Hi Dan, Of course. I'm thinking primarily of composition for Glue given in the CCHM paper you linked, reproduced below. As you know, the single potential issue is that we need pres of a degenerate filling problem and function to be reflexivity. I claim that this holds by regularity of composition in T and A, partly as a consequence of the fact that regularity of composition implies regularity of filling (that fill of a degenerate system is refl), which certainly holds for fill defined by connections (and I believe also holds for fill as defined in ABCFHL). (a) Given i |- B = Glue [ phi |-> (T, f) ] A, with psi, i |- b : B and b0 : B(i0)[ psi |-> b(i0) ], we want to compute b1 = comp^i B [ psi |-> b ] b0 : B(i1)[ psi |-> b(i1) ]. We set a := unglue b and a0 := unglue b0. Set delta := forall i. phi. Then we take: a1' := comp^i A [ psi |-> a ] a0 delta |- t1' := comp^i T [ psi |-> b ] b0 delta |- omega := pres^i f [ psi |-> b ] b0 phi(i1) |- (t, alpha) := equiv f(i1) [ delta |-> (t1', omega), psi |-> (b(i1), refl a1') ] a1' a1 := hcomp^j A(i1) [ phi(i1) |-> alpha j, psi |-> a(i1) ] a1' (note that in the regular setting the psi face is redundant) b1 := glue [ phi(i1) |-> t1 ] a1 With given i |- f : T -> A, with psi, i |- b : T and b0 : T(i0)[ psi |-> b(i0) ], we define pres^i f [ psi |-> b ] b0 = comp^i A [ psi |-> f b, j = 1 |-> f (fill^i T [ psi |-> b ] b0) ] (f(i0) b0). (b) Now consider the regular case, where phi, T, f, and A are independent of i. We want that b1 = b0. We have that a is independent of i, and delta = phi. First consider delta (= phi) |- pres^i f [ psi |-> b ] b0. (This is the explanation of your first dash) Note that if comp^i A is regular, then fill^i A [ psi |-> b ] b0 = b This is comp^i A [ psi |-> f b, j = 1 |-> f (fill^i T [ psi |-> t ] t0) ] (f t0) = comp^i A [ psi |-> f b, j = 1 |-> f t0 ] (f t0) = f t0. Thus pres of a degenerate filling problem and function is reflexivity. Going back to composition of Glue, a1' = a0 phi |- t1' = b0 phi |- omega = refl (f b0) phi |- (t1, alpha) = (t1', omega) (since delta = phi, so we end up in the delta face of equiv) a1 = a1' (the only dependence on j is via (alpha j), but alpha = omega = refl, so this filling problem is degenerate) b1 = glue [ phi |-> t1 ] a1 = glue [ phi |-> b0 ] a0 = glue [ phi |-> b0 ] (unglue b0) = b0 (by eta, see Figure 4 in CCHM) Thus this algorithm for composition of Glue is regular. Other algorithms, such as the one in ABCFHL, may not be, but I am prone to believe that there exist regular algorithms in other settings including Orton-Pitts and Cartesian cubes. Best regards, - Jasper Hugunin On Mon, Sep 16, 2019 at 12:18 PM Licata, Dan wrote: > Hi Jasper, > > It would help me follow the discussion if you could say a little more > about (a) which version of composition for Glue exactly you mean (because > there is at least the one in the CCHM paper and the “aligned” one from > Orton-Pitts, which are intensionally different, as well as other possible > variations), and (b) include some of your reasoning for why you think > things are regular, to make it easier for me and others to reconstruct. > > My current understanding is that > > - For CCHM proper https://arxiv.org/pdf/1611.02108.pdf the potential > issue is with the ‘pres’ path omega, which via the equiv operation ends up > in alpha, so the system in a1 may not be degenerate even if the input is. > Do you think this does work out to be degenerate? > > - For the current version of ABCFHL > https://github.com/dlicata335/cart-cube/blob/master/cart-cube.pdf which > uses aligning “all the way at the outside”, an issue is with the adjust_com > operation on page 20, which is later used for aligning (in that case beta > is (forall i phi)). The potential issue is that adjust_com uses a > *filler*, not just a composition from 0 to 1, so even if t doesn’t depend > on z, the filling does, and the outer com won’t cancel. In CCHM, filling > is defined using connections, so it’s a little different, but I think there > still has to be a dependence on z for it to even type check — it should > come up because of the connection term that is substituted into the type of > the composition problem. So I’d guess there is still a problem in the > aligned algorithm for CCHM. > > However, it would be great if this is wrong and something does work! > > -Dan > > > On Sep 15, 2019, at 10:18 PM, Jasper Hugunin > wrote: > > > > 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 < > anders.mortberg@math.su.se> 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 > > > > > > -- > > > 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. > > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a9oS0CQDGKgj7ghCh8%2BZwAcAefiTg4JJVHemV3HUPcPEg%40mail.gmail.com > . > > > > -- > > 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a8SZx8PiaD-9rq5QWffU75Wz8myrXD1g5P3DCjSO%3DfvOQ%40mail.gmail.com > . > > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. 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