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[209.85.210.48]) by smtp.gmail.com with ESMTPSA id w3sm1999889otg.24.2020.05.10.09.29.05 for (version=TLS1_3 cipher=TLS_AES_128_GCM_SHA256 bits=128/128); Sun, 10 May 2020 09:29:05 -0700 (PDT) Received: by mail-ot1-f48.google.com with SMTP id z17so5617276oto.4 for ; Sun, 10 May 2020 09:29:05 -0700 (PDT) X-Received: by 2002:a9d:170e:: with SMTP id i14mr9979070ota.283.1589128145211; Sun, 10 May 2020 09:29:05 -0700 (PDT) MIME-Version: 1.0 References: <67E9DCCA-F9CA-45B7-9AC8-E5A7E94FE9A9@nottingham.ac.uk> <8C57894C7413F04A98DDF5629FEC90B1652F53A3@Pli.gst.uqam.ca> <20200508064039.GC21473@mathematik.tu-darmstadt.de> <8C57894C7413F04A98DDF5629FEC90B1652F54CC@Pli.gst.uqam.ca> <20200509082829.GA8417@mathematik.tu-darmstadt.de> <8C57894C7413F04A98DDF5629FEC90B1652F55C8@Pli.gst.uqam.ca> <20200509184313.GB28841@mathematik.tu-darmstadt.de> <8C57894C7413F04A98DDF5629FEC90B1652F563A@Pli.gst.uqam.ca> <154D1B4B-6881-4269-8BA1-1CE8EBB0BE0D@nottingham.ac.uk> <3ec2f134-cbd4-7bfd-0a69-efb734a4a231@gmail.com> <9526fc07-cf21-03e8-1c68-acccc2941117@gmail.com> In-Reply-To: From: Michael Shulman Date: Sun, 10 May 2020 09:28:53 -0700 X-Gmail-Original-Message-ID: Message-ID: Subject: Re: [HoTT] Identity versus equality To: "HomotopyT...@googlegroups.com" Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable 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 w= hether "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 wr= ote: >> >> I would guess that "exo-Nat is cofibrant" justifies the coinductive t= ype in question, but not its judgmental computation rules. And the judgment= al computation rules are probably the very reason why one would want this c= oinductive type. But this is just a guess. >> >> -- Nicolai >> >> >> >> On Sun, May 10, 2020 at 3:35 PM Michael Shulman = 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 = 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 t= ypes as a Reedy limit, i.e. an infinite Sigma type >> >>>>> We can certainly *talk* about simplicial types in 2LTT as exofunct= ors >> >>>>> 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 semisimplic= ial >> >>>>> 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 an= d my >> >>>> abstract arXiv:1704.04543. With the assumption "exo-Nat is cofibran= t", >> >>>> 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 "axiomati= c" >> >>>> representations in HoTT have, namely avoiding encodings. (I mean th= e >> >>>> "main advantage" of HoTT compared to traditional approaches, e.g. t= aking >> >>>> bisimplicial sets.) >> >>>> >> >>>>>> You need some extra principles, e.g. that strict Nat is fibrant o= r 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 typ= es >> >>>> with exo-Nat as an index makes it possible to use your paper (Highe= r >> >>>> Stucture Identity Principle, arXiv:2004.06572) and get a constructi= on of >> >>>> semi-simplicial types which is more convenient to use in a proof as= sistant. >> >>>> >> >>>> -- Nicolai >> >> -- >> >> You received this message because you are subscribed to the Google Gr= oups "Homotopy Type Theory" group. >> >> To unsubscribe from this group and stop receiving emails from it, sen= d an email to HomotopyT...@googlegroups.com. >> >> To view this discussion on the web visit https://groups.google.com/d/= msgid/HomotopyTypeTheory/CA%2BAZBBr0Zh-uLfEZCXUapK5KHFDxkzxyvLW22zyjmrB8KmW= tYQ%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/msgi= d/HomotopyTypeTheory/CA%2BAZBBrJF92OezfqYyh9vy-JQNesC8%2BpacAPPrq-xDZN5Y6qN= Q%40mail.gmail.com.