Yes, this is nice. We should add it to the next revision of the paper (arXiv:1705.03307). On Sun, May 10, 2020 at 5:29 PM Michael Shulman wrote: > Okay. But the implication works in the other way, doesn't it? A > product indexed by exo-Nat is the exo-limit of a tower of finite > products. So maybe the tower axiom is the best one. > > On Sun, May 10, 2020 at 9:13 AM Nicolai Kraus > wrote: > > > > I have to correct what I said an hour ago (thanks, Mike). We don't know > whether "exo-Nat is cofibrant" implies that exo-limits of towers are > fibrant. (And probably it doesn't.) > > In other words, we don't know the connection between axioms (A2) and > (A3) in arXiv:1705.03307. > > -- Nicolai > > > > On Sun, May 10, 2020 at 4:23 PM Nicolai Kraus > wrote: > >> > >> Yes, I think that is one main motivation for this axiom (that you've > >> suggested in this form :-) and I also believe that it was Vladimir's > >> main motivation for his axiom "exo-Nat is fibrant". I think the two > >> axioms really serve the same purpose, but the "cofibrant" version is > >> much more harmless. > >> > >> On 10/05/2020 16:16, Michael Shulman wrote: > >> > I forget -- does "exo-Nat is cofibrant" imply that exo-limits of > >> > towers of fibrations are fibrant? That's another useful axiom that > >> > holds in models and might make it easier to construct coinductive > >> > types with judgmental computation rules. > >> > > >> > On Sun, May 10, 2020 at 7:52 AM Nicolai Kraus < > nicola...@gmail.com> wrote: > >> >> I would guess that "exo-Nat is cofibrant" justifies the coinductive > type in question, but not its judgmental computation rules. And the > judgmental computation rules are probably the very reason why one would > want this coinductive type. But this is just a guess. > >> >> -- Nicolai > >> >> > >> >> On Sun, May 10, 2020 at 3:35 PM Michael Shulman < > shu...@sandiego.edu> wrote: > >> >>> Many or all coinductive types can be constructed, at least up to > >> >>> equivalence, using Pi-types and (some kind of) Nat. Is there any > >> >>> chance that "exo-Nat is cofibrant" could be used to justify the > >> >>> existence/fibrancy of the coinductive types you want? > >> >>> > >> >>> On Sun, May 10, 2020 at 7:20 AM Nicolai Kraus < > nicola...@gmail.com> wrote: > >> >>>> On 10/05/2020 15:01, Michael Shulman wrote: > >> >>>>> On Sun, May 10, 2020 at 4:46 AM Thorsten Altenkirch > >> >>>>> wrote: > >> >>>>>> Defining simplicial types isn't entirely straightforward (as you > know I suppose), because Delta is not directed. We can do semi-simplicial > types as a Reedy limit, i.e. an infinite Sigma type > >> >>>>> We can certainly *talk* about simplicial types in 2LTT as > exofunctors > >> >>>>> from the exocategory Delta to the exocategory Type. I assume the > >> >>>>> point you're making is that we don't have a (fibrant) *type of* > >> >>>>> simplicial types, whereas we do have a fibrant type of > semisimplicial > >> >>>>> types (under appropriate axioms)? > >> >>>> Judging from the rest of his message, I believe that Thorsten was > >> >>>> talking about the direct replacement construction in Christian's > and my > >> >>>> abstract arXiv:1704.04543. With the assumption "exo-Nat is > cofibrant", > >> >>>> this gives us a fibrant type that one could call "simplicial > types" (and > >> >>>> Thorsten does). But of course it's an encoding. If we decide to > use such > >> >>>> encodings, I fear we may lose the main advantage that the > "axiomatic" > >> >>>> representations in HoTT have, namely avoiding encodings. (I mean > the > >> >>>> "main advantage" of HoTT compared to traditional approaches, e.g. > taking > >> >>>> bisimplicial sets.) > >> >>>> > >> >>>>>> You need some extra principles, e.g. that strict Nat is fibrant > or maybe better that certain coinductive types exist. > >> >>>>> Personally, I think the best axiom to use here is that exo-Nat is > >> >>>>> *cofibrant*, i.e. Pi-types with domain exo-Nat preserve > fibrancy. We > >> >>>>> don't know how to model "exo-Nat is fibrant" in all higher > toposes, > >> >>>>> but it's easy to interpret "exo-Nat is cofibrant" in such models, > >> >>>>> since Pi-types with domain exo-Nat are just externally-infinite > >> >>>>> products. > >> >>>> I completely agree with your preference for this axiom :-) > >> >>>> But Thorsten does has a point if we consider the "engineering > level" > >> >>>> that was discussed earlier in this thread. Allowing coinductive > types > >> >>>> with exo-Nat as an index makes it possible to use your paper > (Higher > >> >>>> Stucture Identity Principle, arXiv:2004.06572) and get a > construction of > >> >>>> semi-simplicial types which is more convenient to use in a proof > assistant. > >> >>>> > >> >>>> -- Nicolai > >> >> -- > >> >> 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 HomotopyT...@googlegroups.com. > >> >> To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CA%2BAZBBr0Zh-uLfEZCXUapK5KHFDxkzxyvLW22zyjmrB8KmWtYQ%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 HomotopyT...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CA%2BAZBBrJF92OezfqYyh9vy-JQNesC8%2BpacAPPrq-xDZN5Y6qNQ%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 HomotopyT...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQz_JCE8QPTomY7ViOX%3DPgzZ_5aM3t9fx613jyfuOAUtvA%40mail.gmail.com > . >