Re: [HoTT] Weaker forms of univalence

[Steve Awodey <awo...@cmu.edu>]

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
>>> 
>>> --
>>> 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 HomotopyTypeThe...@googlegroups.com.
>>> For more options, visit https://groups.google.com/d/optout.
>> 
>> --
>> 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 HomotopyTypeThe...@googlegroups.com.
>> For more options, visit https://groups.google.com/d/optout.
> 
> -- 
> 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 HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.