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

[Peter LeFanu Lumsdaine <p.l.lu...@gmail.com>]

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> 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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