How Does Alexander-Whitney Duality Relate to Natural Transformations?

[wofsy]

What is Alexander-Whitney duality?

 

[whybother]

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.

 

[Hurkyl]

Since it's a http://en.wikipedia.org/wiki/Natural_transformation" , it could be labeled a duality.

At the risk of derailing the thread... how? 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.