From: Vladimir Voevodsky <vlad...@ias.edu>
To: Matthieu Sozeau <matthie...@inria.fr>
Cc: "Prof. Vladimir Voevodsky" <vlad...@ias.edu>,
homotopytypetheory <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] Is [Equiv Type_i Type_i] contractible?
Date: Thu, 27 Oct 2016 13:08:29 -0400 [thread overview]
Message-ID: <08DC660F-0BA0-400D-821B-D7132A843D8F@ias.edu> (raw)
In-Reply-To: <CALgtc7gp1VhjzF4h5GxpPCcsj0VUiGu4vuwQrrODDQU9u0jO+w@mail.gmail.com>
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In the univalent simplicial model it is not contractible at all. For example Type_i has many connected components that are contractible and any permutation of these components is an equivalence of the ambient type.
This implies that iscontr (weq Type_i Type_i) is not provable in any context that is compatible with the univalent simplicial model.
Vladimir.
> On Oct 27, 2016, at 11:15 AM, Matthieu Sozeau <matthie...@inria.fr> 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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next prev parent reply other threads:[~2016-10-27 17:08 UTC|newest]
Thread overview: 14+ messages / expand[flat|nested] mbox.gz Atom feed top
2016-10-27 15:15 Matthieu Sozeau
2016-10-27 15:19 ` [HoTT] " Martin Escardo
2016-10-27 15:38 ` Martin Escardo
2016-10-27 17:09 ` Nicolai Kraus
2016-10-27 17:08 ` Vladimir Voevodsky [this message]
2016-10-27 17:12 ` Ulrik Buchholtz
2016-10-27 19:44 ` [HoTT] " Richard Williamson
2016-10-27 20:38 ` Ulrik Buchholtz
2016-10-30 20:56 ` Richard Williamson
2016-10-31 10:00 ` Eric Finster
2016-10-31 13:07 ` MLTT with proof-relevant judgmental equality? Neel Krishnaswami
2016-10-31 21:43 ` [HoTT] " Andrej Bauer
2016-10-31 22:01 ` Neel Krishnaswami
2016-10-27 20:18 ` Is [Equiv Type_i Type_i] contractible? nicolas tabareau
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