> No, you don't. Univalence is the axiom that transporting operations across such equivalences _always_ works.
Sure, but the strategy I outlined is risky (as in “may lead to getting suck and having to undo work”) in a language where this isn't guaranteed to work.
> As you say "a monoid is a type plus two operations" -- so fine, we can treat the monoid And as the type bool and the dictionary of operations on it, and all this still works out.
Yup, but Haskell doesn't let you define types parameterized by entire algebraic structures. It only lets you define types parameterized by the carriers of algebraic structures.
Sure, but the strategy I outlined is risky (as in “may lead to getting suck and having to undo work”) in a language where this isn't guaranteed to work.
> As you say "a monoid is a type plus two operations" -- so fine, we can treat the monoid And as the type bool and the dictionary of operations on it, and all this still works out.
Yup, but Haskell doesn't let you define types parameterized by entire algebraic structures. It only lets you define types parameterized by the carriers of algebraic structures.