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Topic: Categorical dualities

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	equivalent categories (Site not responding. Last check: 2007-10-25)
		In lattice theory, there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces.
		One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings.
		Under this duality, every compact Hausdorff space X is associated with the algebra of continuous complex-valued functions on X, and every commutative C*-algebra is associated with the space of its maximal ideals.
	www.yourencyclopedia.net /Equivalent_categories (886 words)

	Stone duality - Wikipedia, the free encyclopedia
		Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.
		Probably the most general duality which is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms.
		The categorical equivalence of Sob and SLoc is the basis for the mathematical area of pointless topology, that is devoted to the study of Loc - the category of all locales of which SLoc is a full subcategory.
	en.wikipedia.org /wiki/Stone_duality (2020 words)

	Stone duality -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-25)
		Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of (additional info and facts about Stone's representation theorem for Boolean algebras) Stone's representation theorem for Boolean algebras.
		Stone-type dualities also provide the foundation for (additional info and facts about pointless topology) pointless topology and are exploited in (additional info and facts about theoretical computer science) theoretical computer science for the study of (additional info and facts about formal semantics) formal semantics.
		The (additional info and facts about categorical equivalence) categorical equivalence of Sob and SLoc is the basis for the mathematical area of (additional info and facts about pointless topology) pointless topology, that is devoted to the study of Loc - the category of all locales of which SLoc is a full subcategory.
	www.absoluteastronomy.com /encyclopedia/s/st/stone_duality.htm (2193 words)

	Semilattice - Wikipedia, the free encyclopedia
		The conditions for homomorphisms of meet-semilattices are the obvious duals of these definitions.
		This gives rise to a number of useful categorical dualities between the categories of all complete semilattices with morphisms preserving all meets or joins, respectively.
		The case of free meet-semilattices is dual, using the opposite subset inclusion as an ordering.
	en.wikipedia.org /wiki/Semilattice (1524 words)

	[No title]
		The latter is nonduality, the former is duality.
		This duality of distinctness is thus maintained in creation both when they form a one (as in the Lord, in angels, and in regenerating persons) and when they are divided (as in those who are in the natural state without the spiritual).
		The Duality Or Distinctness Of All Things The duality or distinctness of all things when united into a one applies at all levels of creation and existence: The love and wisdom acting simultaneously and in harmony, form each and all things, yet in all of them, they are themselves distinct from each other.
	www.soc.hawaii.edu /leonj/v1ch4to5-nonduality.doc (21388 words)

	[No title]
		We give a new formula for the dualizing module of a Gorenstein ring; this formula involves differential graded algebras (or ring spectra) in an essential way and is one instance of a general construction that in another setting gives the Brown-Comenetz dual of the sphere spectrum.
		This is a formula which in one setting constructs the DUALITY 3 injective hull of the residue class field of a local ring (5.1), and in an- other gives the p-primary component of the Brown-Comenetz dual of the sphere spectrum (5.3).
		The Pontriagin dual or Matlis dual of an R-module M (with respect to I) is defined to be Hom R(M, I).
	hopf.math.purdue.edu /Dwyer-Greenlees-Iyengar/duality.txt (13454 words)

	Trespassing limits
		The importance of dual Platonic idealism should not be underestimated: paradoxically Rene Descartes, one of the most important artificers of the Age of Reason, formulated, from his famous dictum cogito ergo sum, the rational continuity of dualistic idealism by incorporating the Platonic Ideas into the realm of the human mind.
		The effects of categorical thinking are quite obvious in language, to such an extent that there are many critical voices from Lacanian to Derridean criticism that support the idea that, as it stands now, language cannot escape binary structures--that is to say, the rule of the Aristotelian categorical Law.
		Categorical language is finally subverted by the suggestion of a symbolic "crying" followed by the blank space on the final page, by the silence that precludes the defeat of the Word or Derridean Logos.
	tarlton.law.utexas.edu /lpop/etext/okla/collado24.htm (10460 words)

