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# Topic: Dual category theory

###### In the News (Mon 20 May 13)

 PlanetMath: dual category More generally, an inverse limit is a direct limit on the opposite category; for this reason, it is sometimes called a colimit. A cokernel is a kernel in the opposite category. This is version 5 of dual category, born on 2002-02-25, modified 2004-03-29. www.planetmath.org /encyclopedia/DualCategory.html   (240 words)

 Category Theory (Stanford Encyclopedia of Philosophy) Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. Category theory is, in this sense, the legitimate heir of the Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis on the axiomatic method and algebraic structures. From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics. plato.stanford.edu /entries/category-theory   (11810 words)

 Encyclopedia: List of category theory topics   (Site not responding. Last check: 2007-10-17) In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... A comma category is a construction in category theory, a branch of mathematics. In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object. www.nationmaster.com /encyclopedia/List-of-category-theory-topics   (2723 words)

 Station Information - Dual (category theory) In category theory, an abstract branch of mathematics, the dual of a category is the category formed by reversing all the morphisms of. The dual of a dual of a category is itself. The Pontryagin duality restricts to the duality between the category of compact Hausdorff abelian topological groups and that of (discrete) abelian groups. www.stationinformation.com /encyclopedia/d/du/dual__category_theory_.html   (206 words)

 Category theory - Wikipedia, the free encyclopedia Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus. Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by "reversing all the arrows". en.wikipedia.org /wiki/Category_theory   (2348 words)

 PlanetMath: dual category   (Site not responding. Last check: 2007-10-17) For example, a coproduct is a product on the opposite category; this can be seen by looking at the commutative diagram that completely specifies a coproduct, and noting that it is the same as the diagram specifying a product with the arrows reversed. cokernel is a kernel in the opposite category. This is version 5 of dual category, born on 2002-02-25, modified 2004-03-29. planetmath.org /encyclopedia/Opposite.html   (226 words)

 Category theory - FreeEncyclopedia   (Site not responding. Last check: 2007-10-17) Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories. Dual vectorspace: an example of a contravariant functor from the category of all real vector spaces to the category of all real vector spaces is given by assigning to every vector space its dual space and to every linear map its dual or transpose. One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings. openproxy.ath.cx /ca/Category_theory.html   (2075 words)

 Dual (category theory) -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-17) Hence, the dual of a dual of a category is itself. The category of (Click link for more info and facts about Stone space) Stone spaces and (Click link for more info and facts about continuous function) continuous functions is equivalent to the opposite of the category of (A system of symbolic logic devised by George Boole; used in computers) Boolean algebras and homomorphisms. A ((geometry) the interchangeability of the roles of points and planes in the theorems of projective geometry) duality between categories C and D is defined as an (Essential equality and interchangeability) equivalence between C and the opposite of D. www.absoluteastronomy.com /encyclopedia/d/du/dual_(category_theory).htm   (452 words)

 Monomorphism - Wikipedia, the free encyclopedia The dual of a monomorphism is an epimorphism (i.e. Early category theorists argued that the correct category-theoretic generalization of injective (one-to-one) was the definition of monomorphism given above, and simply gave the word this new, somewhat different, meaning; this made the term ambiguous. However, whereas the difference is more notable in the case of epimorphisms, in "most" naturally occurring categories of algebras the categorical and algebraic meaning coincide because in any concrete category with a free object on a one element set the categorical monomorphisms are all one-to-one. www.wikipedia.org /wiki/Monic_morphism   (442 words)

 Como Category Theory News Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size. For any two categories that are objects of the metacategory, the category of functors from one to the other exists in the sense that it also is an object in the metacategory (it is unique by exponential adjointness). The category S is itself cartesian closed, and the categories of structures of geometry and analysis are enriched in it. categorytheorynews.blogspot.com   (4369 words)

