From: Peter LeFanu Lumsdaine <p.l.lu...@gmail.com> To: Steve Awodey <awo...@cmu.edu> Cc: Michael Shulman <shu...@sandiego.edu>, Thierry Coquand <Thierry...@cse.gu.se>, homotopy Type Theory <homotopyt...@googlegroups.com> Subject: Re: [HoTT] Semantics of higher inductive types Date: Thu, 1 Jun 2017 19:08:57 +0300 [thread overview] Message-ID: <CAAkwb-nogDyVRwnvDXUphBQb0ttL-TuHfSahnrSMhFnuunc7hw@mail.gmail.com> (raw) In-Reply-To: <9F58F820-A54A-46E7-93DC-F814D4BEE0C6@cmu.edu> [-- Attachment #1: Type: text/plain, Size: 7553 bytes --] On Thu, Jun 1, 2017 at 6:56 PM, Steve Awodey <awo...@cmu.edu> wrote: > > you mean the propositional truncation or suspension operations might lead to cardinals outside of a Grothendieck Universe? Exactly, yes. There’s no reason I know of to think they *need* to, but with the construction of Mike’s and my paper, they do. And adding stronger conditions on the cardinal used won’t help. The problem is that one takes a fibrant replacement to go from the “pre-suspension” to the suspension (more precisely: a (TC,F) factorisation, to go from the universal family of pre-suspensions to the universal family of suspensions); and fibrant replacement blows up the fibers to be the size of the *base* of the family. So the pre-suspension is small, but the suspension — although essentially small — ends up as large as the universe one’s using. So here’s a very precise problem which is as far as I know open: (*) Construct an operation Σ : U –> U, where U is Voevodsky’s universe, together with appropriate maps N, S : Û –> Û over Σ, and a homotopy m from N to S over Σ, which together exhibit U as “closed under suspension”. I asked a related question on mathoverflow a couple of years ago: https://mathoverflow.net/questions/219588/pullback-stable-model-of-fibrewise-suspension-of-fibrations-in-simplicial-sets David White suggested he could see an answer to that question (which would probably also answer (*) here) based on the comments by Karol Szumiło and Tyler Lawson, using the adjunction with Top, but I wasn’t quite able to piece it together. –p. > > > On Jun 1, 2017, at 11:38 AM, Michael Shulman <shu...@sandiego.edu> wrote: > > > > Do we actually know that the Kan simplicial set model has a *universe > > closed under* even simple HITs? It's not trivial because this would > > mean we could (say) propositionally truncate or suspend the generic > > small Kan fibration and get another *small* Kan fibration, whereas the > > base of these fibrations is not small, and fibrant replacement doesn't > > in general preserve smallness of fibrations with large base spaces. > > > > (Also, the current L-S paper doesn't quite give a general syntactic > > scheme, only a general semantic framework with suggestive implications > > for the corresponding syntax.) > > > > > > > > On Thu, Jun 1, 2017 at 8:30 AM, Steve Awodey <awo...@cmu.edu> wrote: > >> > >> On Jun 1, 2017, at 10:23 AM, Thierry Coquand <Thierry...@cse.gu.se > > >> wrote: > >> > >> If we are only interested in providing one -particular- model of HITs, > >> the paper > >> on cubical type theory describes a way to interpret HIT together with a > >> univalent > >> universe which is stable by HIT operations. This gives in particular the > >> consistency > >> and the proof theoretic power of this extension of type theory. > >> > >> > >> but the Kan simplicial set model already does this — right? > >> don’t get me wrong — I love the cubes, and they have lots of nice properties > >> for models of HoTT > >> — but there was never really a question of the consistency or coherence of > >> simple HITs like propositional truncation or suspension. > >> > >> the advance in the L-S paper is to give a general scheme for defining HITs > >> syntactically > >> (a definition, if you like, of what a HIT is, rather than a family of > >> examples), > >> and then a general description of the semantics of these, > >> in a range of models of the basic