From: Jasper Hugunin <jasperh@cs.washington.edu> To: HomotopyTypeTheory@googlegroups.com Subject: [HoTT] A question about the problem with regularity in CCHM cubical type theory Date: Fri, 13 Sep 2019 02:10:55 -0400 Message-ID: <CAGTS-a8fo4FKCLiCVCYm6AGeVc=qB8QNUZ3Weho6kLT8NXtc-Q@mail.gmail.com> (raw) [-- Attachment #1: Type: text/plain, Size: 2138 bytes --] Hello all, I've been trying to understand better why composition for the universe does not satisfy regularity. Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |-> equiv^i E ] A, I would expect regularity to follow from two parts: 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself) 2. That equiv^i (refl A) reduces to equivRefl A I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument. Context: I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly. If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in this thread on the HoTT mailing list <https://groups.google.com/d/msg/homotopytypetheory/oXQe5u_Mmtk/3HEDk5g5uq4J>, the definition of composition for the universe was found to not satisfy regularity. I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in the CCHM paper <http://www.cse.chalmers.se/~coquand/cubicaltt.pdf> (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-> equiv^i refl ] A not reducing to anything). Best regards, - 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. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a8fo4FKCLiCVCYm6AGeVc%3DqB8QNUZ3Weho6kLT8NXtc-Q%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 2723 bytes --] <div dir="ltr">Hello all,<div><br></div><div><div>I've been trying to understand better why composition for the universe does not satisfy regularity.</div><div>Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |-> equiv^i E ] A, I would expect regularity to follow from two parts:</div><div>1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself)</div><div>2. That equiv^i (refl A) reduces to equivRefl A</div><div>I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument.</div><div><br></div><div>Context:</div><div><div>I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly.</div><div>If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in <a href="https://groups.google.com/d/msg/homotopytypetheory/oXQe5u_Mmtk/3HEDk5g5uq4J" target="_blank">this thread on the HoTT mailing list</a>, the definition of composition for the universe was found to not satisfy regularity.</div><div>I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in <a href="http://www.cse.chalmers.se/~coquand/cubicaltt.pdf" target="_blank">the CCHM paper</a> (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-> equiv^i refl ] A not reducing to anything).</div><div><br></div></div></div><div>Best regards,</div><div>- Jasper Hugunin</div></div> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a8fo4FKCLiCVCYm6AGeVc%3DqB8QNUZ3Weho6kLT8NXtc-Q%40mail.gmail.com?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a8fo4FKCLiCVCYm6AGeVc%3DqB8QNUZ3Weho6kLT8NXtc-Q%40mail.gmail.com</a>.<br />

next reply indexThread overview:13+ messages / expand[flat|nested] mbox.gz Atom feed top2019-09-13 6:10 Jasper Hugunin [this message]2019-09-15 5:57 ` [HoTT] " Jasper Hugunin [not found] ` <CAMWCppk4PWzfZ1HKNLMdAZ=RBC-ARxtJXR8okvwO3raea5gC8Q@mail.gmail.com> [not found] ` <CAGTS-a-SvWWF+br6sKxGj6ufVY=4m830FH9BDg06QJR1vbNFsw@mail.gmail.com> 2019-09-16 2:18 ` Fwd: " Jasper Hugunin 2019-09-16 16:18 ` [HoTT] " Licata, Dan 2019-09-16 17:09 ` Jasper Hugunin 2019-09-16 19:01 ` Licata, Dan 2019-09-16 20:17 ` Jasper Hugunin 2019-09-18 12:16 ` Anders Mortberg 2019-09-18 12:52 ` Thierry Coquand 2019-09-15 11:55 ` [HoTT] " Andrew Swan 2019-09-15 22:38 ` Jasper Hugunin 2019-09-16 1:04 ` Jon Sterling [not found] ` <A605E6EE-0101-4390-B50D-A6AEB36FDCC2@icloud.com> 2019-09-16 1:44 ` Jon Sterling

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