 Dual (category theory) - Famous Women   (Site not responding. Last check: 2007-10-17) In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff abelian topological groups and the opposite of the category of (discrete) abelian groups. The category of Stone spaces and continuous functions is equivalent to the opposite of the category of Boolean algebras and homomorphisms. www.famous.tc /Dual_category.html   (344 words)

 Product (category theory) - Wikipedia, the free encyclopedia In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. The product construction given above is actually a special case of a limit in category theory. en.wikipedia.org /wiki/Product_(category_theory)   (399 words)

 eBay — Musical Instruments, Electric Guitar and Pro Audio items on eBay.com. Find IT on eBay. The eBay Musical Instruments category has new, used, and vintage electric and guitars, pro audio equipment, DJ gear, plus band & orchestra. Whatever your musical taste or your band's sound, eBay's Musical Instruments category is the place to buy and sell music gear. Whatever your skill level or interest, eBay's Musical Instruments categories is the ideal location to buy and sell music gear - we make it easy to upgrade your equipment and improve your sound! instruments.ebay.com   (346 words)

 Science Fair Projects - Coproduct It follows that if coproducts exists in a given category (they need not) they are unique up to a unique isomorphism that respect the injections. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors—so much for "dramatically different"). In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point). www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Coproduct_%28category_theory%29   (604 words)

 Luboš Motl's reference frame: Category theory and physics   (Site not responding. Last check: 2007-10-17) String theory is first of all a physical theory, and it should be studied because of physical motivations - the primary physical motivation is to locate the right ideas and equations that describe the real world. Category theory has been used by many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor, many people have claimed that everything that happens in a classical theory has a counterpart in the "corresponding" quantum theory. Category theory often resembles linguistics (or even postmodern literary criticism): it is a science about arrows between different objects and about creating new objects from these arrows, but it does not really care too much whether the objects exist and what are their real properties. motls.blogspot.com /2004/11/category-theory-and-physics.html   (3454 words)

 Dual Coding Theory: Theoretical Overview and Instructional Application   (Site not responding. Last check: 2007-10-17) By the late 1960s, psychological theory and research had undergone a pervasive shift away from behaviorism to an emphasis on cognitive processes and their effects in instruction and learning. DCT proposes that information is much easier to retain and retrieve when dual-coded because of the availability of two mental representations instead of one (Paivio, 1991). DCT claims that pictures are faster and easier to recall since they are coded in both memory systems and the visual system is continuous and parallel in its organization. www.kihd.gmu.edu /immersion/knowledgebase/strategies/cognitivism/DualCodingTheory.htm   (1960 words)

 Pullback (category theory) - Wikipedia, the free encyclopedia In category theory, a branch of mathematics, a pullback (also called a fibered product or cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. In the category of sets the pullback of f and g is the set The categorical dual of a pullback is a called a pushout. www.wikipedia.org /wiki/Fiber_product   (334 words)

 Science Fair Projects - Epimorphism In the category of sets the epimorphisms are exactly the surjective morphisms. In the category of monoids, Mon, the inclusion function N → Z is a non-surjective monoid homomorphism, and hence not an algebraic epimorphism. In the category of rings, Ring, the inclusion map Z → Q is a categorical epimorphism but not an algebraic one. www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Epimorphism   (398 words)

 Re: Category Theory and Physics | The String Coffee Table A state of gauge theory is a principal fiber bundle with connection. Category theory is the language that describes all sorts of gauge transformations. Perturbative RNS string theory is a categorification of supersymmetric quantum mechanics and natural isomorphisms in this context describe gauge and duality transformations of superstring backgrounds. golem.ph.utexas.edu /string/archives/000479.html   (3853 words)

 Literary Theory [Internet Encyclopedia of Philosophy] It is literary theory that formulates the relationship between author and work; literary theory develops the significance of race, class, and gender for literary study, both from the standpoint of the biography of the author and an analysis of their thematic presence within texts. Literary theory and the formal practice of literary interpretation runs a parallel but less well known course with the history of philosophy and is evident in the historical record at least as far back as Plato. The current state of theory is such that there are many overlapping areas of influence, and older schools of theory, though no longer enjoying their previous eminence, continue to exert an influence on the whole. www.iep.utm.edu /l/literary.htm   (4789 words)

