PlanetMath: duality principle

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duality principle

(Definition)

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

Replace each occurrence of “domain” in $\Sigma$ 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 $\mathcal{C}$ :

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

The respective dual statements are

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

The duality principle asserts that if a statement is a theorem, then the dual statment 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 $\mathcal{C}^{*}$ ( $\mathcal{C}^{op}$ ).

If the property $\Sigma$ is the same as its dual, then it is called self-dual.

"duality principle" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Also defines:

self-dual statement

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Cross-references: self-dual, property, dual category, axioms, mean, epi, implies, morphisms, monic, compositions, occurrence, category, theory
There are 6 references to this entry.

This is version 5 of duality principle, born on 2002-02-25, modified 2006-07-07.
Object id is 2688, canonical name is DualityPrinciple.
Accessed 3861 times total.

Classification:

AMS MSC:

18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)

Pending Errata and Addenda

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