From: Dan Christensen <j...@uwo.ca>
To: Homotopy Type Theory List <HomotopyT...@googlegroups.com>
Subject: Re: Joyal's version of the notion of equivalence
Date: Wed, 12 Oct 2016 09:21:10 -0400 [thread overview]
Message-ID: <878ttt3dq1.fsf_-_@uwo.ca> (raw)
In-Reply-To: <CAAkwb-nX167c_hpd2pE7d5VWQ+q30sKJs2V8zf=yEqDpiuL2mw@mail.gmail.com> (Peter LeFanu Lumsdaine's message of "Wed, 12 Oct 2016 11:45:59 +0200")
On Oct 12, 2016, Peter LeFanu Lumsdaine <p.l.lu...@gmail.com> wrote:
> 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!
I'm not sure if this is what you are looking for, but Exercise 11 in
Chapter 0 of Hatcher's "Algebraic Topology" says:
Show that f : X -> Y is a homotopy equivalence if there exist maps
g, h : Y -> X such that fg ≃ 1 and hf ≃ 1. More generally, show that
f is a homotopy equivalence if fg and hf are homotopy equivalences.
I'm pretty sure this was already there in the original 2002 edition.
Of course, this is an easy categorical fact that was well-known, but
this is at least an explicit mention of this in the case of homotopy
equivalences.
Dan
next prev parent reply other threads:[~2016-10-12 13:21 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 [this message]
2016-10-12 22:45 ` Martin Escardo
2016-10-12 22:17 ` Vladimir Voevodsky
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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