From: Ulrik Buchholtz <ulrikbu...@gmail.com> To: Homotopy Type Theory <HomotopyT...@googlegroups.com> Subject: Re: [HoTT] Identity versus equality Date: Sun, 10 May 2020 05:53:21 -0700 (PDT) Message-ID: <e47e6263-d0eb-4882-ab07-e31483095bd4@googlegroups.com> (raw) In-Reply-To: <8C57894C7413F04A98DDF5629FEC90B1652F563A@Pli.gst.uqam.ca> [-- Attachment #1.1: Type: text/plain, Size: 3176 bytes --] Dear André, On Saturday, May 9, 2020 at 10:18:11 PM UTC+2, joyal wrote: > But I find it frustrating that I cant do my everyday mathematics with it. > > For example, I cannot say > > (1) Let X:\Delta^{op}---->Type be a simplicial type; > Let me remind everyone that we don't have a proof that it's impossible to define simplicial types in book HoTT! As long as we haven't settled this question either way, the predicament we're in (and I agree it's a predicament) is more an indictment of type theorists than of type theory. (And I include myself as a type theorist here.) Anyway, from your phrasing of (1), it looks like you're after a directed type theory. Several groups are working on type theories where your (1) is valid syntax, but you have to write Space (or Anima or ...) to indicate the covariant universe of homotopy types and maps, rather than the full universe. (2) Let G be a type equipped with a group structure; > (3) Let BG be the classifying space of a group G; > (4) Let Gr be the category of groups; > Again, let me remind everyone, that groups are precisely one of the few structures we know how to handle (along with grouplike E_n-types, and connective/general spectra): To equip a type G with a group structure is to give a pointed connected type BG and an equivalence of G with the loop type of BG. The type of objects of Gr is the universe of pointed, connected types. When you replace groups by monoids, it gets more embarrassing. Earlier (on May 7) you wrote: > At some level, all mathematics is based on some kind of set theory. > Is it not obvious? > We cannot escape Cantor's paradise! > That is exactly the question, isn't it? What HoTT/UF offers us is another axis of foundational variation, besides the old classical/constructive, impredicative/predicative, infinitist/finitist, namely: everything is based on sets/general homotopy types are not reducible to sets. I don't know of catchy names for these positions, but I think that working with HoTT has a tendency of making the latter position more plausible: Why should there for any type be a set that surjects onto it? A question: Recall that if we assume the axiom of choice (AC) for sets, then there is a surjection from a set onto Set, namely the map that takes a cardinal (or ordinal) to the set of ordinals below it. Is there (with AC for sets) also a surjection from a set to the type of 1-types? To the full universe? The 2-level type theories can be viewed as another way of making a type theory that is faithful to the idea that everything is based on sets. I like to think of the outer layer as an exoskeleton for type theory, giving it a bit of extra strength while we don't know how to use its own powers fully, or whether indeed it is strong enough to effect constructions like that of simplicial types. Every type (which for me is a fibrant type, since that's the mathematically meaningful ones) has a corresponding exotype (indeed an exoset), but there are more exotypes, such as the exonatural numbers, which are sometimes useful. Best wishes, Ulrik [-- Attachment #1.2: Type: text/html, Size: 3944 bytes --]

next prev parent reply indexThread overview:61+ messages / expand[flat|nested] mbox.gz Atom feed top 2020-05-05 8:47 Ansten Mørch Klev 2020-05-06 16:02 ` [HoTT] " Joyal, André 2020-05-06 19:01 ` Steve Awodey 2020-05-06 19:18 ` Michael Shulman 2020-05-06 19:31 ` Steve Awodey 2020-05-06 20:30 ` Joyal, André 2020-05-06 22:52 ` Thorsten Altenkirch 2020-05-06 22:54 ` Thorsten Altenkirch 2020-05-06 23:29 ` Joyal, André 2020-05-07 6:11 ` Egbert Rijke 2020-05-07 6:58 ` Thorsten Altenkirch 2020-05-07 9:04 ` Ansten Mørch Klev 2020-05-07 10:09 ` Thomas Streicher 2020-05-07 16:13 ` Joyal, André 2020-05-07 21:41 ` David Roberts 2020-05-07 23:43 ` Joyal, André 2020-05-07 23:56 ` David Roberts 2020-05-08 6:40 ` Thomas Streicher 2020-05-08 21:06 ` Joyal, André 2020-05-08 23:44 ` Steve Awodey 2020-05-09 2:46 ` Joyal, André 2020-05-09 3:09 ` Jon Sterling [not found] ` <CADZEZBY+3z6nrRwsx9p-HqYuTxAnwMUHv7JasHy8aoy1oaGPcw@mail.gmail.com> 2020-05-09 2:50 ` Steve Awodey 2020-05-09 8:28 ` Thomas Streicher 2020-05-09 15:53 ` Joyal, André 2020-05-09 18:43 ` Thomas Streicher 2020-05-09 20:18 ` Joyal, André 2020-05-09 21:27 ` Jon Sterling 2020-05-10 2:19 ` Joyal, André 2020-05-10 3:04 ` Jon Sterling 2020-05-10 9:09 ` Thomas Streicher 2020-05-10 11:59 ` Thorsten Altenkirch 2020-05-10 11:46 ` Thorsten Altenkirch 2020-05-10 14:01 ` Michael Shulman 2020-05-10 14:20 ` Nicolai Kraus 2020-05-10 14:34 ` Michael Shulman 2020-05-10 14:52 ` Nicolai Kraus 2020-05-10 15:16 ` Michael Shulman 2020-05-10 15:23 ` Nicolai Kraus 2020-05-10 16:13 ` Nicolai Kraus 2020-05-10 16:28 ` Michael Shulman 2020-05-10 18:18 ` Nicolai Kraus 2020-05-10 19:15 ` Thorsten Altenkirch 2020-05-10 19:20 ` Thorsten Altenkirch2020-05-10 12:53 ` Ulrik Buchholtz [this message]2020-05-10 14:01 ` Michael Shulman 2020-05-10 14:27 ` Nicolai Kraus 2020-05-10 15:35 ` Ulrik Buchholtz 2020-05-10 16:30 ` Michael Shulman 2020-05-10 18:56 ` Nicolai Kraus 2020-05-10 18:04 ` Joyal, André 2020-05-11 7:33 ` Thomas Streicher 2020-05-11 14:54 ` Joyal, André 2020-05-11 16:37 ` stre... 2020-05-11 16:42 ` Michael Shulman 2020-05-11 17:27 ` Thomas Streicher 2020-05-10 16:51 ` Nicolai Kraus 2020-05-10 18:57 ` Michael Shulman 2020-05-10 19:18 ` Nicolai Kraus 2020-05-10 20:22 ` Michael Shulman 2020-05-10 22:08 ` Joyal, André

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