RE: [HoTT] Re: Joyal's version of the notion of equivalence

[Egbert Rijke <e.m....@gmail.com>]

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Dear all,

I agree with Vladimir that his notion of equivalence is a very clear one,
but that doesn't take away that on some occasions it is really helpful to
have Joyal's notion around.

For instance, in the definition of localization as a higher inductive type
we use Joyal-equivalence, simply to avoid 2-path constructors.

With kind regards,
Egbert

On Oct 13, 2016 3:14 AM, "Joyal, André" <joyal...@uqam.ca> wrote:

> Dear Vladimir,
>
> I completely agree with you.
>
> André
> ------------------------------
> *From:* homotopyt...@googlegroups.com [homotopytypetheory@
> googlegroups.com] on behalf of Vladimir Voevodsky [vlad...@ias.edu]
> *Sent:* Wednesday, October 12, 2016 6:17 PM
> *To:* Peter LeFanu Lumsdaine
> *Cc:* Prof. Vladimir Voevodsky; Martin Escardo; Joyal, André;
> HomotopyT...@googlegroups.com
> *Subject:* Re: [HoTT] Re: Joyal's version of the notion of equivalence
>
> I think the clearest formulation is my original one - as the condition of
> contractibility of the h-fibers.
>
> This is also the first form in which it was introduced and the first
> explicit formulation and proof of the fact that it is a proposition.
>
> Vladimir.
>
>
>
>
>
> On Oct 12, 2016, at 5:45 AM, Peter LeFanu Lumsdaine <
> p.l.lu...@gmail.com> wrote:
>
> > Although I can see formal proofs that Joyal-equivalence is a proposition
> (or h-proposition), I am still trying to find the best formal proof that
> uncovers the essence of this fact. This is why I asked the question of were
> this formulation of equivalence comes from.
>
> Like André suggested, I feel the nicest viewpoint is the fact that the
> “free (∞,1)-category on a Joyal-equivalence” is contractible.  At least in
> terms of intuition, the conceptually clearest argument I know for that is
> as follows.
>
> Look at the *∞-groupoidification* of this free (∞,1)-category, considered
> as a space.  This is a cell complex which we can easily picture: two points
> x, y, three paths f, g, g' between x and y, and 2-cells giving homotopies f
> ~ g, f ~ g'.  It’s very clear geometrically that this is contractible.
>
> But — the “free (∞,1)-category on a Joyal equivalence” is already an
> ∞-groupoid — and ∞-groupoidification is idempotent, since groupoids are a
> full subcategory.  So the original (∞,1)-category is equivalent to its
> groupoidification, so is contractible.
>
> The same approach works for seeing why half-adjoint equivalences are good,
> but non-adjoint and bi-adjoint equivalences are not.  So as regards
> intuition, I think this is very nice.  However, I suspect that if one looks
> at all the work that goes into setting up the framework needed, then
> somewhere one will have already used some form of “equivalence is a
> proposition”.  So this is perhaps a little unsatisfactory formally, as it
> (a) needs a lot of background, and (b) may need to rely on some more
> elementary proof of the same fact.
>
>
> The earliest explicit discussion I know of this issue
> (i.e.“contractibility of the walking equivalence as a quality criterion for
> structured notions of equivalence) is in Steve Lack’s “A Quillen Model
> Structure for Bicategories”, fixing an error in his earlier “A Quillen
> Model Structure for 2-categories”, where he had used non-adjoint
> equivalences — see http://maths.mq.edu.au/~slack/papers/qmc2cat.html
>  Since it’s just 2-categorical, he’s able to use fully adjoint equivalences
> — doesn’t have to worry about half-adjointness/coherent-adjointness.
> Adjoint equivalences of course go back much further — but I don’t know
> anywhere that this *reason* why they’re better is articulated, before Lack.
>
> And for Joyal-equivalences, I don’t know anywhere they’re explicitly
> discussed at all, before HoTT.  Like Martín, I’d be really interested if
> anyone does know any earlier sources for them!
>
> –p.
>
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