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duality principle
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(Definition)
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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 $\mathcal{C}$ :
- $f:A \to B$
- $f$ 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 epi, 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 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.
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| Also defines: |
self-dual statement |
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Cross-references: self-dual, property, dual category, valid, axioms, mean, theorem, epi, implies, morphisms, monic, category, compositions, conversely, occurrence, abstract category, theory
There are 7 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 5095 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) |
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Pending Errata and Addenda
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