- #1
wofsy
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What is Alexander-Whitney duality?
How Does Alexander-Whitney Duality Relate to Natural Transformations?
- Context: Graduate
- Thread starter wofsy
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SUMMARY
The discussion centers on the Alexander-Whitney duality, specifically the Alexander-Whitney map, which is a natural transformation between the singular chain complexes of topological spaces. The mapping is defined as (X, Y) → Sing(X × Y) and Sing(X) ⊗ Sing(Y), where Sing denotes the total singular chain complex. The conversation also touches on the relationship between natural transformations and dualities, emphasizing that dualities are often represented as contravariant functors, with a natural transformation typically providing an isomorphism from an object to the dual of its dual.
PREREQUISITES- Understanding of topological spaces
- Familiarity with singular chain complexes
- Knowledge of natural transformations in category theory
- Basic concepts of duality in mathematics
- Research the properties of the Alexander-Whitney map in algebraic topology
- Study the concept of natural transformations in category theory
- Explore contravariant functors and their applications
- Investigate the implications of duality in various mathematical contexts
Mathematicians, particularly those specializing in algebraic topology and category theory, as well as students seeking to deepen their understanding of natural transformations and dualities.
- #2
whybother
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It's not a common term, but I assume you are really referring to the Alexander-Whitney map.
If X, Y are topological spaces, then the AW map is the natural transformation between:
(X, Y) \mapsto Sing (X \times Y) and Sing (X) \otimes Sing (Y)
Where Sing is total singular chain complex for the specified top. space.
Since it's a http://en.wikipedia.org/wiki/Natural_transformation" , it could be labeled a duality.
If X, Y are topological spaces, then the AW map is the natural transformation between:
(X, Y) \mapsto Sing (X \times Y) and Sing (X) \otimes Sing (Y)
Where Sing is total singular chain complex for the specified top. space.
Since it's a http://en.wikipedia.org/wiki/Natural_transformation" , it could be labeled a duality.
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- #3
Hurkyl
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At the risk of derailing the thread... how?whybother said:
Dualities are typically expressible as contravariant functors -- the closest "natural transformation" gets to the notion of duality is there is typically a natural transformation (usually isomorphism) from an object to the dual of its dual.
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