From: Martin Escardo <escardo...@googlemail.com>
To: "HomotopyT...@googlegroups.com" <HomotopyT...@googlegroups.com>
Subject: Conjecture
Date: Mon, 27 Mar 2017 22:57:12 +0100 [thread overview]
Message-ID: <1cd04354-59ba-40b4-47ce-9eef3ca3112f@googlemail.com> (raw)
This is a question I would like to see eventually answered.
I posed it a few years ago in a conference (and privately among some of
you), but I would like to have it here in this list for the record.
Definition. A modulus of constancy for a function f:X->Y is a function
(x,y:X)->f(x)=f(y). (Such a function can have zero, one or more moduli
of constancy, but if Y is a set then it can have at most one.)
We know that if Y is a set and f comes with a modulus of constancy, then
f factors through |-|: X -> ||Y||, meaning that we can exhibit an
f':||X||->Y with f'|x| = f(x).
Conjecture. If for all types X and Y and all functions f:X->Y equipped
with a modulus of constancy we can exhibit f':||X||->Y with f'|x| =
f(x), then all types are sets.
For this conjecture, I assume function extensionality and propositional
extensionality, but not (general) univalence. But feel free to play with
the assumptions.
Martin
next reply other threads:[~2017-03-27 21:57 UTC|newest]
Thread overview: 22+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-03-27 21:57 Martin Escardo [this message]
2017-03-29 21:08 ` [HoTT] Conjecture Nicolai Kraus
2017-03-29 22:05 ` Martin Escardo
2017-03-30 10:59 ` Michael Shulman
2017-03-30 19:22 ` Egbert Rijke
2017-03-30 23:02 ` Nicolai Kraus
2017-03-30 22:49 ` Nicolai Kraus
2017-03-31 16:09 ` Martin Escardo
2017-04-05 19:37 ` Martin Escardo
2017-04-06 0:23 ` Jon Sterling
2017-04-06 5:55 ` Martin Escardo
2017-04-06 12:40 ` Vladimir Voevodsky
2017-04-06 13:50 ` Martin Escardo
[not found] ` <81c0782f-9287-4111-a4f1-01cb9c87c7e8@cs.bham.ac.uk>
2017-04-06 16:09 ` Martin Escardo
2017-04-06 11:52 ` Thomas Streicher
2017-04-07 9:49 ` Martin Escardo
2017-04-07 17:11 ` Michael Shulman
2017-04-07 18:10 ` Martin Escardo
2017-04-03 0:35 ` Conjecture Daniel R. Grayson
2017-04-03 2:20 ` [HoTT] Conjecture Favonia
2017-04-03 9:56 ` Nicolai Kraus
2017-04-03 11:50 ` Daniel R. Grayson
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