Re: [HoTT] computing K

[Floris van Doorn <fpvd...@gmail.com>]

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Disregard previous email. I misread Mike's proposed computing K (I thought
it was refl on all loops).

On 26 April 2017 at 16:40, Floris van Doorn <fpvd...@gmail.com> wrote:

> I think (1) is indeed wishful thinking.
>
> I can disprove it assuming the following metatheoretical property. I think
> this property holds for book-HoTT, but I'm not sure.
>
> If X is a HIT and p : A -> X is a path constructor of X, and if p a ≡ p a'
> for terms a a' : A then a ≡ a'. (Here ≡ is judgmental equality.)
>
> I believe that similar properties are true for MLTT if p is a constructor
> of an inductive type.
>
> Now let I be the interval with points 0, 1 : I and consider the following
> HIT:
> HIT X :=
> | * : X
> | p : I -> * = *
> | q : p(0) = refl
> Note that X is the reduced suspension of the interval, which is
> contractible, hence in particular a set. But assuming the above
> metatheoretical property, X cannot have an axiom K with the definitional
> properties Mike described. That would mean that p(0) ≡ p(1) and hence 0 ≡
> 1. But it is impossible that the two points of the interval are
> judgmentally equal (because you can make a function which sends them to
> equal, but not judgmentally equal terms, for example (Σ(X : Type), X =
> empty) and (Σ(X : Type), X = unit)).
>
> Best,
> Floris
>
> On 25 April 2017 at 19:08, Michael Shulman <shu...@sandiego.edu> wrote:
>
>> Favonia, Dan Licata, and I spent a bit of time trying to answer (1)
>> but didn't get anywhere.  I haven't thought much about (2), but I
>> think we were unable to think of any nontrivial types for which we
>> could construct a computing K inside MLTT.
>>
>> On Tue, Apr 25, 2017 at 3:04 PM, Martin Escardo <m.es...@cs.bham.ac.uk>
>> wrote:
>> > Interesting. So, then, here are some questions:
>> >
>> > (1) Suppose, for given type A, a hypothetical
>> >
>> > K: (x : A) (p : x == x) -> p == refl
>> >
>> > is given, internally in a univalent type theory.
>> >
>> > Is it possible to cook-up a K' of the same type, internally in univalent
>> > type theory,  with the definitional behaviour you require?
>> >
>> > We are looking for an endofunction of the type ((x : A) (p : x == x) ->
>> p ==
>> > refl) that performs some sort of definitional improvement.
>> >
>> > We know of an instance of such a phenomenon: if a factor f':||X||->A of
>> some
>> > f:X->A through |-|:X->||X|| is given, then we can find another
>> > *definitional* factor f':||X||->A.
>> >
>> > This is here in agda
>> > http://www.cs.bham.ac.uk/~mhe/truncation-and-extensionality/
>> hsetfunext.html#15508
>> > and also in a paper by Nicolai, Thorsten, Thierry and myself.
>> >
>> > (2) Failing that, can we, given the same hypothetical K, cook-up a type
>> A'
>> > equivalent to A (maybe A'= Sigma(x:A) (x=x)) with K' as above?
>> >
>> > (That is, maybe A itself won't provably have (1), but still will have an
>> > equivalent manifestation which does.)
>> >
>> > Or is this wishful thinking?
>> >
>> > Martin
>> >
>> >
>> > On 25/04/17 10:46, Michael Shulman wrote:
>> >>
>> >> Here is a little observation that may be of interest (thanks to
>> >> Favonia for bringing this question to my attention).
>> >>
>> >> The Axiom K that is provable using unrestricted Agda-style
>> >> pattern-matching has an extra property: it computes to refl on refl.
>> >> That is, if we define
>> >>
>> >> K: (A : Type) (x : A) (p : x == x) -> p == refl
>> >> K A x refl = refl
>> >>
>> >> then the equation "K A x refl = refl" holds definitionally.  As was
>> >> pointed out on the Agda mailing list a while ago, this might be
>> >> considered a problem if one wants to extend Agda's --without-K to
>> >> allow unrestricted pattern-matching when the types are automatically
>> >> provable to be hsets, since a general hset apparently need not admit a
>> >> K satisfying this *definitional* behavior.
>> >>
>> >> However (this is the perhaps-new observation), in good
>> >> model-categorical semantics, such a "computing K" *can* be constructed
>> >> for any hset.  Suppose we are in a model category whose cofibrations
>> >> are exactly the monomorphisms, like simplicial sets or any Cisinski
>> >> model category.  If A is an hset, then the map A -> Delta^*(PA) (which
>> >> type theoretically is A -> Sigma(x:A) (x=x)) is a weak equivalence.
>> >> But it is also a split monomorphism, hence a cofibration; and thus an
>> >> acyclic cofibration.  Therefore, we can define functions by induction
>> >> on loops in A that have definitionally computing behavior on refl,
>> >> which is exactly what unrestricted pattern-matching allows.
>> >>
>> >> Mike
>> >>
>> >
>> > --
>> > Martin Escardo
>> > http://www.cs.bham.ac.uk/~mhe
>>
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>
>

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