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Topic: Equivalence of categories

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Isomorphism of categories

	Equivalence of categories - Wikipedia, the free encyclopedia
		An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor.
		One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings.
		In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.
	en.wikipedia.org /wiki/Equivalence_of_categories (1435 words)

	Equivalence of categories (Site not responding. Last check: 2007-11-06)
		An equivalence of categories consists of a functor between the involved categories which is to have an "inverse" functor.
		In pointless topology the category of spatial locales is to be equivalent to the dual of category of sober spaces.
		D is an equivalence of categories and C is a preadditive category (or additive category or abelian category) then D may be turned into a preadditive (or additive category or abelian category) in a way that F becomes an additive functor.
	www.freeglossary.com /Equivalence_of_categories (1208 words)

	Equivalence of categories - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-06)
		equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same".
		Response transfer between stimuli in generalized equivalence classes: a model for the establishment of natural kind and fuzzy superordinate categories.
		Equivalences in categories, ([University of Oklahoma] Dept. of Mathematics.
	encyclopedia.worldsearch.com /equivalence_of_categories.htm (1470 words)

	Equivalence of categories
		If a category is equivalent to the dual of another category then one speaks of a duality of categories.
		In this situation, we say that the categories C and D are equivalent or dually equivalent, respectively.
		With some additional assumptions, it is indeed possible to determine whether a functor is part of an equivalence of categories, even when the remaining data is not given (see next section).
	pedia.newsfilter.co.uk /wikipedia/e/eq/equivalence_of_categories.html (1520 words)

	Equivalence of categories -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		An equivalence of categories consists of a (Click link for more info and facts about functor) functor between the involved categories, which is required to have an "inverse" functor.
		There is indeed a concept of (Click link for more info and facts about isomorphism of categories) isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.
		C is a (Click link for more info and facts about cartesian closed category) cartesian closed category (or a (A traditional theme or motif or literary convention) topos) iff D is cartesian closed (or a topos).
	www.absoluteastronomy.com /encyclopedia/e/eq/equivalence_of_categories.htm (1733 words)

	Order-embedding - Wikipedia, the free encyclopedia
		A way to understand the different nature of both of these weakenings by exploiting some category theory is discussed at the end of this article.
		The basic category for the study of partial orders is the category of posets and monotone functions.
		It is helpful to consider the abovementioned category as a category of categories: this can be done by noting that any poset is in fact a small-category with at most one arrow between two objects (pointing "upwards" in the order) and that functors between those categories correspond exactly to monotone mappings.
	www.wikipedia.org /wiki/Order-embedding (663 words)

	PlanetMath: equivalence of categories
		In practical terms, two categories are equivalent if there is a fully faithful functor
		Cross-references: isomorphic, faithful functor, equivalent, terms, adjoint, natural isomorphisms, functors, categories
		This is version 4 of equivalence of categories, born on 2003-02-26, modified 2005-05-22.
	planetmath.org /encyclopedia/EquivalenceOfCategories.html (70 words)

	[No title]
		The weak equivalenc* es are a simplicial analogue of the notion of equivalence* of categories.
		Given any model category M, the simplicial localization of M as given in [3] is a simplicial ca* tegory which possesses the homotopy-theoretic information contained in M. Finding a model category* structure on the category of simplicial categories is then the f* *irst step in studying the homotopy theory of homotopy theories.
		Recall that a model category structure on a category C is a choice of three d* is- tinguished classes of morphisms, namely, fibrations, cofibrations, and weak equ *iv- alences.
	hopf.math.purdue.edu /Bergner/SimplicialCategoryMC.txt (5934 words)

	n-Categories - Sketch of a Definition
		In a similarly appropriate sense the category svf(fam(c)^op) of contra-variant set-valued functors on fam(c) is the free symmetric monoidal small-ly co-complete category on c.
		Definition: the monoidal category of "c-signatures", written "sig(c)", is the category svf(prof(c)); it is a monoidal category because the universal property of svf(fam(c)^op) gives an equivalence of categories between sig(c) and the monoidal category of endomorphisms of the symmetric monoidal small-ly co-complete category svf(fam(c)^op).
		Definition: the category of "c-operads" is the category of monoids in the monoidal category sig(c).
	math.ucr.edu /home/baez/ncat.def.html (2249 words)

