From: Michael Shulman <firstname.lastname@example.org> To: "HomotopyTypeTheory@googlegroups.com" <email@example.com> Subject: [HoTT] Recovering an equivalence from univalence in cubical type theory Date: Wed, 18 Sep 2019 08:42:33 -0700 Message-ID: <CAOvivQzzSXNHs+zbPQTyHEuU53aHXJ0sPe4pr+yf0ahLGvUpVA@mail.gmail.com> (raw) 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 HomotopyTypeTheoryfirstname.lastname@example.org. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQzzSXNHs%2BzbPQTyHEuU53aHXJ0sPe4pr%2Byf0ahLGvUpVA%40mail.gmail.com.
next reply index Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-09-18 15:42 Michael Shulman [this message] 2019-09-18 16:15 ` Licata, Dan 2019-09-18 19:23 ` Michael Shulman 2019-09-18 20:35 ` Evan Cavallo 2019-09-19 8:20 ` Anders Mortberg
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