[Agda] Understanding Glue

[Thorsten Altenkirch]

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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