Yoneda Embedding

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Prev: The Yoneda Lemma Next: It’s All About Morphisms

Exercises

1. Express the co-Yoneda embedding in Haskell.
2. Show that the bijection we established between fromY and btoa is an isomorphism (the two mappings are the inverse of each other).
3. Work out the Yoneda embedding for a monoid. What functor corresponds to the monoid’s single object? What natural transformations correspond to monoid morphisms?
4. What is the application of the covariant Yoneda embedding to preorders? (Question suggested by Gershom Bazerman.)
5. Yoneda embedding can be used to embed an arbitrary functor category $[\mathcal{C},\mathcal{D}]$ in the functor category $[[\mathcal{C},\mathcal{D}],\mathbf{Set}]$. Figure out how it works on morphisms (which in this case are natural transformations).