Thanks, Richard!

Of course, this is not directly pertaining to Matthieu, Nicolas and Théo's 
question, but it's trying to capture an intuition that a universe should be 
rigid, at least when considered together with some structure.

How much structure suffices to make the universe rigid, and can we define 
this extra structure in HoTT? (We don't know how to say yet that the 
universe can be given the structure of an infinity-category strongly 
generated by 1, for example.)

Do you know other references that pertain to the inspiring assumption/hypothèse 
inspiratrice?

Best wishes,
Ulrik

On Thursday, October 27, 2016 at 9:44:43 PM UTC+2, Richard Williamson wrote:
>
> I think the earliest proof of some version of Grothendieck's 
> hypothèse inspiratrice is in the following paper of Cisinki. 
>
> http://www.tac.mta.ca/tac/volumes/20/17/20-17abs.html 
>
> It is my belief that Grothendieck's original formulation, which 
> was for the homotopy category itself (as opposed to a lifting of 
> it), is independent of ZFC. A proof of this would be fascinating. 
> I have occasionally speculated about trying to use HoTT to give 
> such an independence proof. Vladimir's comment suggests that one 
> direction of this is already done. 
>
> Best wishes, 
> Richard 
>
> On Thu, Oct 27, 2016 at 10:12:50AM -0700, Ulrik Buchholtz wrote: 
> > This is (related to) Grothendieck's “inspiring assumption” of Pursuing 
> > Stacks section 28. 
> > 
> > I only know of the treatment by Barwick and Schommer-Pries in On the 
> > Unicity of the Homotopy Theory of Higher Categories: 
> > https://arxiv.org/abs/1112.0040 
> > 
> > Theorem 8.12 for n=0 says that the Kan complex of homotopy theories of 
> > (infinity,0)-categories is contractible. Of course this depends on their 
> > axiomatization, Definition 6.8. Perhaps some ideas can be adapted. 
> > 
> > Cheers, 
> > Ulrik 
> > 
> > On Thursday, October 27, 2016 at 5:15:45 PM UTC+2, Matthieu Sozeau 
> wrote: 
> > > 
> > > Dear all, 
> > > 
> > >   we've been stuck with N. Tabareau and his student Théo Winterhalter 
> on 
> > > the above question. Is it the case that all equivalences between a 
> universe 
> > > and itself are equivalent to the identity? We can't seem to prove (or 
> > > disprove) this from univalence alone, and even additional 
> parametricity 
> > > assumptions do not seem to help. Did we miss a counterexample? Did 
> anyone 
> > > investigate this or can produce a proof as an easy corollary? What is 
> the 
> > > situation in, e.g. the simplicial model? 
> > > 
> > > -- Matthieu 
> > > 
> > 
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