Re: [HoTT] A puzzle about "univalent equality"

[Andrew Polonsky <andrew....@gmail.com>]

These are all good points.  I now have an exhaustive answer to my
motivating question.

Thanks,
Andrew

On Tue, Sep 6, 2016 at 2:57 PM, Peter LeFanu Lumsdaine
<p.l.lu...@gmail.com> wrote:
> On Tue, Sep 6, 2016 at 2:32 PM, Michael Shulman <shu...@sandiego.edu>
> wrote:
>>
>> Although, as Voevodsky showed, weak funext implies strong funext.
>
>
> Just to clarify, though, this *doesn’t*  mean that Andews’ original goal
> “proj1 Y = tt” is necesarily inhabited, if the funext witness used early in
> his development is taken just from weak funext.
>
> The proof “weak funext implies strong funext” shows that given some witness
> funext0 of weak funext (i.e. funext0 : (forall X Y f g, f == g -> f = g)),
> then you can construct some new witness funext1, which additionally is a
> (two-sided) inverse for the canonical map the other way (“ap10” in the
> current HoTT library).  (I blogged the details here:
> https://homotopytypetheory.org/2011/12/19/strong-funext-from-weak/)
>
> But it *doesn’t* show that the original witness funext0 is an inverse for
> ap10, and indeed the proof points to how this may fail: funext0 might
> conjugate paths by some family of non-trivial loops in the codomain type.
> Andrew’s original goal “proj1 Y = tt” depends on the witness used earlier
> for funext — so if that witness happens to conjugate paths Bool –> Bool in
> Type by the non-trivial auto-equivalence of Bool, then one could have proj1
> Y = ff.
>
> –p.
>
>
>>
>> On Tue, Sep 6, 2016 at 12:30 AM, Andrew Polonsky
>> <andrew....@gmail.com> wrote:
>> > Thanks, Mike and Dan.  And congratulations on giving essentially
>> > identical solutions at essentially identical times, in two different
>> > languages!
>> >
>> >> I would be very surprised if there was something like this that was not
>> >> provable in "book HoTT”.
>> >
>> > I believe there can't be, either.  But maybe this "belief" is really a
>> > matter of definition, in that the equalities which are "supposed to"
>> > hold, are precisely those which can be derived in book HoTT.
>> >
>> > What I find subtle in the above example is that it apparently cannot
>> > be done with the "pre-HoTT" FunExt axiom; you need to use the stronger
>> > formulation, that the canonical map (f=g -> f==g) is an equivalence,
>> > to make the transports compute.
>> >
>> > Cheers,
>> > Andrew
>> >
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