Re: [HoTT] Identity types of types, and univalence for the entire type theory

[Michael Shulman <shulman@sandiego.edu>]

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It's tempting to think that one can define univalence without universes,
but I don't think this rule makes sense:

On Sat, Oct 8, 2022 at 6:59 AM Madeleine Birchfield <
madeleinebirchfield@gmail.com> wrote:

> The elimination rule says that given a judgment 'A type' in context Gamma,
> a judgment 'B type' in context Gamma, a judgment 'C(p) type' in the context
> 'Gamma, p : A = B', and a judgment 't:C(refl_A)' in the context 'Gamma',
> one could form the conclusion 'J(t, p): C(p)' in the context 'Gamma'


In the ordinary Id-elimination rule, the motive C has to be defined in the
context of two *variable* elements of the type and an equality between
them, not two *specific* elements.  In particular, if A and B are specific
types, then it doesn't make sense to substitute refl_A for p in C, because
you can't substitute A for B.  You can only substitute for a variable.

I think in order to do something like this, you have to augment type theory
by some kind of "polymorphism" that will allow you to hypothesize a "type
variable" in the context.

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