Why do we need natural transformations?

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  • #1
math.geek
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Few days ago, I was thinking about why we need to define V*=Hom(V,K) for a K-vector space when the dimension of V is finite because then V* and V both will have the same dimension and will be isomorphic. So, I couldn't understand why such a thing would be even called a dual vector space if it's the same thing algebraically when the dimension is finite. Then I read that these two vector spaces are isomorphic but there's no natural isomorphism between them.

I'm familiar with some terminology in category theory. I know the definition of a natural transformation in category theory. But I don't understand why natural definitions are important. I remember somewhere I read that Mac Lane has said that he didn't invent category theory to study functors, he invented it to study natural transformations.

Can someone explain to me in layman terms why natural transformations are interesting and how I should think of them?
 

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  • #3
WWGD
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Maybe it is also interesting to understand what happens , i.e., what are the overall advantages when you do have a natural, basis-independent choice, like when you have a non-degenerate bilinear form associated to your space, like, say, a Riemannian metric ( which gives you the natural "musical isomorphism" between tangent and cotangent spaces). I guess in the case of R^n, life becomes simpler in terms of differentiating ( trivial Euclidean connection) , but I can't think now of other areas where having a natural , canonical isomorphism helps, or at least what are other implications of having a natural isomorphism.
 

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