Re: [HoTT] univalence without coherent equivalences

[Nicolai Kraus <nicola...@gmail.com>]

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I had not looked at this from this angle so far. If you want to avoid 
having to come up and justify a notion of equivalence, you could, 
alternatively to ua + uabeta, take the Orton-Pitts "Axioms for univalence"
www.cl.cam.ac.uk/~amp12/papers/axiu/axiu.pdf
Maybe these are even easier to justify than ua + uabeta!
Nicolai

On 10/08/17 21:57, Michael Shulman wrote:
> Thinking about the recently re-mentioned characterization of
> univalence in terms of a map
>    Equiv A B -> (A = B)
> that is only assumed to be a section of the canonical map in the other
> direction, it occurred to me that this gives a way to state the
> univalence axiom without first needing any "coherent" notion of
> equivalence.  For at least if we have funext to start with, then
> equalities in Equiv A B are (for any coherent definition of Equiv A B)
> equivalent to equalities in A -> B, so we can state the retraction
> property in terms of those.
>
> More precisely, let
>    coe : (A = B) -> A -> B
> be coercion along an equality, and let
>    QInv A B := Sigma(f:A->B) Sigma(g:B->A) ( (Pi(x:A) g(f(x)) = x)
> \times (Pi(y:B) f(g(y))=y) )
> be the type of quasi-invertible functions (incoherent equivalences).
> We know that it is inconsistent to ask that the map (A = B) -> QInv A
> B induced by coe is quasi-invertible.  But suppose we instead ask for
> just
>    ua : QInv A B -> (A = B)
> and
>    uabeta : Pi((f,g,a,b) : QInv A B) Pi(a:A), coe (ua (f,g,a,b)) a = f(a)
> If full univalence holds, then such functions certainly exist, since
> any quasi-inverse can be improved to a coherent equivalence.  But
> conversely, if we assume funext to start with, then the full
> univalence axiom can be proven from this ua and uabeta.  (I just
> formalized this in Coq to be sure.)
>
> Maybe other people have already observed this, but I don't think I
> noticed it before.  It means that we don't need to invent or motivate
> a coherent notion of equivalence before stating (or, in cubical type
> theory or a semantic model, proving) univalence.  Instead we can state
> univalence in this way, and then (having already motivated funext,
> which is much easier, and also has a "weak improves to strong"
> theorem) motivate the search for a "good" definition of Equiv A B as
> "can we define more explicitly a type that is equivalent to A = B"?
>
> Mike
>


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