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Hi Thorsten,<br>
<br>
Your implementation is also what Orton and Pitts do in their
internal model of cubical type theory:<br>
<a class="moz-txt-link-freetext" href="https://doi.org/10.23638/LMCS-14(4:23)2018">https://doi.org/10.23638/LMCS-14(4:23)2018</a><br>
<br>
They rely on an additional axiom, the strictness axiom, to strictify
MyGlue so that it actually extends T.<br>
<br>
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.<br>
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"?<br>
<br>
Best regards,<br>
Andreas<br>
<br>
<div class="moz-cite-prefix">On 20/05/2020 11:34, Thorsten
Altenkirch wrote:<br>
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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
style="font-size:11.0pt" lang="DE">MyGlue A i1 (T , e) = T
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt" lang="DE"><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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