Discussion of Homotopy Type Theory and Univalent Foundations
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From: Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>
To: Thomas Streicher <stre...@mathematik.tu-darmstadt.de>
Cc: "homotopyt...@googlegroups.com" <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] "Identifications" ?
Date: Mon, 4 May 2020 12:12:04 +0000	[thread overview]
Message-ID: <31C5A7B0-8512-4ABD-8DAA-EF3A1FA0EEFF@nottingham.ac.uk> (raw)
In-Reply-To: <20200504120620.GA14623@mathematik.tu-darmstadt.de>

Hi Thomas

But I am not "interpreting HoTT in SSet". I am just working in HoTT. You are doing Metamathematics, I am just talking about Mathematics.

Sure when you consider models of your theory you can say that certain equalities are just metatheoretic equalities and others are non-trivial equalities. 

Do you think of a model of set theory when you do set theory?

Cheers,
Thorsten

On 04/05/2020, 13:06, "Thomas Streicher" <stre...@mathematik.tu-darmstadt.de> wrote:

    Hi Thorsten,

    >     This is really different in the respective topos models as eg simplicial
    >     sets or already the topos of reflexive graphs as exemplified by N. Kraus's
    >     puzzling counterexample.
    > 
    > What puzzling counterexample is this?

    I learnt it from Nicolai when he gave a little talk at CSL 2017 when
    given a prize for his thesis.
    The counterexample or rather my appreciation of it I have made a little note
    on which I have put for convenience at
    https://www.mathematik.tu-darmstadt.de/~streicher/kraus.pdf

    The point is that interpreting HoTT in sSet is very different from the
    traditional interpretation of logic in sSet. The same happens already
    the topos of reflexive graphs. Propositions are connected reflexive
    graphs and thee are very man nonisomorphic such guys but HoTT
    identifies them all.

    But this has nothing to do with HoTT per se. For example you can
    interpret HOL in assemblies in two very different ways once `a la
    topos (proof irrelevant) and once `a la  model of TT (proof relevant).

    In mathematics equality is a very relative notion. But one may perform
    quotients to reconcile them...

    Thomas




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  reply	other threads:[~2020-05-04 12:12 UTC|newest]

Thread overview: 26+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2020-05-04  9:35 Thorsten Altenkirch
2020-05-04 10:59 ` [HoTT] " stre...
2020-05-04 11:04   ` Steve Awodey
2020-05-04 11:17   ` Thorsten Altenkirch
2020-05-04 11:42     ` Nicolai Kraus
2020-05-04 12:04       ` Thorsten Altenkirch
2020-05-04 12:06     ` Thomas Streicher
2020-05-04 12:12       ` Thorsten Altenkirch [this message]
2020-05-04 12:39         ` Thomas Streicher
2020-05-04 13:16 ` Michael Shulman
2020-05-04 14:17   ` Thorsten Altenkirch
2020-05-04 14:48   ` Stefan Monnier
2020-05-04 15:46     ` Nicolai Kraus
2020-05-04 15:57       ` Thorsten Altenkirch
2020-05-04 15:59     ` Michael Shulman
2020-05-04 16:07       ` Steve Awodey
2020-05-04 16:17         ` Thorsten Altenkirch
2020-05-04 16:53           ` Steve Awodey
2020-05-04 17:25             ` Thorsten Altenkirch
2020-05-04 17:43               ` Michael Shulman
2020-05-04 17:55               ` Steve Awodey
2020-05-04 16:21         ` Peter LeFanu Lumsdaine
2020-05-04 16:16       ` Joyal, André
2020-05-04 20:38         ` Joyal, André
2020-05-07 19:43 ` Martín Hötzel Escardó
2020-05-08 10:41   ` [HoTT] " Thorsten Altenkirch

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