	Re: Linear logic semantics (Barwise) (353 lines)
		Here D constitutes the composite logical-nonlogical structure, the duality expresses conservation of information, and C is the view of D from a traditional logic perspective.
		A logician might identify this duality with word-object duality, the Galois connection between theories and classes of structures, with position naturally identified with word and momentum with object.) This symmetry of observation means that not all observers can be perfect.
		The dual A' of a CSLat A is the cslat A->2 of all CSLat maps from A to the 2-element CSLat, while the dual f':B'->A' of a map f:A->B is the map taking maps g:B->2 to gf:A->2.
	www.cis.upenn.edu /~bcpierce/types/archives/1992/msg00047.html (2031 words)

	Semilattice
		In this situation, the meet of x and y is denoted by x'y.
		A join-semilattice is an algebraic structure (S,), where satisfies the same axioms as above -- the difference in notation is only for convenience, since one has of cause dual interpretations in mind for the two semilattice operations.
		Similar conclusion hold for join-semilattices, where one just derives the order ≤ in a dual way.
	encyclopedia.codeboy.net /wikipedia/s/se/semilattice.html (1564 words)

	[No title]
		We encode categorical attributes, based on their labels, as an integer and break up the integer into bit planes.
		By this scheme we can encode a categorical attribute that has a domain of 2n values in n bits without losing any of the distance properties of the standard encoding (which uses one Boolean attribute for each of the 2n domain values).
		It is important to note that the standard encoding of categorical attributes that uses one Boolean attribute for each domain value can easily be regained by a bit-wise "AND" operation on a combination of Boolean attributes and their complements.
	www.cs.ndsu.nodak.edu /~datasurg/papers/02_PAKDD_Foundations.doc (1298 words)

	Equivalence of categories - InfoSearchPoint.com (Site not responding. Last check: 2007-10-25)
		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.
		In lattice theory, there are a number of famous dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces.
		A categorical equivalence of the above form, connecting classes of ordered sets to classes of topological spaces, is sometimes called Stone's duality.
	www.infosearchpoint.com /display/Equivalence_of_categories (618 words)

	Declarative Continuations and Categorical Duality - Filinski (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		It points out how values and continuations can be seen as categorically dual concepts, and that this symmetry extends to not only data types, but also control structures, evaluation strategies and higher-order constructs.
		The central idea is a view of continuations as a declarative concept, rather than an imperative one, and the implications of this make up the spine of the presentation.
		His categorical view of a continuation is a morphism A 0 where 0 is a weak initial object.
	citeseer.ist.psu.edu /filinski89declarative.html (684 words)

	Miroslav Plo\v s\v cica (Site not responding. Last check: 2007-10-25)
		Abstract:We establish categorical dualities between varieties of isotropic median algebras and suitable categories of operational and relational topological structures.
		We follow a general duality theory of B.A. Davey and H. Werner.
		The duality results are used to describe free isotropic median algebras.
	www.zblmath.fiz-karlsruhe.de /exx/journals/CMUC/cmuc9203/abs/plos.htm (56 words)

	Socially Marginal, Culturally Central ... (Site not responding. Last check: 2007-10-25)
		The dual injunctions of Romans 11:28-9, "secundum evangelium quidem inimici propter vos, secundum electionem autem clarissimi propter patres" ("As concerning the gospel, indeed, they are enemies for your sake: but as touching election, they are most dear for the sake of the fathers"),
		This dual perspective and the eschatological role imposed upon the Jews in the Pauline writings are developed by Augustine into what is called the doctrine of relative toleration.
		The dualities and contradictions that inform ecclesiastical and civil responses to the Jews continue into the fourteenth century with a noticeable deterioration in the position of the Jews in Christendom to that point: increasing erosion of their rights, and a serious attentuation of the long-observed doctrine of relative toleration.
	web.english.ufl.edu /exemplaria/preprints/narin.htm (7351 words)

