From: Egbert Rijke <e.m....@gmail.com>
To: "Martín Hötzel Escardó" <"escardo..."@gmail.com>
Cc: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] Univalence <-> equivalence induction
Date: Fri, 18 May 2018 09:04:57 -0400 [thread overview]
Message-ID: <CAGqv1ODKNMkyOxScnO4SkMEr8Qh8f+LbSE18DK-Z-gFr1f_FTQ@mail.gmail.com> (raw)
In-Reply-To: <4f23688f-af7c-43cb-946c-988f9d476848@googlegroups.com>
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Hi Martin,
I think it was known. I taught this in my intro to HoTT class this semester:
http://www.andrew.cmu.edu/user/erijke/hott/univalence.pdf
Best wishes,
Egbert
On Fri, May 18, 2018 at 2:36 AM, Martín Hötzel Escardó <
escardo...@gmail.com> wrote:
> Equivalence induction says that in order to prove something for all
> equivalences, it is enough to prove it for all identity equivalences for
> all types.
>
> This follows from univalence. But also, conversely, univalence follows
> from it:
>
> http://www.cs.bham.ac.uk/~mhe/agda-new/UF-Univalence.html#JEq
>
> Is this known? Some years ago it was claimed in this list that equivalence
> induction would be strictly weaker than univalence.
>
> To prove the above, I apply a technique I learned from Peter Lumsdaine,
> that given an abstract identity system (Id, refl , J) with no given
> "computation rule" for J, produces another identity system (Id, refl , J' ,
> J'-comp) with
> a "propositional computation rule" J'-comp for J'.
>
> http://www.cs.bham.ac.uk/~mhe/agda-new/Lumsdaine.html
>
> Martin
>
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--
egbertrijke.com
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next prev parent reply other threads:[~2018-05-18 13:04 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2018-05-18 6:36 Martín Hötzel Escardó
2018-05-18 13:04 ` Egbert Rijke [this message]
2018-05-18 15:40 ` [HoTT] " Michael Shulman
2018-05-18 21:03 ` Martín Hötzel Escardó
2018-05-19 18:09 ` Nicolai Kraus
2018-05-19 19:38 ` Thierry Coquand
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