Did you mean: dual, dual (grammatical number), dual (electronics), Dual (brand), Dual (film), duality (mathematics), Dual! Parallel Trouble Adventure, DUAL (cognitive architecture) More...

Results for dual

On this page:

Dictionary:

dual

(dū'əl, dyū'-) pronunciation

adj.

Composed of two usually like or complementary parts; double: dual controls for pilot and copilot; a car with dual exhaust pipes.
Having a double character or purpose: a belief in the dual nature of reality.
Grammar. Of, relating to, or being a number category that indicates two persons or things, as in Greek, Sanskrit, and Old English.

n. Grammar.

The dual number.
An inflected form of a noun, adjective, pronoun, or verb used with two items or people.

[Latin duālis, from duo, two.]

dually du'al·ly adv.

#SearchBtn{margin-top:14px !important;}

Library

Thesaurus: dual

adjective

Composed of two parts or things: biform, binary, double, duple, duplex, duplicate, geminate, twofold. See part/whole.
Consisting of two identical or similar related things, parts, or elements: double, paired, twin. See same/different/compare.

Antonyms: dual

adj

Definition: double
Antonyms: single

Word Tutor: dual

IN BRIEF: Made up of or having two parts or elements.

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, an abstract branch of mathematics, the dual category or opposite category C^op of a category C is the category formed by reversing all the morphisms of C. That is, we take C^op to be the category with objects that are those of C, but with the morphisms from X to Y in C^op being the morphisms from Y to X in C. Hence, the dual category of the dual category of a category is the original category itself.

Examples

An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤_new by

x ≤_new y if and only if y ≤ x.

For example, there are opposite pairs child/parent, or descendant/ancestor.

The category of Boolean algebras and Boolean homomorphisms can be shown to be the opposite of the category of Stone spaces and continuous functions.

Formal definition

Let $Σ$ be any statement of the elementary theory of an abstract category. We form the dual of $Σ$ as follows:

Replace each occurrence of "domain" in $Σ$ with "codomain" and conversely.
Replace each occurrence of $g \circ f =h$ with $f \circ g = h$

Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions. For example, consider the following statements about a category $\mathcal{C}$ :

$f : A \to B$
$f$ is monic, i.e. for all morphisms $g, h$ for which composition makes sense, $f \circ g = f \circ h$ implies $g = h .$

The respective dual statements are

$f : B \to A$
$f$ is epic, i.e. for all morphisms $g, h$ for which composition makes sense, $g \circ f = h \circ f$ implies $g = h .$

The duality principle asserts that if a statement is a theorem, then the dual statement is also a theorem. We take "theorem" here to mean provable from the axioms of the elementary theory of an abstract category. In practice, for a valid statement about a particular category $\mathcal{C}$ , the dual statement is valid in the dual category $\mathcal{C}^{*}$ ( $\mathcal{C}^{op}$ ).

Duality

The 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.

Generalising that observation, limits and colimits are interchanged when one passes to the opposite category. This is immediately useful, when one can identify the opposite category in concrete terms. For example the category of affine schemes is equivalent to the opposite of the category of commutative rings. 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.

Dualities

A duality between categories C and D is defined as an equivalence between C and the opposite of D. The above are all examples of dualities. A self-dual category is a category equivalent to its opposite. An example of a self-dual category is the category of finite abelian groups.

Covariance and contravariance of functors

One other way in which the concept is used is to remove the distinction between covariant and contravariant functors: a contravariant functor to D is equally a functor to the opposite of D.

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		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
		Antonyms. © 1999-2008 by Answers Corporation. All rights reserved. Read more
		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
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Dual (category theory)". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

On this page: Print Link

dual	Dual Lu40pw
Dual Dvi	Dual Xdvd180

	How do you establish paternity with out the father?
	What is the life cycle of flatworms?
	Will cherry tomato plants cross-pollinate with other varieties of tomato plant?
	How do you install a household entry door?
	What is the role of the seahorse in the food chain?

dual

Examples

Formal definition

Duality

Dualities

Covariance and contravariance of functors

See also

Keep Reading

Do you have the answers?