Re: [HoTT] Injective types

[Michael Shulman <shulman@sandiego.edu>]

On Tue, Apr 30, 2019 at 11:25 PM <escardo.martin@gmail.com> wrote:
> It is nice, in practice, that universes are uniquely
> determined. Insted of thousands of level constraints, we have a
> unique, small, (polymorphic) level assignment.
> ...
> I think it is fair to the
> reader to let them know what the universe level assignments are rather
> than just say that they exist. In some cases, as injectivity, the
> universe levels are crucial, and if we ignore them we get false
> claims.

Yes, this is a good point in favor of Agda-style non-cumulative
Russell universes over Coq-style cumulative Russell universes.

But isn't there a middle ground with Tarski universes?  Suppose we
have explicit lifting operators Lift : U_i -> U_{i+1}, so that as in
Agda we have unique small polymorphic level assignments.  But then
suppose we have *definitional* equalities El(Lift(A)) == El(A).  Then
on the (rare) occasions when we do have to explicitly lift types from
one universe to another, we would get stronger cumulativity behavior.
(And I could imagine a proof assistant that implements this and sugars
away the El to look like Russell universes to the user.)

-- 
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.