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Topic: Duality of categories

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	Duality for Simple $\omega$-Categories and Disks
		The category $\cal S$ of simple $\omega$-categories is a full subcategory of the category, with strict $\omega$-functors as morphisms, of all $\omega$-categories.
		The category $\cal S$ is a key ingredient in another concept of weak $\omega$-category, called protocategory.
		We prove that $\cal D$ and $\cal S$ are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories.
	www.emis.de /journals/TAC/volumes/8/n7/8-07abs.html (212 words)

	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)

	categories: symmetry vs. duality
		In the vernacular, symmetry and duality are sometimes used interchangeably, both referring to 2-ness: the quality, character or condition of being two or twofold - a dichotomy.
		Indeed it has been suggested that visual symmetry, in the sense of exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis, is a source of beauty as a result of this balance or harmonious arrangement.
		The categorical notion of duality is captured precisely by adjointness, or the unity and identity of opposites.
	north.ecc.edu /alsani/ct01(5-8)/msg00048.html (425 words)

	Articles - Category theory (Site not responding. Last check: 2007-10-27)
		Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
		Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1945, in connection with algebraic topology.
		Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by "reversing all the arrows".
	www.lastring.com /articles/Category_theory?mySession=212cf973f817176716c66f3d09ad7c9c (2292 words)

	Inventiveness, Categories, Duality
		I asked a category theorist more or less why his field had not noticed the different physical and mathematical applications of subraction [or subtraction-addition] and division [or multiplication-division], and his reply was that the distributive law distinguishes them by a[b plus c] = ab plus ac, [b plus c]a = ba plus ca.
		Duality was Inventive, and yet there are some signs already that people are explaining this not by urging an Inventive Axiom [something like Almost Nothing is Answered which is not Inventive] but by appealing to Necessity.
		Duality was certainly Necessary because scientists did not know how to create masses from knowledge of non-massive theories, how to understand non-perturbation from perturbation, how to understand weak coupling from strong coupling and vice versa, how to understand different energy regimes, etc.
	superstringtheory.com /forum/dualboard/messages8/75.html (595 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		The "duality" in Joyal's paper is the "bottom level" case of our result; this is a well-known fact, and it is mentioned in Joyal's paper for expository purposes.
		This is intended as a finitary equivalent to the notion of multitopic category (the latter is a version of the Baez/Dolan concept of opetopic category; the main part of the definition of "multitopic category" is described in a paper by Claudio Hermida, the speaker and John Power, entitled "On higher dimensional categories I").
		Joyal's "theta category" is a well-identifble, albeit proper, part of the notion of "protocategory".
	www.math.mcgill.ca /rags/seminar/makkai.txt (380 words)

	Dual Reading Light Floor Lamps
		This Duality combination Torchiere will add a modern flair to any room.
		This stylish antique brass torchiere will be a lovely addition to any room.
		Finished in polished brass this torchiere lamp from the Duality collection also serves your read...
	www.lampsontheweb.com /NoFrame/Products/Dual-Reading-Light-Floor-Lamps1.html (634 words)

	Nikshych (Site not responding. Last check: 2007-10-27)
		A fusion category is a semisimple tensor category with duality.
		Such categories arise in several areas of mathematics and physics - conformal field theory, operator algebras, representation theory of quantum groups, and others.
		We show that the global dimension of a fusion category is always positive and that fusion categories and functors between them are undeformable (in particular, the number of categories realizing a given fusion rule is finite).
	www.imath.kiev.ua /~snmp2003/abstract2003/Nikshych.html (106 words)

	@CAT 2002-2003
		Abstract: The quadrality square combines (involutive) duality in linear structures such as the categories of abelian groups and sup-lattices with Girard's linear decomposition of cartesian closed categories and Stone duality for those categories.
		Moreover, we identified \K^R as categories with regular factorizations in which regular epimorphisms are closed with respect to composition.
		This suggests an extension of the category of commutative rings to a category of commutative rings and logical morphisms.
	www.mscs.dal.ca /~pare/Sem02-03.html (1493 words)

	Duality Based on Galois Connection. Part I
		In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [15, p.
		The duality is based on the concept of the Galois connection.
		Duality of Subcategories of {\it INF} and {\it SUP}
	www.mizar.org /JFM/Vol13/waybel34.html (128 words)

