Discussion of Homotopy Type Theory and Univalent Foundations
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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


             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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