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[209.85.213.180]) by smtp.gmail.com with ESMTPSA id l6sm3293392ywf.16.2017.06.06.12.36.53 for (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Tue, 06 Jun 2017 12:36:54 -0700 (PDT) Received: by mail-yb0-f180.google.com with SMTP id 4so17611029ybl.1 for ; Tue, 06 Jun 2017 12:36:53 -0700 (PDT) X-Received: by 10.37.171.230 with SMTP id v93mr4256802ybi.242.1496777813202; Tue, 06 Jun 2017 12:36:53 -0700 (PDT) MIME-Version: 1.0 Received: by 10.37.18.215 with HTTP; Tue, 6 Jun 2017 12:36:32 -0700 (PDT) In-Reply-To: References: <1128BE39-BBC4-4DC6-8792-20134A6CAECD@chalmers.se> <292DED31-6CB3-49A1-9128-5AFD04B9C2F2@cmu.edu> <9F58F820-A54A-46E7-93DC-F814D4BEE0C6@cmu.edu> <2efaa818-9ed1-459f-a3a5-a274d19e6a96@googlegroups.com> <1c2cb641-89e3-444d-aa0c-cb8ccb79cf3c@googlegroups.com> From: Michael Shulman Date: Tue, 6 Jun 2017 13:36:32 -0600 X-Gmail-Original-Message-ID: Message-ID: Subject: Re: [HoTT] Semantics of higher inductive types To: Andrew Swan Cc: Homotopy Type Theory , Steve Awodey , Thierry Coquand Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable What class of homotopy theories can be presented by such models? On Tue, Jun 6, 2017 at 10:22 AM, Andrew Swan wrote: > I don't know how general this is exactly in practice, but I think it shou= ld > work in the setting that appears in van de Berg, Frumin A homotopy-theore= tic > model of function extensionality in the effective topos, which regardless= of > the title is not just the effective topos, but any topos together with a > interval object with connections, a dominance satisfying certain conditio= ns, > with fibrations defined as maps with the rlp against pushout product of > endpoint inclusions and elements of the dominance (& in addition there > should be some more conditions to ensure that free monads on pointed > endofunctors exist). > > > I'm a bit more confident that it works now. The class of weak fibrations = is > not cofibrantly generated in the usual sense (as I claimed in the first > post), but they are in the more general sense by Christian Sattler in > section 6 of The Equivalence Extension Property and Model Structures. The= n a > version of step 1 of the small object argument applies to Christian's > definition, which gives a pointed endofunctor whose algebras are the weak > fibrations. The same technique can also be used to describe "box flatteni= ng" > (which should probably be called something else, like "cylinder flattenin= g" > in the general setting). > > > Andrew > > On Tuesday, 6 June 2017 15:35:36 UTC+2, Michael Shulman wrote: >> >> I'll be interested to see if you can make it work! >> >> But I'd be much more interested if there is something that can be done >> in a general class of models, rather than a particular one like >> cubical or simplicial sets. >> >> On Tue, Jun 6, 2017 at 4:03 AM, Andrew Swan wrote: >> > Actually, I've just noticed that doesn't quite work - I want to say th= at >> > a >> > map is a weak fibration if it has a (uniform choice of) diagonal fille= rs >> > 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 i= n >> >> 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 pusho= ut >> >> 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 anoth= er >> >> 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. The= n >> >> I >> >> think the resulting awfs of "weak fibrant replacement" should be stab= le >> >> under pullback (although of course, the right maps in the factorisati= on >> >> 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, (May= be >> >> one >> >> way to do this would be to note that the cubical set argument is most= ly >> >> 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 mig= ht >> >>> > lead to cardinals outside of a Grothendieck Universe? >> >>> >> >>> Exactly, yes. There=E2=80=99s no reason I know of to think they *ne= ed* to, >> >>> but >> >>> with the construction of Mike=E2=80=99s and my paper, they do. And = adding >> >>> stronger >> >>> conditions on the cardinal used won=E2=80=99t help. The problem is = that one >> >>> takes a >> >>> fibrant replacement to go from the =E2=80=9Cpre-suspension=E2=80=9D = to the suspension >> >>> (more >> >>> precisely: a (TC,F) factorisation, to go from the universal family o= f >> >>> 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 =E2=80=94 althoug= h >> >>> essentially >> >>> small =E2=80=94 ends up as large as the universe one=E2=80=99s using= . >> >>> >> >>> So here=E2=80=99s a very precise problem which is as far as I know o= pen: >> >>> >> >>> (*) Construct an operation =CE=A3 : U =E2=80=93> U, where U is Voevo= dsky=E2=80=99s >> >>> universe, >> >>> together with appropriate maps N, S : =C3=9B =E2=80=93> =C3=9B over = =CE=A3, and a homotopy m >> >>> from N >> >>> to S over =CE=A3, which together exhibit U as =E2=80=9Cclosed under = suspension=E2=80=9D. >> >>> >> >>> I asked a related question on mathoverflow a couple of years ago: >> >>> >> >>> https://mathoverflow.net/questions/219588/pullback-stable-model-of-f= ibrewise-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 Szumi= =C5=82o >> >>> and >> >>> Tyler Lawson, using the adjunction with Top, but I wasn=E2=80=99t qu= ite able >> >>> to >> >>> piece it together. >> >>> >> >>> =E2=80=93p. >> >>> >> >>> > >> >>> > > 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, where= as >> >>> > > 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 =E2=80=94 ri= ght? >> >>> > >> don=E2=80=99t get me wrong =E2=80=94 I love the cubes, and they= have lots of nice >> >>> > >> properties >> >>> > >> for models of HoTT >> >>> > >> =E2=80=94 but there was never really a question of the consiste= ncy 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 =E2=80=9Cflattening an open= box=E2=80=9D, >> >>> > >> which >> >>> > >> solves >> >>> > >> in >> >>> > >> this case the issue of interpreting HIT with parameters (such a= s >> >>> > >> propositional >> >>> > >> truncation or suspension) without any coherence issue. >> >>> > >> Since the syntax used in this paper is so close to the semantic= s, >> >>> > >> 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 =E2=80=9Cflattening boxes=E2= =80=9D 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 >> >>> > >> (=E2=88=9E,1)-category is presented by some model category to w= hich 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 i= t, >> >>> > >> 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 i= t, >> >>> > >> 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 i= t, >> >>> > >> 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 Goog= le >> >>> > > 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.