	On the algebraic K-theory of model categories, by Steffen Sagave (Site not responding. Last check: 2007-11-06)
		In this short paper we prove a generalization of Waldhausen's Approximation Theorem which applies for example to categories with cofibrations and weak equivalences arising as subcategories of model categories.
		Using this, we show that an exact functor inducing an equivalence of homotopy categories also induces an equivalence in the algebraic K-theory.
		Furthermore, we state conditions on a model category under which the inclusion of the subcategory of finite cofibrant objects into the subcategory of homotopy finite cofibrant objects is a K-theory equivalence.
	www.math.uiuc.edu /K-theory/0655 (102 words)

	equivalence of categories (Site not responding. Last check: 2007-11-06)
		Formally, given two categories C and D, an equivalence of categories is a functor F : C
		In this situation, we say that the categories C and D are equivalent.
		If F and G are contravariant functors, then one speaks instead of a ''duality of categories''.
	www.yourencyclopedia.net /equivalence_of_categories.html (1169 words)

	Equivalence of categories
		In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between categories that are "essentially the same".
		Establishing such an equivalence usually means to discover strong similarities between mathematical structures that formerly where considered to be unrelated or where the relation was not understood properly.
		A categorical equivalence of the above form, connecting classes of ordered sets to classes of topological spaces, is sometimes called Stone's duality.
	www.infomutt.com /e/eq/equivalence_of_categories.html (594 words)

	categories: adjoint equivalence (Site not responding. Last check: 2007-11-06)
		By this, I mean the 2-category Equiv which is freely generated by objects a and b, morphisms L: a -> b and R: b -> a, and isomorphisms i: 1_b => RL and e: LR => 1_a.
		Well, any equivalence in any 2-category C is just the same as a 2-functor F: Equiv -> C The functor F turns the "abstract" equivalence in Equiv into a "concrete" equivalence in C! This is reminiscent of Plato's theory of ideas and how they get manifested in concrete situations.
		Anyway, the walking equivalence is a weak 2-groupoid: a 2-category where every 2-morphism is invertible and every morphism is invertible up to 2-isomorphism.
	north.ecc.edu /alsani/ct01(9-12)/msg00093.html (974 words)

	Open problems on model categories
		This clearly ought to be the homotopy category of a model structure on the category of chain complexes of comodules, but we have been unable to build such a model structure.
		Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model category (with underlying fibrations and weak equivalences).
		The obvious candidate for this is the simplest nontrivial model category, the one on chain complexes where weak equivalences are chain homotopy equivalences.
	claude.math.wesleyan.edu /~mhovey/problems/model.html (1291 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The interest and the problem stem from the observation that the nerve functor N induces an equivalence between Cat and the category of simplicial sets, after inverting weak equivalences.
		Golasinski's proposed structure failed to satisfy the factorization axiom, but does lead to a closed model category on the category of pro-objects in Cat, as shown in Golasinski, On closed models on the precategory of small categories and simplicial schemes, Uspekhi Mat.
		The upshot of all this is that Thomason's is still the only model category structure known (or, at least, published) on Cat making it equivalent to the category of simplicial sets.
	www.lehigh.edu /~dmd1/sc55 (664 words)

	Concrete Categories
		In the paper, we develop the notation of duality and equivalence of categories and concrete categories based on [22].
		As the main result of this paper it is shown that every category is isomorphic to its concretization (the concrete category with the same objects).
		Categories without uniqueness of \rm cod and \rm dom.
	www.mizar.org /JFM/Vol13/yellow18.html (194 words)

	Behavioural Satisfaction and Equivalence in Concrete Model Categories - Bidoit, Tarlecki (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		We use the well-known framework of concrete categories to show how much of standard universal algebra may be done in an abstract and still rather intuitive way.
		This is used to recast the unifying view of behavioural semantics of speci cations based on behavioural satisfaction and, respectively, on behavioural equivalence of models abstracting away from many particular features of standard algebras.
		...of di erent equivalence de nable by means of open maps.
	citeseer.lcs.mit.edu /bidoit96behavioural.html (634 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The equivalence of the categories of vector bundles and projective modules of finite type over a Banach algebra.
		Equivalence of categories of finitely generated projective modules and vector bundles over a compact for C
		Vanishing of Hochschild, cyclic and periodic homologies on the category of Fredholm modules.
	www.rmi.acnet.ge /atestacia/staff/kandel-list.htm (137 words)

	The Ultimate Category of vector spaces - American History Information Guide and Reference
		In mathematics, the category K-Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms.
		There is a forgetful functor from K-Vect to Ab, the category of abelian groups, which takes each vector space to its additive group.
		K-Vect is a monoidal category with K (as a one dimensional vector space over K) as the identity and the tensor product as the monoidal product.
	www.historymania.com /american_history/K-Vect (200 words)

