From: Vladimir Voevodsky <vlad...@ias.edu>
To: Peter LeFanu Lumsdaine <p.l.lu...@gmail.com>
Cc: "Prof. Vladimir Voevodsky" <"vlad..."@ias.edu>,
"Martin Escardo" <"escardo..."@googlemail.com>,
"\"Joyal, André\"" <"joyal..."@uqam.ca>,
"HomotopyT...@googlegroups.com" <"HomotopyT..."@googlegroups.com>
Subject: Re: [HoTT] Re: Joyal's version of the notion of equivalence
Date: Wed, 12 Oct 2016 18:17:42 -0400 [thread overview]
Message-ID: <CBA53C75-423D-4E98-9C59-BE98166A8FB4@ias.edu> (raw)
In-Reply-To: <CAAkwb-nX167c_hpd2pE7d5VWQ+q30sKJs2V8zf=yEqDpiuL2mw@mail.gmail.com>
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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 <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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next prev parent reply other threads:[~2016-10-12 22:17 UTC|newest]
Thread overview: 14+ messages / expand[flat|nested] mbox.gz Atom feed top
[not found] <963893a3-bfdf-d9bd-8961-19bab69e0f7c@googlemail.com>
2016-10-07 23:51 ` Martin Escardo
2016-10-08 0:21 ` [HoTT] " Martin Escardo
2016-10-08 17:34 ` Joyal, André
2016-10-09 18:31 ` Martin Escardo
2016-10-09 18:56 ` Joyal, André
2016-10-11 22:54 ` Martin Escardo
2016-10-12 9:45 ` Peter LeFanu Lumsdaine
2016-10-12 13:21 ` Dan Christensen
2016-10-12 22:45 ` [HoTT] " Martin Escardo
2016-10-12 22:17 ` Vladimir Voevodsky [this message]
2016-10-12 23:55 ` Martin Escardo
2016-10-13 10:14 ` Thomas Streicher
2016-10-13 7:14 ` Joyal, André
2016-10-13 12:48 ` Egbert Rijke
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