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spf=pass (google.com: domain of thorsten.altenkirch@nottingham.ac.uk designates 128.243.43.125 as permitted sender) smtp.mailfrom=Thorsten.Altenkirch@nottingham.ac.uk Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , --_000_7EB528F2E4034706B89B6A46C7176A7Dexmailnottinghamacuk_ Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Don=E2=80=99t you need at least some sort of quotients? How do you define t= he Cauchy Reals otherwise? Ok using resizing (not recommended) you can encode quotients (as in a topos= ). However quotients are not enough. In https://arxiv.org/abs/1705.07088, Lumsdaine and Shulman Section 9 given= an example based on a construction by Blass which shows that there are QIT= s (set truncated HITs) that are not definable using quotients. I say =E2=80=9Clikely=E2=80=9D because I think that their construction does= n=E2=80=99t allow for univalence. On the other hand I don=E2=80=99t see a w= ay how to define their counterexample using univalence either. Thorsten From: on behalf of Jasper Hugunin Date: Friday, 7 September 2018 at 04:56 To: "homotopytypetheory@googlegroups.com" Subject: [HoTT] Looking for a reference that HITs are a strict extension of= HoTT Hello all, Many ways of doing HoTT (Coq + Univalence Axiom, Cubical Type Theory) make = sense without including support for defining Higher Inductive Types. The po= ssibility of defining small, closed types which are not hsets (like the cir= cle) or have infinite h-level (like the 2-sphere, conjectured?) makes const= ructing HITs from other types seem difficult, since all the type formers ex= cept universes preserve h-level. Does anyone know a proof that it is impossible to construct some HITs from = basic type formers (say 0, 1, 2, Sigma, Pi, W, and a hierarchy of univalent= universes U_n), up to equivalence? - Jasper Hugunin -- You received this message because you are subscribed to the Google Groups "= Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. This message and any attachment are intended solely for the addressee and may contain confidential information. If you have received this message in error, please contact the sender and delete the email and attachment.=20 Any views or opinions expressed by the author of this email do not necessarily reflect the views of the University of Nottingham. Email communications with the University of Nottingham may be monitored=20 where permitted by law. --=20 You received this message because you are subscribed to the Google Groups "= Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. --_000_7EB528F2E4034706B89B6A46C7176A7Dexmailnottinghamacuk_ Content-Type: text/html; charset="UTF-8" Content-ID: Content-Transfer-Encoding: quoted-printable

Don=E2=80= =99t you need at least some sort of quotients? How do you define the Cauchy= Reals otherwise?

&nbs= p;

Ok using = resizing (not recommended) you can encode quotients (as in a topos).

&nbs= p;

However q= uotients are not enough.

&nbs= p;

In https://arxiv.org/abs/1705.07088,  Lumsdaine and Shulman Section 9= given an example based on a construction by Blass which shows that there a= re QITs (set truncated HITs) that are not definable using quotients.

&nbs= p;

I say =E2= =80=9Clikely=E2=80=9D because I think that their construction doesn=E2=80= =99t allow for univalence. On the other hand I don=E2=80=99t see a way how = to define their counterexample using univalence either.

&nbs= p;

Thorsten<= o:p>

&nbs= p;

From= : <homotopytypet= heory@googlegroups.com> on behalf of Jasper Hugunin <jasperh@cs.washi= ngton.edu>
Date: Friday, 7 September 2018 at 04:56
To: "homotopytypetheory@googlegroups.com" <homotopytype= theory@googlegroups.com>
Subject: [HoTT] Looking for a reference that HITs are a strict exten= sion of HoTT

 

Hello all, =

=  

Many = ways of doing HoTT (Coq + Univalence Axiom, Cubical Type Theory) m= ake sense without including support for defining Higher Inductive Types. Th= e possibility of defining small, closed types which are not hsets (like the circle) or have infinite h-level (like the 2= -sphere, conjectured?) makes constructing HITs from other types seem diffic= ult, since all the type formers except universes preserve h-level.

=  

Does = anyone know a proof that it is impossible to construct some HITs from basic= type formers (say 0, 1, 2, Sigma, Pi, W, and a hierarchy of univalent univ= erses U_n), up to equivalence?

=  

- Jas= per Hugunin

=  

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