*[HoTT] Looking for a reference that HITs are a strict extension of HoTT@ 2018-09-07 3:56 Jasper Hugunin2018-09-07 6:14 ` Nicolai Kraus 2018-09-07 12:38 ` Thorsten Altenkirch 0 siblings, 2 replies; 5+ messages in thread From: Jasper Hugunin @ 2018-09-07 3:56 UTC (permalink / raw) To: HomotopyTypeTheory [-- Attachment #1: Type: text/plain, Size: 926 bytes --] 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. [-- Attachment #2: Type: text/html, Size: 1204 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: [HoTT] Looking for a reference that HITs are a strict extension of HoTT2018-09-07 3:56 [HoTT] Looking for a reference that HITs are a strict extension of HoTT Jasper Hugunin@ 2018-09-07 6:14 ` Nicolai Kraus2018-09-07 6:30 ` Nicolai Kraus 2018-09-07 12:38 ` Thorsten Altenkirch 1 sibling, 1 reply; 5+ messages in thread From: Nicolai Kraus @ 2018-09-07 6:14 UTC (permalink / raw) To: Jasper Hugunin, HomotopyTypeTheory [-- Attachment #1: Type: text/plain, Size: 2238 bytes --] Hi Jasper, here's an argument: Without HITs, it's consistent to assume that every type in U_n is an n-type (since, as you said, all type formers preserve h-level). But with HIT's, consider the type Sigma (k: Nat), S^k. This is not a k-type for any k since the k-th fundamental group is nontrivial if you choose the base point correctly (see Licata-Brunerie CPP 2013). Remarks: 1. If we knew that S^2 is not a k-type for any k, then this would work as well for the second step, but as you said, we don't know so far whether this can be shown in HoTT. 2. For more general universe hierarchies than the one you use, for example indexed over omega+1 or indexed over any poset of arbitrary height, my argument won't work; I can't think of a proof for that situation off the top of my head. Nicolai On 07/09/18 04:56, Jasper Hugunin wrote: > 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 > <mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com>. > For more options, visit https://groups.google.com/d/optout. -- 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. [-- Attachment #2: Type: text/html, Size: 3334 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: [HoTT] Looking for a reference that HITs are a strict extension of HoTT2018-09-07 6:14 ` Nicolai Kraus@ 2018-09-07 6:30 ` Nicolai Kraus2018-09-07 10:30 ` Nicolai Kraus 0 siblings, 1 reply; 5+ messages in thread From: Nicolai Kraus @ 2018-09-07 6:30 UTC (permalink / raw) To: Jasper Hugunin, HomotopyTypeTheory [-- Attachment #1: Type: text/plain, Size: 1957 bytes --] Small addition to my first remark: On 07/09/18 07:14, Nicolai Kraus wrote: > Remarks: 1. If we knew that S^2 is not a k-type for any k, then this > would work as well for the second step, but as you said, we don't know > so far whether this can be shown in HoTT. Since we don't need an internal argument, it should be possible to use S^2 together with Thierry's result in Christian's post https://groups.google.com/forum/#!topic/homotopytypetheory/imPb56IqxOI But this is only for CCHM type theory. Nicolai > On 07/09/18 04:56, Jasper Hugunin wrote: >> 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 >> <mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com>. >> For more options, visit https://groups.google.com/d/optout. > -- 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. [-- Attachment #2: Type: text/html, Size: 3503 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*Re: [HoTT] Looking for a reference that HITs are a strict extension of HoTT2018-09-07 6:30 ` Nicolai Kraus@ 2018-09-07 10:30 ` Nicolai Kraus0 siblings, 0 replies; 5+ messages in thread From: Nicolai Kraus @ 2018-09-07 10:30 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 1038 bytes --] The addition from my second email didn't make sense, sorry. The first email with the easy argument should be correct. Nicolai On Friday, September 7, 2018 at 7:30:23 AM UTC+1, Nicolai Kraus wrote: > > Small addition to my first remark: > > On 07/09/18 07:14, Nicolai Kraus wrote: > > Remarks: 1. If we knew that S^2 is not a k-type for any k, then this would > work as well for the second step, but as you said, we don't know so far > whether this can be shown in HoTT. > > > Since we don't need an internal argument, it should be possible to use S^2 > together with Thierry's result in Christian's post > https://groups.google.com/forum/#!topic/homotopytypetheory/imPb56IqxOI > But this is only for CCHM type theory. > Nicolai > -- 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. [-- Attachment #1.2: Type: text/html, Size: 1960 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

*2018-09-07 3:56 [HoTT] Looking for a reference that HITs are a strict extension of HoTT Jasper Hugunin 2018-09-07 6:14 ` Nicolai KrausRe: [HoTT] Looking for a reference that HITs are a strict extension of HoTT@ 2018-09-07 12:38 ` Thorsten Altenkirch1 sibling, 0 replies; 5+ messages in thread From: Thorsten Altenkirch @ 2018-09-07 12:38 UTC (permalink / raw) To: Jasper Hugunin, homotopytypetheory [-- Attachment #1: Type: text/plain, Size: 2714 bytes --] 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: <homotopytypetheory@googlegroups.com> on behalf of Jasper Hugunin <jasperh@cs.washington.edu> Date: Friday, 7 September 2018 at 04:56 To: "homotopytypetheory@googlegroups.com" <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<mailto: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. [-- Attachment #2: Type: text/html, Size: 7506 bytes --] ^ permalink raw reply [flat|nested] 5+ messages in thread

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