Re: [HoTT] Weaker forms of univalence

[Bas Spitters <b.a.w.s...@gmail.com>]

>It was observed previously on this list,
Maybe we should be using our wiki more?
https://ncatlab.org/homotopytypetheory/


On Wed, Jul 19, 2017 at 7:19 PM, Michael Shulman <shu...@sandiego.edu> wrote:
> It was observed previously on this list, I think, that full univalence
> (3) is equivalent to
>
> (4)  forall A, IsContr( Sigma(B:U) (A ≃ B) ).
>
> This follows from the fact that a fiberwise map is a fiberwise
> equivalence as soon as it induces an equivalence on total spaces, and
> the fact that based path spaces are contractible.  But the
> contractibility of based path spaces also gives (2) -> (4), and hence
> (2) -> (3).
>
> I am not sure about (1).  It might be an open question even in the
> case when A and B are propositions.
>
>
> On Wed, Jul 19, 2017 at 9:26 AM, Ian Orton <ri...@cam.ac.uk> wrote:
>> Consider the following three statements, for all types A and B:
>>
>>   (1)  (A ≃ B) -> (A = B)
>>   (2)  (A ≃ B) ≃ (A = B)
>>   (3)  isEquiv idtoeqv
>>
>> (3) is the full univalence axiom and we have implications (3) -> (2) -> (1),
>> but can we say anything about the other directions? Do we have (1) -> (2) or
>> (2) -> (3)? Can we construct models separating any/all of these three
>> statements?
>>
>> Thanks,
>> Ian
>>
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