Don’t you need at least some sort of quotients? How do you define the 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 QITs (set truncated HITs) that are not definable using quotients. I say “likely” because I think that their construction doesn’t allow for univalence. On the other hand I don’t see a way 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 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 difficult, 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 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 email 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. 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 where permitted by law. -- 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 email to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout.