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<p class="MsoNormal"><span style="font-size:11.0pt">Hi,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">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<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">MyGlue : (A : Set) {φ : I}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> → Partial φ (Σ[ T </span>
<span style="font-size:11.0pt;font-family:"Cambria Math",serif">∈</span><span style="font-size:11.0pt"> Set ] T
</span><span style="font-size:11.0pt;font-family:"Cambria Math",serif">≃</span><span style="font-size:11.0pt">' A) → Set<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">MyGlue A {φ} Te = Σ (PartialP φ λ o → fst (Te o)) (λ t → A [ φ
</span><span style="font-size:11.0pt;font-family:"Cambria Math",serif">↦</span><span style="font-size:11.0pt"> (λ p → fst (snd (Te p)) (t p)) ])<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">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<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">ua : {A B : Set} → (p : A </span>
<span style="font-size:11.0pt;font-family:"Cambria Math",serif">≃</span><span style="font-size:11.0pt">' B) → A ≡ B<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">because we need the extra definitional equality
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-indent:36.0pt"><span lang="DE" style="font-size:11.0pt">MyGlue A i1 (T , e) = T
<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Is there a way to avoid this? This is not a provable equality either, I think.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Cheers,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Thorsten<o:p></o:p></span></p>
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