From: Michael Shulman <shu...@sandiego.edu> To: Andrew Swan <wakeli...@gmail.com> Cc: Homotopy Type Theory <HomotopyT...@googlegroups.com>, Steve Awodey <awo...@cmu.edu>, Thierry Coquand <Thierry...@cse.gu.se> Subject: Re: [HoTT] Semantics of higher inductive types Date: Wed, 7 Jun 2017 17:06:39 -0600 [thread overview] Message-ID: <CAOvivQwJ_Dd=HCVmsYL2sPtHCv4USrwHTBzP6E50aX-iuruoFg@mail.gmail.com> (raw) In-Reply-To: <ed7ad345-85e4-4536-86d7-a57fbe3313fe@googlegroups.com> On Wed, Jun 7, 2017 at 12:34 PM, Andrew Swan <wakeli...@gmail.com> wrote: > The technique used in cubical type theory seems fairly flexible. I'm not > sure exactly how flexible, but I think I can get suspension to work in > simplicial sets. In the below, throughout I use the characterisation of > fibrations as maps with the rlp against the pushout product of each > monomorphism with endpoint inclusion into the interval. WLOG there is also a > uniformity condition - we have a choice of lift and "pulling back the > monomorphism preserves the lift." Why is there no LOG in this? > Given a fibration X -> Y, you first freely add elements N and S together > with a path from N to S for each element of X (I think this is the same as > what you called pre suspension). Although the pre suspension is not a > fibration in general, it does have some of the properties you might expect > from a fibration. Given a path in Y, and an element in the fibre of an > endpoint, one can transport along the path to get something in the fibre of > the other endpoint. There should also be a "flattening" operation that takes > a path q in presuspension(X) over p in Y, and returns a path from q(1) to > the transport along p of q(0) that lies entirely in the fibre of p(1). > > You then take the "weak fibrant replacement" of the pre suspension. A map > in simplicial sets is a fibration if and only if it has the rlp against each > pushout product of a monomorphism with an endpoint inclusion into the > interval. In fibrant replacement you freely add a diagonal lift for each > such lifting problems. In weak fibrant replacement you only add fillers for > some of these lifting problems. The pushout product of a monomorphism A -> B > with endpoint inclusion always has codomain B x I - then only consider those > lifting problems where the bottom map factors through the projection B x I > -> B. I think there are two ways to ensure that the operation of weak > fibrant replacement is stable under pullback - one way is carry out the > operation "internally" in simplicial sets (viewed as a topos), and the other > to use the algebraic small object argument, ensuring that uniformity > condition above is in the definition. The intuitive reason why this should > be stable is that the problem that stops the usual fibrant replacement from > being stable is that e.g. when we freely add the transport of a point along > a path, p we are adding a new element to the fibre of p(1) which depends on > things outside of that fibre, whereas with weak fibrant replacement we only > add a filler to an open box to a certain fibre if the original open box lies > entirely in that fibre. > > In order to show that the suspension is fibrant one has to use both the > structure already present in pre suspension (transport and flattening) and > the additional structure added by weak fibrant replacement. The idea is to > follow the same proof as for cubical type theory. It is enough to just show > composition and then derive filling. So to define the composition of an open > box, first flatten it, then use the weak fibration structure to find the > composition. (And I think that last part should be an instance of a general > result along the lines of "if the monad of transport and flattening > distributes over a monad, then the fibrant replacement monad distributes > over the coproduct of that monad with weak fibrant replacement"). > > > Best, > Andrew > > > On Wednesday, 7 June 2017 11:40:12 UTC+2, Peter LeFanu Lumsdaine wrote: >> >> On Thu, Jun 1, 2017 at 6:08 PM, Peter LeFanu Lumsdaine >> <p.l....@gmail.com> wrote: >>> >>> 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. >> >> >> I realise I was a bit unclear here: it’s only suspension that I meant to >> suggest is problematic, not propositional truncation. The latter seems a >> bit easier to do by ad hoc constructions; e.g. the construction below does >> it in simplicial sets, and I think a similar thing may work also in cubical >> sets. (I don’t claim originality for this construction; I don’t think I >> learned it from anywhere, but I do recall discussing it with people who were >> already aware of it or something similar (I think at least Mike, Thierry, >> and Simon Huber, at various times?), so I think multiple people may have >> noticed it independently.) >> >> So suspension (or more generally pushouts/coequalisers) is what would make >> a really good test case for any proposed general approach — it’s the >> simplest HIT which as far as I know hasn’t been modelled without a size >> blowup in any infinite-dimensional model except cubical sets, under any of >> the approaches to modelling HIT’s proposed so far. (Am I right in >> remembering that this has been given for cubical sets? I can’t find it in >> any of the writeups, but I seem to recall hearing it presented at >> conferences.) >> >> Construction of propositional truncation without size blowup in simplicial >> sets: >> >> (1) Given a fibration Y —> X, define |Y| —> X as follows: >> >> an element of |Y|_n consists of an n-simplex x : Δ[n] —> X, together with >> a “partial lift of x into Y, defined at least on all vertices”, i.e. a >> subpresheaf S ≤ Δ[n] containing all vertices, and a map y : S —> Y such that >> the evident square commutes; >> >> reindexing acts by taking pullbacks/inverse images of the domain of the >> partial lift (i.e. the usual composition of a partial map with a total map). >> >> (2) There’s an evident map Y —> |Y| over X; and the operation sending Y to >> Y —> |Y| —> X is (coherently) stable up to isomorphism under pullback in X. >> (Straightforward.) >> >> (3) In general, a fibration is a proposition in the type-theoretic sense >> iff it’s orthogonal to the boundary inclusions δ[n] —> Δ[n] for all n > 0. >> (Non-trivial but not too hard to check.) >> >> (4) The map |Y| —> X is a fibration, and a proposition. (Straightforward, >> given (3), by concretely constructing the required liftings.) >> >> (5) The evident map Y —> |Y| over X is a cell complex constructed from >> boundary inclusions δ[n] —> Δ[n] with n > 0. >> >> To see this: take the filtration of |Y| by subobjects Y_n, where the >> non-degenerate simplices of Y_n are those whose “missing” simplices are all >> of dimension ≤n. Then Y_0 = Y, and the non-degenerate simplices of Y_{n+1} >> that are not in Y_n are all {n+1}-cells with boundary in Y_n, so the >> inclusion Y_n —> Y_{n+1} may be seen as gluing on many copies of δ[n+1] —> >> Δ[n+1]. >> >> (6) The map Y —> |Y| is orthogonal to all propositional fibrations, stably >> in X. (Orthogonality is immediate from (3) and (5); stability is then by >> (2).) >> >> (7) Let V be either the universe of “well-ordered α-small fibrations”, or >> the universe of “displayed α-small fibrations”, for α any infinite regular >> cardinal. Then V carries an operation representing the construction of (1), >> and modelling propositional truncation. (Lengthy to spell out in full, but >> straightforward given (2), (6).) >> >> >> –p. >> >> >> > -- > 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.

next prev parent reply other threads:[~2017-06-07 23:07 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 Awodey 2017-06-01 16:08 ` Peter LeFanu Lumsdaine 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 [this message]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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