Discussion of Homotopy Type Theory and Univalent Foundations
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From: Thomas Streicher <stre...@mathematik.tu-darmstadt.de>
To: Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>
Cc: "homotopyt...@googlegroups.com" <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] "Identifications" ?
Date: Mon, 4 May 2020 14:06:20 +0200	[thread overview]
Message-ID: <20200504120620.GA14623@mathematik.tu-darmstadt.de> (raw)
In-Reply-To: <AD9A399C-4C95-4CEF-A106-88B68CAD15A8@nottingham.ac.uk>

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

  parent reply	other threads:[~2020-05-04 12:06 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 [this message]
2020-05-04 12:12       ` Thorsten Altenkirch
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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