theory. > >> > >> Steve > >> > >> > >> The approach uses an operation of “flattening an open box”, which solves > >> in > >> this case the issue of interpreting HIT with parameters (such as > >> propositional > >> truncation or suspension) without any coherence issue. > >> Since the syntax used in this paper is so close to the semantics, we > >> limited > >> ourselves to a syntactical presentation of this interpretation. But it can > >> directly > >> be transformed to a semantical interpretation, as explained in the following > >> note > >> (which also incorporates a nice simplification of the operation of > >> flattering > >> an open box noticed by my coauthors). I also try to make more explicit in > >> the note > >> what is the problem solved by the “flattening boxes” method. > >> > >> Only the cases of the spheres and propositional truncation are described, > >> but one > >> would expect the method to generalise to other HITs covered e.g. in the HoTT > >> book. > >> > >> On 25 May 2017, at 20:25, Michael Shulman <shu...@sandiego.edu> wrote: > >> > >> The following long-awaited paper is now available: > >> > >> Semantics of higher inductive types > >> Peter LeFanu Lumsdaine, Mike Shulman > >> https://arxiv.org/abs/1705.07088 > >> > >> From the abstract: > >> > >> We introduce the notion of *cell monad with parameters*: a > >> semantically-defined scheme for specifying homotopically well-behaved > >> notions of structure. We then show that any suitable model category > >> has *weakly stable typal initial algebras* for any cell monad with > >> parameters. When combined with the local universes construction to > >> obtain strict stability, this specializes to give models of specific > >> higher inductive types, including spheres, the torus, pushout types, > >> truncations, the James construction, and general localisations. > >> > >> Our results apply in any sufficiently nice Quillen model category, > >> including any right proper simplicial Cisinski model category (such as > >> simplicial sets) and any locally presentable locally cartesian closed > >> category (such as sets) with its trivial model structure. In > >> particular, any locally presentable locally cartesian closed > >> (∞,1)-category is presented by some model category to which our > >> results apply. > >> > >> -- > >> 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 HomotopyTypeThe...@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 HomotopyTypeThe...@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 HomotopyTypeThe...@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 HomotopyTypeThe...@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 HomotopyTypeThe...@googlegroups.com. > For more options, visit https://groups.google.com/d/optout. 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next prev parent reply other threads:[~2017-06-01 16:08 UTC|newest]Thread overview:25+ messages / expand[flat|nested] mbox.gz Atom feed top 2017-05-25 18:25 Michael Shulman 2017-05-26 0:17 ` [HoTT] " Emily Riehl 2017-06-01 14:23 ` Thierry Coquand 2017-06-01 14:43 ` Michael Shulman 2017-06-01 15:30 ` Steve Awodey 2017-06-01 15:38 ` Michael Shulman 2017-06-01 15:56 ` Steve Awodey2017-06-01 16:08 ` Peter LeFanu Lumsdaine [this message]2017-06-06 9:19 ` Andrew Swan 2017-06-06 10:03 ` Andrew Swan 2017-06-06 13:35 ` Michael Shulman 2017-06-06 16:22 ` Andrew Swan 2017-06-06 19:36 ` Michael Shulman 2017-06-06 20:59 ` Andrew Swan 2017-06-07 9:40 ` Peter LeFanu Lumsdaine 2017-06-07 9:57 ` Thierry Coquand [not found] ` <ed7ad345-85e4-4536-86d7-a57fbe3313fe@googlegroups.com> 2017-06-07 23:06 ` Michael Shulman 2017-06-08 6:35 ` Andrew Swan 2018-09-14 11:15 ` Thierry Coquand 2018-09-14 14:16 ` Andrew Swan 2018-10-01 13:02 ` Thierry Coquand 2018-11-10 15:52 ` Anders Mörtberg 2018-11-10 18:21 ` Gabriel Scherer 2017-06-08 4:57 ` CARLOS MANUEL MANZUETA 2018-11-12 12:30 ` Ali Caglayan

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