This doesn't seem right; as far as I can tell, composition for Glue types
in CCHM preserves regularity and reduces to composition in A on phi.
- Jasper Hugunin
On Sun, Sep 15, 2019 at 3:28 AM Anders Mortberg
wrote:
> Hi Jasper,
>
> Indeed, the problem is to construct an algorithm for comp (or even
> coe/transp) for Glue that reduces to the one of A when phi is true
> while still preserving regularity. It was pointed out independently by
> Sattler and Orton around 2016 that one can factor out this step in our
> algorithm in a separate lemma that is now called "alignment". This is
> Thm 6.13 in Orton-Pitts and discussed in a paragraph in the end of
> section 2.11 of ABCFHL. Unless I'm misremembering this is exactly
> where regularity for comp for Glue types break down. In this step we
> do an additional comp/hcomp that inserts an additional forall i. phi
> face making the comp/coe irregular.
>
> One could imagine there being a way to modify the algorithm to avoid
> this, maybe by inlining the alignment step... But despite considerable
> efforts no one has been able to figure this out and I think Swan's
> recent paper (https://arxiv.org/abs/1808.00920v3) shows that this is
> not even possible!
>
> Another approach would be to have weak Glue types that don't strictly
> reduce to A when phi is true, but this causes problems for the
> composition in the universe and would be weird for cubical type
> theory...
>
> In light of Swan's negative results I think we need a completely new
> approach if we ever hope to solve this problem. Luckily for you Andrew
> Swan is starting as a postdoc over in Baker Hall in October, so he can
> explain his counterexamples to you in person.
>
> Best,
> Anders
>
> On Sun, Sep 15, 2019 at 7:57 AM Jasper Hugunin
> wrote:
> >
> > Offline, Carlo Angiuli showed me that the difficulty was in part 1,
> because of a subtlety I had been forgetting.
> >
> > Since types are *Kan* cubical sets, we need that the Kan operations
> agree as well as the sets.
> > So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A,
> compGlue) = (A, compA), and getting that the Kan operations to agree was/is
> difficult.
> > (Now that I know what the answer is, it is clear that this was already
> explained in the initial discussion.)
> >
> > Humbly,
> > - Jasper Hugunin
> >
> > On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <
> jasperh@cs.washington.edu> wrote:
> >>
> >> Hello all,
> >>
> >> I've been trying to understand better why composition for the universe
> does not satisfy regularity.
> >> Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |->
> equiv^i E ] A, I would expect regularity to follow from two parts:
> >> 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of
> regularity condition for the Glue type constructor itself)
> >> 2. That equiv^i (refl A) reduces to equivRefl A
> >> I'm curious as to which (or both) of these parts was the issue, or if
> regularity for the universe was supposed to follow from a different
> argument.
> >>
> >> Context:
> >> I've been studying and using CCHM cubical type theory recently, and
> often finding myself wishing that J computed strictly.
> >> If I understand correctly, early implementations of ctt did have strict
> J for Path types, and this was justified by a "regularity" condition on the
> composition operation, but as discussed in this thread on the HoTT mailing
> list, the definition of composition for the universe was found to not
> satisfy regularity.
> >> I don't remember seeing the regularity condition defined anywhere, but
> my understanding is that it requires that composition in a degenerate line
> of types, with the system of constraints giving the sides of the box also
> degenerate in that direction, reduces to just the bottom of the box. This
> seems to be closed under the usual type formers, plus Glue, but not the
> universe with computation defined as in the CCHM paper (for trivial reasons
> and non-trivial reasons; it gets stuck at the start with Glue [ phi |->
> equiv^i refl ] A not reducing to anything).
> >>
> >> Best regards,
> >> - Jasper Hugunin
> >
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