From: "Licata, Dan" <email@example.com> To: Michael Shulman <firstname.lastname@example.org> Cc: "HomotopyTypeTheory@googlegroups.com" <email@example.com> Subject: Re: [HoTT] Recovering an equivalence from univalence in cubical type theory Date: Wed, 18 Sep 2019 16:15:19 +0000 Message-ID: <1DF8E802-2959-4BEF-A85A-3C6E5E7B9595@wesleyan.edu> (raw) In-Reply-To: <CAOvivQzzSXNHs+zbPQTyHEuU53aHXJ0sPe4pr+yf0ahLGvUpVA@mail.gmail.com> In ABCFHL, even the function fst(coe(ua(e))) : A -> B is only path-equal to fst(e) : A -> B. If I recall correctly, the issue is that composition in the Glue type that you use to implement ua doesn’t judgementally give you f; instead there is some morally-the-identity-composition (that would cancel with regularity) that gets stuck in. This is because the general algorithm for composition in the glue type has to coerce in the “base” of the glue type (B in Glue [alpha -> T] B), which in the case of ua(e) = Glue [x = 0 -> (A,e), x=1 -> (B,id)] B is degenerate in x, but in general might not be. I don’t recall any cubical type theories solving this, but I don’t remember the details of all of the other variations that have been explored well enough to say for sure. > On Sep 18, 2019, at 11:42 AM, Michael Shulman <firstname.lastname@example.org> wrote: > > Let Equiv(A,B) denote the type of half-adjoint equivalences, so that > an e:Equiv(A,B) consists of five data: a function A -> B, a function B > -> A, two homotopies, and a coherence 2-path. Using univalence, we > can make e into an identification ua(e) : A=B, and then back into an > equivalence coe(ua(e)) : Equiv(A,B), which is typally equal to e. > > Now we might wonder whether coe(ua(e)) might be in fact *judgmentally* > equal to e; or at least whether this might be true of some, if not > all, of its five components. In Book HoTT this is clearly not the > case, since univalence is posited as an axiom about which we know > nothing else. But what about cubical type theories? Can any of the > components of an equivalence e be recovered, up to judgmental > equality, from coe(ua(e))? (My guess would be that at least the > function A -> B, and probably also the function B -> A, can be > recovered, but perhaps not more.) > > -- > 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 HomotopyTypeTheoryemail@example.com. > To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQzzSXNHs%2BzbPQTyHEuU53aHXJ0sPe4pr%2Byf0ahLGvUpVA%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 HomotopyTypeTheoryfirstname.lastname@example.org. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/1DF8E802-2959-4BEF-A85A-3C6E5E7B9595%40wesleyan.edu.
next prev parent reply index Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-09-18 15:42 Michael Shulman 2019-09-18 16:15 ` Licata, Dan [this message] 2019-09-18 19:23 ` Michael Shulman 2019-09-18 20:35 ` Evan Cavallo 2019-09-19 8:20 ` Anders Mortberg
Reply instructions: You may reply publically to this message via plain-text email using any one of the following methods: * Save the following mbox file, import it into your mail client, and reply-to-all from there: mbox Avoid top-posting and favor interleaved quoting: https://en.wikipedia.org/wiki/Posting_style#Interleaved_style * Reply using the --to, --cc, and --in-reply-to switches of git-send-email(1): git send-email \ --in-reply-to=1DF8E802-2959-4BEF-A85A-3C6E5E7B9595@wesleyan.edu \ --email@example.com \ --firstname.lastname@example.org \ --email@example.com \ /path/to/YOUR_REPLY https://kernel.org/pub/software/scm/git/docs/git-send-email.html * If your mail client supports setting the In-Reply-To header via mailto: links, try the mailto: link
Discussion of Homotopy Type Theory and Univalent Foundations Archives are clonable: git clone --mirror http://inbox.vuxu.org/hott Newsgroup available over NNTP: nntp://inbox.vuxu.org/vuxu.archive.hott AGPL code for this site: git clone https://public-inbox.org/ public-inbox