Products and Coproducts

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Prev: Kleisli Categories Next: Simple Algebraic Data Types

Exercises

1. Show that the terminal object is unique up to unique isomorphism.
2. What is a product of two objects in a poset? Hint: Use the universal construction.
3. What is a coproduct of two objects in a poset?
4. Implement the equivalent of Haskell Either as a generic type in your favorite language (other than Haskell).
5. Show that Either is a “better” coproduct than int equipped with two injections:

int i(int n) { return n; }
int j(bool b) { return b ? 0: 1; }

Hint: Define a function

int m(Either const & e);

that factorizes i and j.

6. Continuing the previous problem: How would you argue that int with the two injections i and j cannot be “better” than Either?
7. Still continuing: What about these injections?

int i(int n) {
    if (n < 0) return n;
    return n + 2;
}
 
int j(bool b) { return b ? 0: 1; }

8. Come up with an inferior candidate for a coproduct of int and bool that cannot be better than Either because it allows multiple acceptable morphisms from it to Either.