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Math Help - Natural

  1. #1
    Senior Member slevvio's Avatar
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    Natural

    I know what a chain homotopy is, but can anyone tell me what a natural chain homotopy is?

    I know this means that it must arise from a functor but what category is a natural chain homotopy a morphism in?


    I understand what a natural chain map is, since those are morphisms in the category of chain complexes.

    Thanks for any help !
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by slevvio View Post
    I know what a chain homotopy is, but can anyone tell me what a natural chain homotopy is?

    I know this means that it must arise from a functor but what category is a natural chain homotopy a morphism in?


    I understand what a natural chain map is, since those are morphisms in the category of chain complexes.

    Thanks for any help !
    Isn't it just a functor \Phi:\textbf{Top}\to\textbf{Top}?
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  3. #3
    Senior Member slevvio's Avatar
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    well my book (Dold) has a diagram and says F0,F1 and s satisfy the naturality diagram, (Where F0,F1 are natural chain maps and s is a natural chain homotopy)

    but I didn't understand this since F,G are chain maps, and s is not a chain map. Also chain homotopies go between abelian groups, not topological spaces. Thanks for your answer, I will try to upload a picture so you can see what im on about
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  4. #4
    Senior Member slevvio's Avatar
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    I am just wondering how s makes sense here. Thanks
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