	Index 1-26
		It is know that the category PRTOP (of pretopological spaces and continuous maps) is not cartesian closed, and thus the same holds for the category PRAP of pre-approach spaces and contractions, introduced by E. Lowen and R. Lowen.
		Using the approach of Eckmann and Hilton to the spectral sequence in an abelian category, he considers exact couples in a semi-abelian category and shows the possibility of derivation if the endomorphism of the exact couple is strict and, consequently, the existence of the spectral sequence of the couple if all its morphisms are strict.
		For a C-indexed category, A, an A-descent equivalence is a morphism of bundles in C which induces an equivalence between the A-descent categories of its domain and codomain.
	perso.wanadoo.fr /vbm-ehr/CT/CT22.htm (6236 words)

	Maurice Auslander Distinguished Lectures (Site not responding. Last check: 2007-10-27)
		We are going to link this question with the topic of hereditary noetherian categories which, due to the work of a number of authors, has attracted a lot of attention in recent years.
		In the case of a, possibly graded, surface singularity we investigate the category of coherent sheaves on the punctured (graded) spectrum of the singularity.
		Each such category is a hereditary, in particular abelian, category whose objects are noetherian and which satisfies Serre duality, in a sense to be specified.
	styx.math.neu.edu /~alexmart/MADL/2004/2004lectures.html (474 words)

	[No title]
		It does not appear to be Poincare duality since there is no separate treatment of the torsion and torsion free parts.
		In fact, the former is dual to the N simplex and the latter to the N sphere.
		Do this in the stable-homotopy category (obtained by forcing the suspension functor on the homotopy category to become an automorphism) and normalize the dimensions so that the dual of a space has the negative of its dimension.
	www.mta.ca /~cat-dist/catlist/1999/algtop-barr (1088 words)

	Math 961 : Algebraic Theory of Tensor Categories (Fall 2004) (Site not responding. Last check: 2007-10-27)
		This course is an introduction to the theory of tensor categories.
		Such categories arise in several areas of mathematics and physics - conformal field theory, operator algebras, representation theory, and others.
		Non-degeneracy theorem for fusion categories and its consequences.
	www.math.unh.edu /~nikshych/961.html (298 words)

	Abstract Stone Duality (Site not responding. Last check: 2007-10-27)
		Abstract Stone Duality (ASD) is a type theory in which the topology on a space is an exponential with a lambda-calculus, not an infinitary lattice.
		By Stone duality, the opposite of the category C of ``spaces'' is to be a category of ``algebras'', but defined by a monad over C rather than over sets.
		An object of the category can then be defined using finitary data about a basis of open subspaces and a related family of compact ones, maybe obtained from a space that is locally compact in the external sense.
	www.cs.man.ac.uk /~pt/ASD/manifesto.html (2776 words)

	Duality for Some Categories of Coalgebras - Goldblatt (ResearchIndex) (Site not responding. Last check: 2007-10-27)
		Duality for Some Categories of Coalgebras - Goldblatt (ResearchIndex)
		Abstract: A contravariant duality is constructed between the category of coalgebras of a given signature, and a category of Boolean algebras with operators, including modal operators corresponding to state transitions in coalgebras, and distinguished elements abstracting the sets of states defined by observable equations.
		This duality is used to give a new proof that a class of coalgebras is definable by Boolean combinations of observable equations if it is closed under disjoint unions, domains and...
	citeseer.ist.psu.edu /goldblatt01duality.html (516 words)

	\bf The Duality Between Aglebraic Posets and Bialgebraic Frames: A Lattice Theoretic Perspective
		The need for categories domains to be cartesian-closed stems from the special demands of modelling recursion; it is not necessary for a general order theoretic treatment of the subject.
		Theorem 5 The category PCoh is dually equivalent to the category PCohFrm, and the category Coh is dually equivalent to the category CohFrm.
		Theorem 10 The category AlgTree is dually equivalent to the category BIRN, and the category AlgTree
	www.mtsu.edu /~jhart/ALGFRM.html (9751 words)

	[No title]
		Given Poincare spaces M and X, we study the pos- sibility of compressing embeddings of Mx I in Xx I down to em- beddings of M in X. This results in a new approach to embedding in the metastable range both in the smooth and Poincare duality categories.
		The homotopy category of R(X), denoted hoR(X), is the category whose objects are those of R(X) and in which the hom-set from an object Y to an object Z is given by homotopy classes of morphisms Y c!
		This admits the structure of a model category in which a morphism Y -!Z is a weak equivalence if (and only if) it is a weak homotopy equivalence of spaces.
	hopf.math.purdue.edu /Klein/compress1.txt (6272 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		Such classical notions as quadratic forms, hyper- and metabolic forms, and Witt groups can be introduced and studied over exact or triangulated categories with duality.
		We discuss a possibility of introducing analogous invariants for quadratic forms over exact categories with duality, with values in suitable subquotients of the higher K-groups.
		Over triangulated categories with duality, Witt theory has been developed recently by P. Balmer, with interesting applications to the Witt groups of algebraic varieties.
	www.math.mcgill.ca /rags/seminar/nenashev.txt (130 words)

