_0387977104.html (1338 words)

	Background and genesis of topos theory (Site not responding. Last check: 2007-10-16)
		Generalproblems of so-called 'descent' in algebraic geometry were considered, at the same period when the fundamental group was generalised to the algebraic geometry setting (as a pro-finite group).
		The theory rounded itself out, by establishing that a Grothendieck topos was acategory of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.
		The topos concept arose in algebraic geometry, as a consequence of combining the concept of sheafand closure under categorical operations.
	www.therfcc.org /background-and-genesis-of-topos-theory-206657.html (1408 words)

	Categorical logic (Site not responding. Last check: 2007-10-16)
		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.
		The founders of elementary topos theory were Lawvere and Tierney.
	www.worldhistory.com /wiki/C/Categorical-logic.htm (833 words)

	Mathematical topos : Topos
		See introduction to topos theory for an account of the genesis of this concept.
		A topos is a category which has the following additional properties:
		John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html (http://math.ucr.edu/home/baez/topos.html).
	www.fastload.org /to/Topos.html (438 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		He defined it as a category IB with pullbacks such that each slice is an elementary topos and the reindexing functors f* are logical and have right adjoints Pi_f.
		Nevertheless this definition of partial topos as a category all whose slices are toposes has a certain defect - pointed out by Benabou himself - namely that ANY groupoid is a partial topos as it has pullbacks and any slice is the trivial topos.
		EXAMPLES of Partial Toposes (i) (Benabou 1980) Let IE be an elementary topos and F a nonempty downward closed subset of Sub_IE(1).
	www.mta.ca /~cat-dist/catlist/1999/partial-topos (579 words)

	[No title]
		Of course, for some people, it would be a perfectly sensible point of view that topos theory is interesting precisely because of results to the effect that a couple of elementary constructs are sufficient to justify the fairly general use of set-theoretic notation, such as refering to a `set of sets of ordered pairs'.
		A topos that satisfies both an existential condition concerning sections of epis and a disjunctive condition concerning subsets of 1 is an important attempted extreme case of constancy and non-cohesion, that usually in mathematics becomes a more determinate category of variation and cohesion, modelled via structures sketched by diagrams of specified shapes.
		An itopos, or topos with inclusions, extends the notion of identity to a preorder on the objects as a subcategory of the topos.
	www.mta.ca /~cat-dist/catlist/1999/set-memb-func-comp (15844 words)

	Amazon.de: English Books: Sketches of an Elephant: A Topos Theory Compendium. 2 Volume Set. (Site not responding. Last check: 2007-10-16)
		Sketches of an Elephant: A Topos Theory Compendium.
		Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the overall unity of the subject.
		Zum Seitenanfang : Sketches of an Elephant: A Topos Theory Compendium.
	www.amazon.de /exec/obidos/ASIN/019852496X (273 words)

	[No title]
		This chapter will introduce topos theory, which arose from two separate explorations; first, Lawvere's proposal to replace the membership axioms for set theory by axioms on the composition of functions; and second, the Grothendieck initiatives in algebraic geometry.
		But a topos is not only a generalized space; it can also be viewed as a generalized ``universe'' of sets -- as indeed the sheaves on a one-point space form the classical category of sets.
		Any such topos is a universe in which one can do mathematics, classical except for the restriction that the logic in such a topos is in general ``constructive'' or ``intuitionistic''.
	www.elsevier.com /homepage/saj/523281/h14.htm (667 words)

	Abstract Stone Duality (Site not responding. Last check: 2007-10-16)
		A new definition of sobriety is formulated in terms of lambda calculus and elementary category theory, with no reference to lattice structure, but, for topological spaces, this coincides with the standard lattice-theoretic definition.
		This subcategory is then a topos, and the whole category is characterised in the minimal situation as that of locally compact locales over that topos.
		We present an elementary axiomatisation of synthetic domain theory and show that it is sufficient to deduce the fixed point property and solve domain equations.
	www.cs.man.ac.uk /~pt/ASD/index.html (3053 words)

	topos
		Okay, you wanna know what a topos is? First I'll give you a hand-wavy vague explanation, then an actual definition, then a few consequences of this definition, and then some examples.
		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.
		This is a great introduction to category theory via the topos of sets: it describes ordinary set theory in topos-theoretic terms, making it clear which axioms will be dropped when we go to more general topoi, and why.
	math.ucr.edu /home/baez/topos.html (1773 words)

	Dictionary of Meaning www.mauspfeil.net
		A topos is a type of category theory category that behaves like the category of sheaves of sets on a topological space.
		* John Baez: ''Topos theory in a nutshell'', [http://math.ucr.edu/home/baez/topos.html http://math.ucr.edu/home/baez/topos.html].
		There you find a list of all editors and the possibility to edit the original text of the article Topos.
	www.mauspfeil.net /Topos.html (305 words)

