From: Thierry Coquand <Thierry...@cse.gu.se>
To: Vladimir Voevodsky <vlad...@ias.edu>
Cc: "univalent-...@googlegroups.com" <univalent-...@googlegroups.com>,
homotopytypetheory <homotopyt...@googlegroups.com>
Subject: Re: [UniMath] [HoTT] about the HTS
Date: Mon, 27 Feb 2017 18:58:22 +0000 [thread overview]
Message-ID: <1806f2f9c6484a438935d6c2ed6d7fc2@cse.gu.se> (raw)
In-Reply-To: <9803E33B-06CD-427D-8485-F817390ED527@ias.edu>
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Exactly.
In my note, I consider three universes (all fibrant)
sProp
sSet
bSet
sProp sub-presheaf of sSet sub-presheaf bSet sub-presheaf U
They correspond to 3 properties
G |- A sprop
G |- A sset
G |- A bset
that can be described semantically in a simple way.
For sprop and sset, to have a fibration structure is a property
and
G |- A sset and fibrant
should correspond to the notion of covering space.
But sSet does not correspond to a decidable type system, while
it should be the case that sProp and bSet corresponds to a decidable
type system.
At least with bSet we validate axiom K and all developments that have
been done using this axiom.
________________________________
From: Vladimir Voevodsky <vlad...@ias.edu>
Sent: Monday, February 27, 2017 7:53 PM
To: Thierry Coquand
Cc: Prof. Vladimir Voevodsky; univalent-...@googlegroups.com; homotopytypetheory
Subject: Re: [UniMath] [HoTT] about the HTS
BTW, even if the universe of strict sets is not fibrant we can still have a judgement that something is a strict set and the rule that a = b implies a is definitionally equal to b if a and b are elements of a strict set.
It is such a structure that would make many things very convenient.
It is non- clear to, however, why typing would be decidable in such a system.
Vladimir.
On Feb 27, 2017, at 1:50 PM, Vladimir Voevodsky <vlad...@ias.edu<mailto:vlad...@ias.edu>> wrote:
On Feb 25, 2017, at 2:19 PM, Thierry Coquand <Thierry...@cse.gu.se<mailto:Thierry...@cse.gu.se>> wrote:
"Bishop set" which corresponds
to the fact that any two paths between the same end points are -judgmentally- equal.
This is not what I mean by a strict set. A strict set is a Bishop set where any two points connected by a path are judgmentally equal.
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next prev parent reply other threads:[~2017-02-27 18:58 UTC|newest]
Thread overview: 22+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-02-23 14:47 Vladimir Voevodsky
2017-02-23 14:57 ` [HoTT] " Benedikt Ahrens
2017-02-23 18:08 ` Vladimir Voevodsky
2017-02-23 18:52 ` Benedikt Ahrens
2017-02-23 21:45 ` Vladimir Voevodsky
[not found] ` <87k28fek09.fsf@capriotti.io>
2017-02-24 14:36 ` [UniMath] " Vladimir Voevodsky
2017-02-24 15:06 ` Paolo Capriotti
2017-02-24 15:10 ` Vladimir Voevodsky
2017-03-10 13:35 ` HIT Thierry Coquand
2017-02-24 14:36 ` [HoTT] about the HTS Paolo Capriotti
2017-02-25 19:19 ` Thierry Coquand
2017-02-27 18:50 ` [UniMath] " Vladimir Voevodsky
2017-02-27 18:53 ` Vladimir Voevodsky
2017-02-27 18:58 ` Thierry Coquand [this message]
2017-02-28 2:17 ` Vladimir Voevodsky
2017-03-01 20:23 ` Thierry Coquand
2017-03-20 15:12 ` Matt Oliveri
2017-03-22 16:49 ` [HoTT] " Thierry Coquand
2017-03-22 21:01 ` Vladimir Voevodsky
2017-03-23 11:22 ` Matt Oliveri
2017-03-23 11:33 ` Michael Shulman
2017-03-23 12:16 ` Matt Oliveri
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