[Agda] Understanding Glue

[Thorsten Altenkirch]

Thank you for your reply, Andreas.

I am now puzzled by the fact that this means that you can define a non-Kan type in cubical Agda. I guess "Partial" is a non-fibrant type former? Does Agda somehow prevent the user to exploit Kan-fibrancy of types constructed using "Partial"?

Yes, in the moment agda puts it into the universe Setω which is a stop gap for having a non-fibrant universe. As I mentioned I switched off universe checking which means I can pretend that this is fibrant.

Jesper implemented a better more principled solution where you can have both a hierarchy of fibrant and non-fibrant universes but the cubical library currently doesn’t use this.


From: Andreas Nuyts <andreas.nuyts at kuleuven.be>
Date: Wednesday, 20 May 2020 at 12:06
To: Thorsten Altenkirch <psztxa at exmail.nottingham.ac.uk>, Thierry Coquand <Thierry.Coquand at cse.gu.se>, Anders Mortberg <andersmortberg at gmail.com>, Andrea Vezzosi <sanzhiyan at gmail.com>
Cc: Agda mailing list <agda at lists.chalmers.se>
Subject: Re: [Agda] Understanding Glue

Hi Thorsten,

Your implementation is also what Orton and Pitts do in their internal model of cubical type theory:
https://doi.org/10.23638/LMCS-14(4:23)2018

They rely on an additional axiom, the strictness axiom, to strictify MyGlue so that it actually extends T.

Note that this implementation only relies on one direction of the equivalence. This is the only direction needed to form Glue (up to iso or relying on the strictness axiom) as a pre-type. The other direction is needed to make it Kan fibrant.
I am now puzzled by the fact that this means that you can define a non-Kan type in cubical Agda. I guess "Partial" is a non-fibrant type former? Does Agda somehow prevent the user to exploit Kan-fibrancy of types constructed using "Partial"?

Best regards,
Andreas
On 20/05/2020 11:34, Thorsten Altenkirch wrote:
Hi,

I am trying to understand the Glue construction better. I was wondering whether Glue just means that a certain pre-type is fibrant (i.e. admits composition, i.e. transp and hcomp). I was trying in agda

MyGlue : (A : Set) {φ : I}
     → Partial φ (Σ[ T ∈ Set ] T ≃' A) → Set
MyGlue A {φ} Te = Σ (PartialP φ λ o → fst (Te o)) (λ t → A [ φ ↦ (λ p → fst (snd (Te p)) (t p)) ])

I have switched off universe checking, hence Setw = Set. Basically I have used the domain of glue as the definition of Glue. However, this doesn’t seem enough to derive

ua : {A B : Set} → (p : A ≃' B) → A ≡ B

because we need the extra definitional equality

MyGlue A i1 (T , e) = T

Is there a way to avoid this? This is not a provable equality either, I think.

Cheers,
Thorsten

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