Re: [HoTT] Re: Is [Equiv Type_i Type_i] contractible?

[Richard Williamson <rwilli...@gmail.com>]

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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