Clarification of the proof of Lemma 6.2.9. (universal property of the circle) about book HOT 10 CLOSED

I think you're right, this was the intent (and so the invocation of the uniqueness principle is perhaps overly glib). I don't know offhand the best way to do this; probably you could deduce it from the construction of p in the proof of the uniqueness principle.

from book.

I've been thinking about this, and doing path induction on p and eqf in (ap φ p)⁻¹ ∙ eqf = eqg simplifies the goal to refl (φ f) ≡ eqg, which is false, so while the proof should be able to carried out, I'm suspecting it is not possible to carry it out directly, as was implied. Would you welcome PR using the usual bi-invertible maps?

from book.

from book.

Thanks, you're right. I've been thinking about this. We can not induct on p but we can induct on eqg, it is also enough to prove it for the case where f and eqf are the ones from recursion, i.e. f is 𝕊¹-rec A b l and eqf is given by pairing refl b with the computation principle of f. With this assumptions, we need

ap φ (funext (𝕊¹-ind-uniq f g (refl b) (𝕊¹-rec-comp A b l))) = eqf

Where 𝕊¹-ind-uniq is the function from Lemma 6.2.8.

This seems true but I see no way to prove this...

from book.

I think it is much easier to prove that this map has a quasi-inverse, and apply adjointification, than to prove directly that it has contractible fibers. Having contractible fibers, like being a half-adjoint equivalence, contains an extra coherence 2-path that makes it a proposition, but which is harder to construct in examples, and I think that's what's causing the pain here.

from book.

Yup I definetely agree, managing 2-paths is still very hard (to me). I suggest again maybe changing to the usual quasi-inverses proof, I can PR if you want.

from book.

Yes, I was going to suggest that once you worked out the details. (I also just formalized it in HoTT/Coq-HoTT#1664.) If you've done it already, I think a PR would be great. Thanks!

from book.

I've just finished it! (I've also formalized it in my agda repo).

I'll be closing this issue :)

Thank you!

from book.

The usual github-y thing to do would be to leave the issue open until the PR is merged.

from book.

Oh I didn't know, thanks for reopening then

from book.