Re: [HoTT] "Identifications" ?

[Peter LeFanu Lumsdaine <p.l.l...@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"
> >>
> >
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