Actually, I've just noticed that doesn't quite work - I want to say that a map is a weak fibration if it has a (uniform choice of) diagonal fillers for lifting problems against generating cofibrations where the bottom map factors through the projection I x V -> V, but that doesn't seem to be cofibrantly generated. Maybe it's still possible to do something like fibrant replacement anyway. Andrew On Tuesday, 6 June 2017 11:19:37 UTC+2, Andrew Swan wrote: > > I've been thinking a bit about abstract ways of looking at the HITs in > cubical type theory, and I don't have a complete proof, but I think > actually the same sort of thing should work for simplicial sets. > > We already know that the fibrations in the usual model structure on > simplicial sets can be defined as maps with the rlp against the pushout > product of generating cofibrations with interval endpoint inclusions (in > Christian's new paper on model > structures he cites for this result Chapter IV, section 2 of P. Gabriel and > M. Zisman. Calculus of fractions and homotopy theory, but I'm not familiar > with the proof myself). > > Now a generating trivial cofibration is the pushout product of a > generating cofibration with endpoint inclusion, so its codomain is of the > form I x V, where V is the codomain of the generating cofibration (which > for cubical sets and simplicial sets is representable). Then we get another > map by composing with projection I x V -> V, which is a retract of the > generating trivial cofibration and so also a trivial cofibration. If a map > has the rlp against all such maps, then call it a weak fibration. Then I > think the resulting awfs of "weak fibrant replacement" should be stable > under pullback (although of course, the right maps in the factorisation are > only weak fibrations, not fibrations in general). > > Then eg for propositional truncation, construct the "fibrant truncation" > monad by the coproduct of truncation monad with weak fibrant replacement. > In general, given a map X -> Y, the map ||X|| -> Y will only be a weak > fibration, but if X -> Y is fibration then I think the map ||X|| -> Y > should be also. I think the way to formulate this would be as a > distributive law - the fibrant replacement monad distributes over the > (truncation + weak fibrant replacement) monad. It looks to me like the same > thing that works in cubical sets should also work here - first define a > "box flattening" operation for any fibration (i.e. the operation labelled > as "forward" in Thierry's note), then show that this operation lifts > through the HIT constructors to give a box flattening operation on the HIT, > then show that in general weak fibration plus box flattening implies > fibration, (Maybe one way to do this would be to note that the cubical set > argument is mostly done internally in cubical type theory, and simplicial > sets model cubical type theory by Orton & Pitts, Axioms for Modelling > Cubical Type Theory in a Topos > > ) > > Best, > Andrew > > > > On Thursday, 1 June 2017 18:08:58 UTC+2, Peter LeFanu Lumsdaine wrote: >> >> On Thu, Jun 1, 2017 at 6:56 PM, Steve Awodey 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 >> 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 wrote: >> > >> >> > >> On Jun 1, 2017, at 10:23 AM, Thierry Coquand >> > >> 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 >> 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. >> >