	Citebase - Lifted closure operators
		Categorical structure of closure operators, volume 346 of Mathematics and its Applications.
		In Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), pages 284-296.
		In Categorical topology (Toledo, Ohio, 1983), volume 5 of Sigma Ser.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0502410 (497 words)

	\bf The Duality Between Aglebraic Posets and Bialgebraic Frames: A Lattice Theoretic Perspective
		The categorical correspondences considered are usually derived from a topological viewpoint (see, for example Abramsky and Jung [2], Abramsky [1], or Mislove [21]) and trace their lineage to the approach in Lawson [17].
		(D) is order-isomorphic to D. The correspondence extends to a categorical duality which assures that axiomatic semantics is compatible with dentotational semantics - in the sense that the denotation of a program is identified with the set of specifications which are true of it.
		Bialgebraic (that is, algebraic and dually algebraic), distributive lattices should be familiar enough to algebraists, but this may not be the case with domains.
	www.mtsu.edu /~jhart/ALGFRM.html (9751 words)

	[No title]
		For non-numeric attribute domains, such as categorical attribute domains, we either code them numeric or construct separate, individual, vertical bitmaps for each category.
		If the categorical domain is hierarchical, we simply use composite bitmaps to accommodate the higher levels in that concept hierarchy.
		Both the “partition - equivalence relation” duality and the “partition - label function” duality will be exploited in this treatment - namely, every partition generates an equivalence relation and vice versa, and every labeled partition generates a function from the partitioned set to the label set and vice versa.
	www.cs.ndsu.nodak.edu /~perrizo/classes/765/buildingptree.doc (2058 words)

	Forming The New Church Mind
		From this rational perspective the medical model of mental functioning rests on a nonduality model of the mind and is therefore fundamentally opposed to all the dualities of the Writings.
		The inflow is not into the body directly but into the “cupidities and falsities pertaining to man.” Second, note that medical and natural remedies heal the disease, or stop the action of the evil spirits, when the Lord heals the disease.
		The first duality we need to hold up to ourselves is the separation between the outer self and inner self, as explained in the quote above (CL 272; see also AC 5354).
	theisticpsychology.org /books/nonduality/v1ch6to7-nonduality.html (11227 words)

	Kestrel Institute - Research Staff - Dusko Pavlovic - Math and Logic papers
		In categorical proof theory, propositions and proofs are presented as objects and arrows in a category.
		It thus embodies the strong constructivist paradigms of propositions-as-types and proofs-as-constructions, which lie in the foundation of computational logic.
		Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logical-operations-as-adjunctions.
	www.kestrel.edu /home/people/pavlovic/math.html (435 words)

	Stone duality - Term Explanation on IndexSuche.Com
		The category CohSp of coherent sober spaces is equivalent to the category CohLoc of coherent locales.
		The duality of sober spaces and spacial locales
		At this point we got already more than enough data to obtain the desired result: the functors Ω and pt define an adjunction between the categories Top and Loc = Frmop, where pt is right adjoint to Ω and the natural_transformations ψ and φop provide the required unit and counit, respectively.
	www.indexsuche.com /Stone_duality.html (1801 words)

	Category Theory
		Indeed, from a categorical point of view, a set-theoretical cartesian product, a direct product of groups, a direct product of abelian groups, a product of topological spaces and a conjunction of propositions in a deductive system are all instances of a categorical concept: the categorical product.
		First, using the axiomatic method and the categorical language, Grothendieck defined abstractly types of categories, e.g., additive and abelian categories, showed how to perform various constructions in these categories and proved various results for them.
		Indeed, categorical logic, the study of logic with the help of categorical means, has been around for about 30 years now and is still vigorous.
	plato.stanford.edu /entries/category-theory (7029 words)

	MOB - New Books (Site not responding. Last check: 2007-10-25)
		The theory of natural dualities, as presented in this text, is broad enough to encompass many known dualities through a rich assortment of substantive theorems yet concrete enough to be used to generate an array of previously undiscovered dualities.
		This text will serve as a user manual for algebraists, for category theorists and for those who use algebra in their work, particularly mathematicians and computer scientists interested in non-classical logics.
		As the first text devoted to the theory of Natural Dualities, it provides an efficient path through a large body of results, examples and applications in this subject which is otherwise available only in scattered research papers.
	www.mathbookshelf.com /new/new3_22.htm (4521 words)