	Journal of the American Mathematical Society (Site not responding. Last check: 2007-10-27)
		As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.
		As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.
		A. Bondal and M. Van den Bergh, Generators of triangulated categories and representability of functors, in preparation.
	80-www.ams.org.library.uor.edu /jams/2002-15-02/S0894-0347-02-00387-9/home.html (664 words)

	CoACT.2002.html
		Day and Street believe their discovery [34] that star-autonomy is the appropriate concept of duality required to quantize the theory of groupoids is quite exciting.
		Apart from their appearance in mathematics, star-autonomous categories have been known for over a decade to model the linear logic of Girard used extensively in computer science as a way of coping with resources and resource control.
		He furthermore constructed an action of a categorical analogue of the little n-cubes operad on the category of extensions in a monoidal abelian category; this is an interesting weak form of the generalised Deligne hypothesis.
	www.maths.mq.edu.au /~street/CoACT.2002.html (2701 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		By definition the dependency structure is a tree: there is a head uniquely identifiable as the root, and, a head may govern various dependents but a dependent may only modify a single head.
		Control Categories and Duality Peter Selinger, University of Michigan and BRICS In this talk, I will describe a class of categorical models for functional languages with control operators, and specifically for Parigot's lambda mu calculus.
		I will introduce the class of "control categories", which is based on Power and Robinson's premonoidal categories and closely related to Thielecke's tensor-not categories.
	www.cs.bham.ac.uk /news/notice-board/theorysem/theory.summer98 (2806 words)

	M. Montserrat Bruguera
		A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches.
		Reflexivity in this category was defined and studied by E. Binz and H. Butzmann.
		E. Binz and H. Butzmann have succeeded to extend Pontryagin duality theory to the category of convergence abelian groups and continuous homomorphisms, CONABGRP.
	www.utm.edu /~jschomme/topology/c/a/a/h/10.htm (924 words)

	Products And Duality In Categories With Cofibrations And Weak Equivalences (ResearchIndex) (Site not responding. Last check: 2007-10-27)
		Corollary: The Tate cohomology of Z=2 acting on the K--theory of any ring with involution is a generalized Eilenberg--MacLane spectrum, and it is 4--periodic.
		Introduction Categories with cofibrations and weak equivalences were introduced in [Wald].
		0.2: On The Karoubi Filtration Of A Category - Cárdenas, Pedersen
	citeseer.ist.psu.edu /249777.html (264 words)

	Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics): ... (Site not responding. Last check: 2007-10-27)
		More recently, many authors (including the authors of this book) have investigated relationships between categories of modules over a pair of rings that are induced by both covariant and contravariant representable functors, in particular, by tilting and cotilting theories.
		He is responsible for the definition of generalized Morita duality and was one of the first to consider the tilting and cotilting theory of finite dimensional algebras in the more general setting of general ring theory.
		Kent R. Fuller is a professor of mathematics at the University of Iowa.
	www.newyorkwebhosting.us /stuff-0521838215.html (341 words)

	TAC Seminars Home Page
		If you are interested in attending, you may find useful to look at information about the departments or about hotels in Genoa.
		Categorie d'Azumaya e gruppo di Brauer di una categoria monoidale chiusa
		On algebraically exact categories and essential localizations of varieties
	www.disi.unige.it /seminars/tac (198 words)

	AMCA: Duality for Convergence Abelian Groups by M. Montserrat Bruguera
		Binz and H. Butzmann have succeeded to extend Pontryagin duality theory to the category of convergence abelian groups and continuous homomorphisms, CONABGRP.
		If G is a LCA group, the continuous convergence structure in \GammaG is precisely the convergence given by the compact open topology [3], thus, the ''convergence dual'' and the ordinary dual are identical.
		G is an isomorphism in the category CONABGRP.
	at.yorku.ca /c/a/a/h/10.htm (846 words)

	ALGEBRA SEMINAR
		Semi--dualizing complexes for $R$ are defined by generalizing the above definition to the derived category of $R$--modules.
		The aim is to show that \SD\ complexes provide a suitable language for a unified treatment of some notions and techniques usually regarded as being of different nature.
		The choice of the term \emph{semi--dualizing} should seem fairly natural to those familiar with dualizing complexes, however, it has been pointed out to me that it is not in line with the spirit of the times and an alternative has been suggested, but this will only be revealed at the end of the talk.
	www.math.ku.dk /cal/events/122.htm (229 words)

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