Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / mirror / Atom feed
From: "Licata, Dan" <dlicata@wesleyan.edu>
To: "Martín Hötzel Escardó" <escardo.martin@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Propositional Truncation
Date: Thu, 7 Mar 2019 14:10:53 +0000	[thread overview]
Message-ID: <CD53FDB0-C16B-4E95-A796-1DC91C523046@wesleyan.edu> (raw)
In-Reply-To: <12cd6b73-7ca6-481c-9503-250af28b8113@googlegroups.com>

Just in case anyone reading this thread later is confused about a more beginner point than the ones Nicolai and Martin made, one possible stumbling block here is that, if someone means “is inhabited” in an external sense (there is a closed term of that type), then the answer is yes (at least in some models): if ||A|| is inhabited then A is inhabited.  For example, in cubical models with canonicity, it is true that a closed term of type ||A|| evaluates to a value that has as a subterm a closed term of type A (the other values of ||A|| are some “formal compositions” of values of ||A||, but there has to be an |a| in there at the base case).  This is consistent with what Martin and Nicolai said because “if A is inhabited then B is inhabited” (in this external sense) doesn’t necessarily mean there is a map A -> B internally.  

-Dan

> On Mar 5, 2019, at 6:07 PM, Martín Hötzel Escardó <escardo.martin@gmail.com> wrote:
> 
> Or you can read the paper https://lmcs.episciences.org/3217/ regarding what Nicolai said.
> 
> Moreover, in the HoTT book, it is shown that if || X||->X holds for all X, then univalence can't hold. (It is global choice, which can't be invariant under equivalence.)
> 
> The above paper shows that unrestricted ||X||->X it gives excluded middle. 
> 
> However, for a lot of kinds of types one can show that ||X||->X does hold. For example, if they have a constant endo-function. Moreover, for any type X, the availability of ||X||->X is logically equivalent to the availability of a constant map X->X (before we know whether X has a point or not, in which case the availability of a constant endo-map is trivial).
> 
> Martin
> 
> On Tuesday, 5 March 2019 22:47:55 UTC, Nicolai Kraus wrote:
> You can't have a function which, for all A, gives you ||A|| -> A. See the exercises 3.11 and 3.12!
> -- Nicolai
> 
> On 05/03/19 22:31, Jean Joseph wrote:
>> Hi,
>> 
>> From the HoTT book, the truncation of any type A has two constructors:
>> 
>> 1) for any a : A, there is |a| : ||A||
>> 2) for any x,y : ||A||, x = y. 
>> 
>> I get that if A is inhabited, then ||A|| is inhabited by (1). But is it true that, if ||A|| is inhabited, then A is inhabited? 
>> -- 
>> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
>> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
>> For more options, visit https://groups.google.com/d/optout.
> 
> 
> -- 
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.

-- 
You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

  reply	other threads:[~2019-03-07 14:10 UTC|newest]

Thread overview: 14+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-03-05 22:31 Jean Joseph
2019-03-05 22:47 ` Nicolai Kraus
2019-03-05 23:07   ` Martín Hötzel Escardó
2019-03-07 14:10     ` Licata, Dan [this message]
2019-03-07 16:16       ` Martín Hötzel Escardó
2019-03-07 16:35         ` Ben Sherman
2019-03-07 21:52         ` Anders Mörtberg
2019-03-07 22:41           ` Martín Hötzel Escardó
2019-03-07 22:51             ` Licata, Dan
2019-03-07 23:01               ` Martín Hötzel Escardó
2019-03-07 23:23                 ` Martín Hötzel Escardó
2019-03-08 14:59                   ` Anders Mortberg
2019-03-08 15:13                     ` Licata, Dan
2019-03-08 22:28                       ` Martín Hötzel Escardó

Reply instructions:

You may reply publicly to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to=CD53FDB0-C16B-4E95-A796-1DC91C523046@wesleyan.edu \
    --to=dlicata@wesleyan.edu \
    --cc=HomotopyTypeTheory@googlegroups.com \
    --cc=escardo.martin@gmail.com \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).