	Publications, Lecture Notes etc. - Thomas Streicher
		This topos was introduced independently at the same time by Dana Scott under the name "relative realizability topos" and studied in great detail by his student Lars Birkedal in his Thesis.
		In this short note we prove that for a bounded geometric morphism from E to S the fibration of maps which are locally quotients of discrete maps is a definable subfibration of the fundamental fibration of E.
		However, the topos induced by this tripos doesn't validate Markov's principle for higher types in contrast to (the Diller-Nahm variant of) Gödel's Dialectica Interpretation.
	www.mathematik.tu-darmstadt.de /~streicher (2266 words)

	Topology Seminar (Site not responding. Last check: 2007-10-16)
		The categorical axioms are elementary, and a corresponding type theory has been described.
		One is to develop a type theory for inductive and coinductive types that would be interpreted respectively as overt discrete and compact Hausdorff objects in ASD and so translate into a very simple lambda calculus.
		Roughly speaking, from this point of view a (Grothendieck) topos is a topological space for which there are "not enough opens" and instead the topological structure has to be defined through what are known as its sheaves.
	www.cs.bham.ac.uk /research/events/topology-seminar/topology.html (2602 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)

	SDRL: Systems Design Research Lab
		In contrast to elementary topos theory, such categories of sets model elementary set theory directly.
		We use the methods developed by Joyal and Moerdijk, employing an axiomatic notion of a system of small maps in a category of classes, which serve to limit comprehension in a new and flexible way.
		We show that every elementary topos arises as such a category of sets, and that (a certain) elementary set theory is deductively complete with respect to such topos models.
	www.cis.upenn.edu /sdrl/sem02.php3 (2444 words)

	A Guided Tour in the Topos of Graphs (ResearchIndex) (Site not responding. Last check: 2007-10-16)
		Abstract: In this paper we survey the fundamental constructions of a presheaf topos in the case of the elementary topos of graphs.
		Keywords: topos theory, graph theory, automata theory, transition systems 1 Introduction Sheaf topoi are usually associated to continuous...
		12 a topological topos (context) - Johnstone - 1979
	sherry.ifi.unizh.ch /5902.html (325 words)

	Introduction to topos theory: Definition and links.
		The 'open set' discussion had effectively been summed up in the conclusion that varieties had a rich enough site of open sets in unramified covers of their (normal) Zariski-open sets.
		They have also produced a spin-off in pointless topology, where the locale concept isolates some of more accessible insights found by treating topos as a significant development of topological space.
		Definition / meaning of Introduction to topos theory:
	www.encyclopedian.com /in/Introduction-to-topos-theory.html (1502 words)

	eWiC: Towards an Override in Topoi (Site not responding. Last check: 2007-10-16)
		The operations are developed in an elementary topos.
		This is achieved by constructing each operation in the topos Set, of sets and total functions, and then using these constructions as the definition of the operations in an elementary topos.
		As an example the operation of override is considered in the topos Set.
	ewic.bcs.org /conferences/1998/2ndirish/papers/paper6.htm (135 words)

	Mob Software
		"Topos" literally means a place or location in Greek, and a topos is a place from which similar stories can be woven.
		The Jini topos speaks of a world of "simply connect" and "spontaneous networks" accompanied by a set of home, office, and automobile scenarios derived from this topos.
		The Jini topos was highly effective and created a thriving community and associated technology—as hoped—along with thriving E-Speak, UPnP, and SOAP communities and associated technologies.
	www.dreamsongs.com /MobSoftware.html (11649 words)

	week68
		But instead of "time-dependent sets", let's act like topos theorists and call them simply "objects", and instead of talking about one being a "subset" of another, let's say one is a "subobject" of another.
		After studying this sort of thing a while, it's rather hard to go on pretending that the Zermelo-Fraenkel axioms of set theory, which were cooked up in the early 20th century to escape the logical paradoxes of Russell and others, are the last word on "foundations".
		One can develop topos theory within set theory if one wishes, but one can also set up topos theory from scratch, as a kind of pluralistic foundation of mathematics.
	math.ucr.edu /home/baez/week68.html (2650 words)

	Infinitary Logic
		A topos may be conceived, then, as a universe of "variable" sets.
		Any topos may be conceived as possible "universe of discourse" in which mathematical assertions may be interpreted and mathematical constructions may be performed.
		In other words two structures are topos isomorphic if their canonical representatives are isomorphic in the internal language of some topos.
	plato.stanford.edu /entries/logic-infinitary (6682 words)

	Oxford University Press
		This book covers elementary aspects of category theory and topos theory.
		It gives a clear exposition of key concepts and gives complete elementary proofs of theorems, including the fundamental theorem of toposes and the sheafification theorem.
		This book will be essential reading for third year undergraduates and graduates studying logics and category theory as part of a course on mathematics, computer science, or philosophy.
	www.oup.com /ca/isbn/0-19-851473-5 (225 words)

	Amazon.fr : Livres en anglais: Sketches Of An Elephant, tome 1 et 2 : A Topos Theory Compendium (Site not responding. Last check: 2007-10-16)
		Amazon.fr : Livres en anglais: Sketches Of An Elephant, tome 1 et 2 : A Topos Theory Compendium
		Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics.
		Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientsts, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continum physics.
	www.amazon.fr /exec/obidos/ASIN/019852496X (420 words)

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