	Poetry. Media. Dickinson & Co.: A User's Guide: Writing Machines
		Breaking down dualities, body/mind, woman/man, emotion/reason, is so standard in women’s studies/feminist studies it isn’t even remotely new—how did text/medium, or content/form (where “form” isn’t long lines vs. short, but screen vs. manuscript vs. book) get left alone for so long?
		Hayles' focus on the "sharp line between representation and the technologies producing them" (19) is, in some ways, the distillation of all that we've covered so far.
		Technologies are not always metal and circuitry; they are also mechanisms of control, be it legislative, socially exclusionary or physically categorical.
	www.mith2.umd.edu /engl748a/archives/writing_machines.php (4865 words)

	[No title]
		A Counterpoint is designed to address Gould's focus: the problem of apparent dichotomies and dualities which are really continua.
		As a musical term, a counterpoint is defined as a combination of two or more related but independent melodies within a harmonic texture; a foil, a contrasting element or item; a device using significant contrast or interplay of distinguishable elements.
		In addition, we try to teach students to resist what we call the certainty principle and the hardening of the categories, to be open to problems, new ideas, and new challenges.
	www.chss.montclair.edu /inquiry/spr95/melchior.html (1679 words)

	Quotations from the Works of Leon James with Diane Nahl Part 1
		Because Cartesian and Kantian rationalism posit a scientifically empty duality, which is to say that the second, underlying world is unobservable, therefore it is that Kates rejects GT oriented research today.
		When he was in a dual mode of consciousness, Swedenborg seemed to himself phenomenologically walking around in a spiritual body, recognizably similar in appea rance to his physical body, and seeing and talking to others whom he knew from history or as acquaintances.
		Constitutive exchanges are sufficiently characterized by two features which are also necessary: (1) the mutual observance of a common (jointly held) turn taking procedure (Y. "the ritual of talk"), and (2) the mutual adherence to a common (jointly held) dialectics of moves and reply moves (Y. "discourse units", mentioning something, doing something).
	www.soc.hawaii.edu /leonj/leonj/leonpsy/qqlj.html (13378 words)

	Welcome to Categoricalanalysis
		ABSTRACT: An integration of dualist and non-dualist viewpoints both in terms of mind/body dualities as well as in terms of mortal mind/Divine Mind dualities showing the relation of plank's constant to free will.
		It is then demonstrated how certain developments in modern logic have not only invalidated Kant’s rationale for banishing the noumenal universe from rational scientific analysis but actually mandate bringing the noumenal universe and therefore the Phoenix into the realm of Science.
		ABSTRACT: Categoreal Relativity is derived from a theophanic approach to knowledge which takes as its starting point, that the laws of matter, energy, space and time are reflections on the space/time manifold of awareness, of the laws of awareness itself.
	www.categoricalanalysis.com (628 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		conventional tantric progress-schemes do seem to integrate the value of self-control, restraining oneself so as to pursue the categorical shift from animal (pasu) to vira (hero, if memory serves).
		It's the state beyond all of these dualities...
		so now you say that tantra is something beyond dualities, rather than that it is the search for union with the divine.
	www.luckymojo.com /avidyana/dallun/files/0004.nondual.ny (1250 words)

	PART II (Site not responding. Last check: 2007-10-25)
		Because of this paradoxical occurrence we have also questioned in part Norbert Elias' categorical view of the civilizing process as a process of increasing restraint.
		This dual agenda of the movement cannot be stressed enough.
		The downgrading of the supposedly upright citizen as an 'uptight square' was not a frontal attack on decency as much as it was a rejection of adjustment to social conventions that were seen to be producing destructive consequences.
	www.bdavetian.com /CH8.htm (16207 words)

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