Discussion of Homotopy Type Theory and Univalent Foundations
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From: Peter LeFanu Lumsdaine <p.l.l...@gmail.com>
To: Steve Awodey <steve...@gmail.com>
Cc: Michael Shulman <shu...@sandiego.edu>,
	Stefan Monnier <mon...@iro.umontreal.ca>,
	 Thorsten Altenkirch <Thorsten....@nottingham.ac.uk>,
	 "homotopyt...@googlegroups.com" <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] "Identifications" ?
Date: Mon, 4 May 2020 18:21:21 +0200	[thread overview]
Message-ID: <CAAkwb-npDgX=T18N5MskKpix76E3PnbV42N19nbFF4n60yv73A@mail.gmail.com> (raw)
In-Reply-To: <E31B538B-E0C9-4D4F-A4F1-4335E59CAE0D@gmail.com>

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Ah, I just thought the whole system was having a clock malfunction and
re-sending a discussion from 2016…  :-P

On the serious side: one motivation for “identification” not yet mentioned
is that it’s more in line with traditional mathematical usage.  Saying that
the circle defined as R/Z or as a subset of R^2 are “equal” clashes badly
with the traditional mathematical usage of words, and (for at least some
mathematicians) their mental picture of mathematics — but saying that we
can identify them, and that all nice properties respect such
identifications, is completely consistent with traditional writing and
worldviews.  Also, the fact that we may identify things in multiple ways,
and so have to be a little careful about how we make use of such
identifications, fits with traditional intuition and practice.

So saying “identifications” may require less rewiring in our brains than
saying “equalities” or “paths”, at least in types of algebraic structures
and similar.  (For synthetic homotopy theory, of course, “paths” often fits
closer to established usage.)

–p.

On Mon, May 4, 2020 at 6:07 PM Steve Awodey <steve...@gmail.com> wrote:

> I’m afraid that someone may have hacked Thorsten’s email account. The real
> Thorsten went through all this with us many years ago.
> : - )
>
>
> > On May 4, 2020, at 12:00, Michael Shulman <shu...@sandiego.edu> wrote:
> >
> > The word "path" is closely tied to the homotopy interpretation, and to
> > the classical perspective of oo-groupoids presented via topological
> > spaces, which has various problems.  This is particularly an issue in
> > cohesive type theory, where there is a separate "point-set level"
> > notion of path that it is important to distinguish from
> > identifications.
> >
> >> On Mon, May 4, 2020 at 7:48 AM Stefan Monnier <mon...@iro.umontreal.ca>
> wrote:
> >>
> >>> I don't think using "identification" necessarily implies any
> >>> difference between "identification" and "equality".  I don't think of
> >>> it that way.  For me the point is just to have a word that refers to
> >>> an *element* of an identity type.  Calling it "an equality" can have
> >>> the wrong connotation because classically, an equality is just a
> >>> proposition (or a true proposition), whereas an element of an identity
> >>> type carries information.  Calling it "an identification" suggests
> >>> exactly the information that it carries: a way of identifying two
> >>> things.
> >>
> >> I thought that's what "path" was for?
> >>
> >>
> >>        Stefan "who really doesn't know what he's talking about"
> >>
> >
> > --
> > 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 HomotopyT...@googlegroups.com.
> > To view this discussion on the web visit
> https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQx_2TinRHBrmOAZFnmFp8VVQ-yMcPvtKFtX-d9wGsD%2B2Q%40mail.gmail.com
> .
>
> --
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> To view this discussion on the web visit
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> .
>

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  parent reply	other threads:[~2020-05-04 16:21 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
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 [this message]
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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