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 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 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 >> 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 >> >> -- >> You received this message because you are subscribed to the Google Groups >> "Homotopy Type Theory" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to HomotopyTypeThe...@googlegroups.com. >> For more options, visit https://groups.google.com/d/optout. >> > >