Natural Transformations

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Prev: Function Types Next: Declarative Programming

Exercises

1. Define a natural transformation from the Maybe functor to the list functor. Prove the naturality condition for it.
2. Define at least two different natural transformations between Reader () and the list functor. How many different lists of () are there?
3. Continue the previous exercise with Reader Bool and Maybe.
4. Show that horizontal composition of natural transformation satisfies the naturality condition (hint: use components). It’s a good exercise in diagram chasing.
5. Write a short essay about how you may enjoy writing down the evident diagrams needed to prove the interchange law.
6. Create a few test cases for the opposite naturality condition of transformations between different Op functors. Here’s one choice:

op :: Op Bool Int
op = Op (\x -> x > 0)

and

f :: String -> Int
f x = read x