Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / mirror / Atom feed
From: Michael Shulman <shulman@sandiego.edu>
To: "HomotopyTypeTheory@googlegroups.com"
Subject: [HoTT] Recovering an equivalence from univalence in cubical type theory
Date: Wed, 18 Sep 2019 08:42:33 -0700	[thread overview]
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 HomotopyTypeTheory+unsubscribe@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQzzSXNHs%2BzbPQTyHEuU53aHXJ0sPe4pr%2Byf0ahLGvUpVA%40mail.gmail.com.

             reply	other threads:[~2019-09-18 15:42 UTC|newest]

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

Reply instructions:

You may reply publicly 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:

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to=CAOvivQzzSXNHs+zbPQTyHEuU53aHXJ0sPe4pr+yf0ahLGvUpVA@mail.gmail.com \
    --to=shulman@sandiego.edu \
    --cc=homotopytypetheory@googlegroups.com \


* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).