wofsy
- 724
- 0
What is Alexander-Whitney duality?
How Does Alexander-Whitney Duality Relate to Natural Transformations?
- Context: Graduate
- Thread starter wofsy
- Start date
-
- Tags
- Duality
Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
2 replies · 3K views
whybother
- 166
- 0
It's not a common term, but I assume you are really referring to the Alexander-Whitney map.
If [tex]X[/tex], [tex]Y[/tex] are topological spaces, then the AW map is the natural transformation between:
[tex](X, Y) \mapsto Sing (X \times Y)[/tex] and [tex]Sing (X) \otimes Sing (Y)[/tex]
Where [tex]Sing[/tex] 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 [tex]X[/tex], [tex]Y[/tex] are topological spaces, then the AW map is the natural transformation between:
[tex](X, Y) \mapsto Sing (X \times Y)[/tex] and [tex]Sing (X) \otimes Sing (Y)[/tex]
Where [tex]Sing[/tex] 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.
Last edited by a moderator:
- 14,922
- 28
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.
Last edited by a moderator: