1. ## 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 !

2. Originally Posted by slevvio
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 $\Phi:\textbf{Top}\to\textbf{Top}$?

3. 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

4. I am just wondering how s makes sense here. Thanks