	Tim Hodges - Hodges-Smith 1 (Site not responding. Last check: 2007-11-06)
		T.J. Hodges and S. Smith, Sheaves of noncommutative algebras and the Bernstein-Beilinson equivalence of categories, Proc.
		This says that the category of modules over certain primitive factors of the enveloping algebra of a semi-simple Lie algebra is equivalent to the category of quasi-coherent sheaves over a sheaf of twisted differential operators on the flag variety.
		We show that the equivalence of categories will hold (roughly) if the direct sum of the local sections on an affine cover is faithfully flat over the ring of global sections.
	math.uc.edu /~hodgestj/research/citations/hs-shv.htm (457 words)

	Equivalence Of Categories (Site not responding. Last check: 2007-11-06)
		The western countries have used a concept called `Substantial equivalenceâ to determine the safety of GM...
		The amount will be risk adjusted for eligibility categories as well as...
		The following were the award categories and the respective award winners...
	www.wikiverse.org /equivalence-of-categories (1144 words)

	The Ultimate Simplicial set - American History Information Guide and Reference
		Using the language of category theory, a simplicial set is a simplicial object in
		, that is, a contravariant functor from the simplicial category Δ to
		There is an equivalence of categories between it and the subcategory with objects
	www.historymania.com /american_history/Simplicial_set (156 words)

	Professor Mark Johnson Abstract (Site not responding. Last check: 2007-11-06)
		Each will be invariant under an equivalence of homotopy categories, and will suffice to detect trivial homotopy categories.
		Analogous "stabilizations" of the category of topological spaces with respect to a chosen space are readily constructed as localizations of certain topological diagram categories.
		This sheds some light on the recent attempts by Po Hu to compute the Picard group of Voevodsky's stable category, which is a similar "stabilization" of "k-spaces" with respect to projective space.
	www.math.uiuc.edu /hilda/htmlcalendars/Jan31_00/johnson_feb01-00.html (178 words)

	The Psychological Record : A METHODOLOGICAL INTEGRATION OF GENERALIZED EQUIVALENCE CLASSES, NATURAL CATEGORIES, AND ... (Site not responding. Last check: 2007-11-06)
		A generalized equivalence class contains some stimuli that are perceptually disparate and others that resemble one another.
		They all function as members of a generalized equivalence class when all of them occasion the selection of the others in the set, and a response trained to one or a few are occasioned by all other stimuli in the set.
		The structural and functional properties that characterize generalized equivalence classes also characterize naturally occurring categories, natural kinds, semantic superordinate categories, and cross-modal perceptual classes.
	static.elibrary.com /t/thepsychologicalrecord/january012001/amethodologicalintegrationofgeneralizedequivalence/index.html (261 words)

	Equivalence and Duality for Module Categories (with Tilting and Cotilting for Rings) -- Robert R. Colby Kent R. Fuller (Site not responding. Last check: 2007-11-06)
		This book provides a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years.
		In particular, during the past dozen or so years many authors (including the authors of this book) have investigated relationships between categories of modules over a pair rings that are induced by both covariant and contravariant representable functors, in particular by tilting and cotilting theories.
		By here collecting and unifying the basic results of these investigations with innovative and easily understandable proofs, the authors' aim is to provide an aid to further research in this central topic in abstract algebra, and a reference for all whose research lies in this field.
	www.frontlist.com /detail/0521838215 (166 words)

	Citations: Multiple categories: the equivalence between a globular and cubical approach - AL-Agl, BROWN, STEINER ... (Site not responding. Last check: 2007-11-06)
		Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence between a globular and cubical approach, Advances in Math., 170 (2002) 71--118.
		, but none of the current definitions of n categories are expressed in the cubical terms that appear naturally in the n fold groupoid approach to homotopy theory.
		AL-Agl, R. BROWN and R. Multiple categories: the equivalence between a globular and cubical approach.
	citeseer.ist.psu.edu /context/1883768/0 (319 words)

	Topology and Computability (Site not responding. Last check: 2007-11-06)
		Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a Cartesian closed category with extended definability.
		Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a Cartesian closed category.
		We show how several familiar examples of categories of partial equivalence relations fit into the general framework.
	www.cs.cmu.edu /~birkedal/ltc/abstracts/exact-completion.html (83 words)

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