Re: [HoTT] Injective types

[Andreas Nuyts <andreasnuyts@gmail.com>]

Even with cumulativity, that sounds suspicious, because cumulativity of 
U in a bigger universe V does not obviously give you a map

(A B : U) -> Id V A B -> Id U A B.

The J-rule does not allow you to build this map.

On 7/05/19 14:42, Nils Anders Danielsson wrote:
> On 02/05/2019 22.46, escardo.martin@gmail.com wrote:
>> I can confirm from a 26k line Agda development (with comments and
>> repeated blank lines removed in this counting of the number of lines)
>> that not once did I need to embed a universe into a larger universe,
>> except when I wanted to state the theorem that any universe is a
>> retract of any larger universe if one assumes the propositional
>> resizing axiom (any proposition in a universe U has an equivalent copy
>> in any universe V). So I would say that such situations are *extremely
>> rare* in practice.
>
> I once wrote some code where I made use of univalence at three different
> levels. Does anyone know if one can prove that univalence at one level
> implies univalence at lower levels, without making use of cumulativity?
>

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