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Dictionary:
du·al (dū'əl, dyū'-)  |
[Latin duālis, from duo, two.]
dually du'al·ly adv.| Thesaurus: dual |
| Antonyms: dual |
| Word Tutor: dual |
Education today, more than ever before, must see clearly the dual objectives: Educating for living and educating for making a living.
— James Mason Wood.
Tutor's tip: The character's "dual" (made up of two parts; having two functions) role of father and brother made it impossible for him to fight in a "duel" (combat) when he was challenged.
| Wikipedia: Dual (category theory) |
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about C, then its dual statement is true about Cop. Also, if a statement is false about C, then its dual has to be false about Cop.
Given a concrete category C, it is often the case that the opposite category Cop per se is abstract. Cop need not be a category that arises from mathematical practice. In this case, another category D is also termed to be in duality with C if D and Cop are equivalent as categories.
In the case when C and its opposite Cop are equivalent, such a category is self-dual.
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We define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms.
Let σ be any statement in this language. We form the dual σop as follows:
with 
Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions.
Duality is the observation that σ is true for some category C if and only if σop is true for Cop.
is a monomorphism if 
implies g = h. Performing the dual operation, we get the statement that 
implies g = h. for a morphism 
. This is precisely what is means for f to be an epimorphism. In short, the property of being a monomorphism is dual to the property of being an epimorphism.Applying duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category Cop is an epimorphism.
This example on orders is a special case, since partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
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| Translations: Dual |
Dansk (Danish)
adj. - todelt, dobbelt-
n. - dualis
v. tr. - ombygge til dobbelt kørebane
idioms:
Nederlands (Dutch)
tweevoudig, tweeledig, dubbel, tweevoud
Français (French)
adj. - double, binaire, à deux, à double, (Psych) dédoublement
n. - (Ling) duel
v. tr. - constituer/former deux objets, avoir un double (caractère, nature)
idioms:
Deutsch (German)
adj. - zweifach, doppelt, zweigliedrig
n. - (Ling.) Dual
v. - eine Straße zweispurig machen
idioms:
Ελληνική (Greek)
adj. - διπλός, διττός, διχασμένος, δυϊκός
n. - δυϊκός τύπος
idioms:
idioms:
Português (Portuguese)
adj. - duplo
n. - dual (m)
idioms:
Русский (Russian)
двойной, двоичный
idioms:
Español (Spanish)
adj. - doble, dual, binario
n. - doble, dual, binario
v. tr. - convertir en doble vía
idioms:
Svenska (Swedish)
adj. - som gäller två
n. - gram. dualism
中文（简体）(Chinese (Simplified))
两的, 双的, 双重的, 双倍的, 双数, 双数词, 改道路为双向道
idioms:
中文（繁體）(Chinese (Traditional))
adj. - 兩的, 雙的, 雙重的, 雙倍的
n. - 雙數, 雙數詞
v. tr. - 改道路為雙向道
idioms:
한국어 (Korean)
adj. - 이중의, 두 사람으로 이루어진, 양수의
n. - 양수형
v. tr. - 이중으로 만들다
idioms:
العربيه (Arabic)
(صفه) ثنائي, مزدوج (الاسم) المثنى
עברית (Hebrew)
adj. - דו-, כפול, זוגי
n. - צורה כפולה של שם-עצם או פועל
v. tr. - הפך דרך לכביש דו-מסלולי
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 | Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more |
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| Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more |
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| Antonyms. © 1999-2009 by Answers Corporation. All rights reserved. Read more |
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| Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved. eSpindle provides personalized spelling and vocabulary tutoring online; free trial. Read more |
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| Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Dual (category theory)". Read more |
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