Re: [HoTT] Weaker forms of univalence

[Michael Shulman <shu...@sandiego.edu>]

But is it known that this is definitely weaker?  E.g. are there models
that satisfy invariance but not the computation rule?

On Thu, Jul 20, 2017 at 4:59 AM, Steve Awodey <awo...@cmu.edu> wrote:
> I think we’ve been through this before:
>
>  (1)  (A ≃ B) -> (A = B)
>
> is logically equivalent to what may be called “invariance”:
>
>         if P(X) is any type depending on a type variable X, then given any equivalence e : A ≃ B , we have P(A) ≃ P(B).
>
> if we add to this a certain “computation rule”, we get something logically equivalent to UA:
> assume p : A ≃ B → A = B; then given e : A ≃ B, we have p(e) : A = B is a path in U.
> Since we can transport along this path in any family of types over U, and transport is always an equivalence,
> there is a transport p(e)∗ : A ≃ B in the identity family.
> The required “computation rule” states that p(e)∗ = e.
>
> Steve
>
>
>
>> On Jul 20, 2017, at 8:56 AM, Bas Spitters <b.a.w.s...@gmail.com> wrote:
>>
>>> 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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