Re: [HoTT] Recovering an equivalence from univalence in cubical type theory

["Licata, Dan" <dlicata@wesleyan.edu>]

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 <shulman@sandiego.edu> 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.)
> 
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