 MATHS: Category Theory Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic. A Category is a mixture of an algebra and a directed graph. Category theory shows that in most known algebras and logistic systems, there is a way to construct an equivalent. www.csci.csusb.edu /dick/maths/math_25_Categories.html   (3607 words)

 Category Theory Categories are algebraic structures with many different complementary nature, e.g., geometric, logical, computational, combinatorial, just as groups are many-faceted algebraic structures. Another crucial aspect of category theory is that it allows to see how different kind of structures are related to one another. In the very spirit of category theory, what should matter here are the morphisms between categories. www.science.uva.nl /~seop/archives/win2004/entries/category-theory   (7032 words)

 Application of Category Theory in Model-Based Diagnostic Reasoning - LI, PEREIRA (ResearchIndex)   (Site not responding. Last check: 2007-10-17) System descriptions are regarded as categories, in which each object is a system model and each morphism represents changes and transformations of system models. A dual way would consist in beginning with the domain to be modeled as the basic category and defining morphisms from that domain... 3 Theory diagnosis: A concise characterization of faulty syste.. citeseer.ist.psu.edu /li95application.html   (604 words)

 Limit (category theory)   (Site not responding. Last check: 2007-10-17) In category theory, the limit of a functor generalizes the notions of inverse limit and product used in various parts of mathematics. If J is a small category and every functor from J to C has a limit, then the limit operation forms a functor from the functor category (see category theory) C In the category Ab of abelian groups, this for example shows that the kernel of a product of homomorphisms is naturally identified with the product of the kernels. www.city-search.org /li/limit-(category-theory).html   (945 words)

 Colimit   (Site not responding. Last check: 2007-10-17) In category theory the colimit of a functor, also known as a direct limit, is dual to the notion of a limit (inverse or projective limit). The general definition of a colimit is given on the limit page. Special cases of colimits, each dual to a special case of limits, include coproduct, coequaliser, pushout. www.theezine.net /c/colimit.html   (72 words)

 Product (category theory) -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-17) Let C be a category and let be an (Click link for more info and facts about indexed family) indexed family of objects in C. The product construction given above is actually a special case of a (The greatest possible degree of something) limit in category theory. The product can be defined as the limit of any (Click link for more info and facts about discrete subcategory) discrete subcategory in C. www.absoluteastronomy.com /encyclopedia/p/pr/product_(category_theory).htm   (611 words)

 week83 There is an interesting analogy between the dual of a vector space and the inverse of a number which I would like to explain now. Well, the whole point of the dual vector space y is that a vector in y is a linear functional from x to 1. So we see that dual vector spaces are a bit like inverse numbers, except that we don't have yx = 1 and 1 = xy, and we don't even have that yx is *isomorphic* to 1 and 1 is *isomorphic* to xy. math.ucr.edu /home/baez/week83.html   (1752 words)

 math lessons - Subobject The dual concept to a subobject is a quotient object; that is, to define quotient object replace monic by epic above and reverse arrows. In the category Sets, a subobject of A corresponds to a subset B of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in Sets is just its subset lattice. Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object (element) to another if the first is less than or equal to the second. www.mathdaily.com /lessons/Subobject   (309 words)

 Category Theory Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-10-17) Looking For category theory - Find category theory and more at Lycos Search. Find category theory - Your relevant result is a click away! Look for category theory - Find category theory at one of the best sites the Internet has to offer! www.karr.net /encyclopedia/Category:Category_theory   (190 words)

 dual - OneLook Dictionary Search DUAL : Stammtisch Beau Fleuve Acronyms [home, info] Phrases that include dual: dual currency bond, dual purpose, dual trading, dual boot, dual carriageway, more... Words similar to dual: twofold, double, dually, duple, threefold, treble, more... www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=dual   (313 words)

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