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In the concluding statement of "universe hierarchies", Conor McBride calls it

[...] that key lesson which I learned from James McKinna: never resort to polymorphism when initiality will do.

How can you recognize when you could use initiality instead of polymorphism, and why should you prefer to use initiality?

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Initiality comes with a powerful universal property which allows you to, internally, prove statements about the constructions you perform. If you give me an element of data Nat = Z | S Nat, I can perform induction on it.

Polymorphism on the other hand requires you to appeal to external principles such as parametricity (*) to recover similar reasoning principles. If you give me a function of type (a -> a) -> a -> a, I will know it is bound to be a natural number in disguise but will be powerless to exploit that fact.

Additionally in predicative systems with a tower of universes, initiality is typically size-respecting whereas an encoding via polymorphism will land you a definition that lives one level up in the hierarchy due to the universal quantification. If you want to iterate these constructions, that's bad news.

(*) caveat: there are type theories attempting to internalise parametricity

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    $\begingroup$ Wish I could up-vote multiple times. This answer is so incredibly obvious (post facto) I'm kind of ashamed that I didn't see it myself. I think the deep explanation (prefer size-respecting internal reasoning over external, potentially size-increasing reasoning) is one that can likely be instantiated multiple times to good effect. $\endgroup$
    – Jacques Carette
    Commented Mar 25, 2022 at 12:35
  • $\begingroup$ In the third paragraph, do you mean "in predicative systems"? $\endgroup$
    – Mike Shulman
    Commented Mar 25, 2022 at 15:06
  • $\begingroup$ @MikeShulman Good catch. Fixed it now. Thanks! $\endgroup$
    – gallais
    Commented Mar 25, 2022 at 16:26

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