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<p class="MsoNormal">Thank you for your reply, Andreas.<o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt">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"?<o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">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.<br>
<br>
<span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><b><span style="color:black">From:
</span></b><span style="color:black">Andreas Nuyts <andreas.nuyts@kuleuven.be><br>
<b>Date: </b>Wednesday, 20 May 2020 at 12:06<br>
<b>To: </b>Thorsten Altenkirch <psztxa@exmail.nottingham.ac.uk>, Thierry Coquand <Thierry.Coquand@cse.gu.se>, Anders Mortberg <andersmortberg@gmail.com>, Andrea Vezzosi <sanzhiyan@gmail.com><br>
<b>Cc: </b>Agda mailing list <agda@lists.chalmers.se><br>
<b>Subject: </b>Re: [Agda] Understanding Glue</span><span style="color:black;mso-fareast-language:EN-GB"><o:p></o:p></span></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal" style="mso-margin-top-alt:0cm;margin-right:0cm;margin-bottom:12.0pt;margin-left:36.0pt">
Hi Thorsten,<br>
<br>
Your implementation is also what Orton and Pitts do in their internal model of cubical type theory:<br>
<a 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<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">On 20/05/2020 11:34, Thorsten Altenkirch wrote:<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">Hi,</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><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</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">MyGlue : (A : Set) {φ : I}</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> → Partial φ (Σ[ T
</span><span style="font-size:11.0pt;font-family:"Cambria Math ,serif",serif">∈</span><span style="font-size:11.0pt"> Set ] T
</span><span style="font-size:11.0pt;font-family:"Cambria Math ,serif",serif">≃</span><span style="font-size:11.0pt">' A) → Set</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><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",serif">↦</span><span style="font-size:11.0pt"> (λ p → fst (snd (Te p)) (t p)) ])</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><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</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">ua : {A B : Set} → (p : A
</span><span style="font-size:11.0pt;font-family:"Cambria Math ,serif",serif">≃</span><span style="font-size:11.0pt">' B) → A ≡ B</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">because we need the extra definitional equality
</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt;text-indent:36.0pt"><span lang="DE" style="font-size:11.0pt">MyGlue A i1 (T , e) = T
</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span lang="DE" style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">Is there a way to avoid this? This is not a provable equality either, I think.</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"> </span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">Cheers,</span><o:p></o:p></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">Thorsten</span><o:p></o:p></p>
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