```
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: Wed, 7 Jun 2017 11:40:11 +0200 [thread overview]
Message-ID: <CAAkwb-k1GnZKkrEoszNQQ0zyA3HSFh1LXmKVp-+3juRk5s8taQ@mail.gmail.com> (raw)
In-Reply-To: <CAAkwb-nogDyVRwnvDXUphBQb0ttL-TuHfSahnrSMhFnuunc7hw@mail.gmail.com>
[-- Attachment #1: Type: text/plain, Size: 4309 bytes --]
On Thu, Jun 1, 2017 at 6:08 PM, Peter LeFanu Lumsdaine <
p.l.lu...@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.
[-- Attachment #2: Type: text/html, Size: 5150 bytes --]
```

next prev parent reply other threads:[~2017-06-07 9:40 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 Swan2017-06-07 9:40 ` Peter LeFanu Lumsdaine [this message]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

Be sure your reply has aReply instructions:You may reply publicly to this message via plain-text email using any one of the following methods: * Save the following mbox file, import it into your mail client, and reply-to-all from there: mbox Avoid top-posting and favor interleaved quoting: https://en.wikipedia.org/wiki/Posting_style#Interleaved_style * Reply using the--to,--cc, and--in-reply-toswitches of git-send-email(1): git send-email \ --in-reply-to=CAAkwb-k1GnZKkrEoszNQQ0zyA3HSFh1LXmKVp-+3juRk5s8taQ@mail.gmail.com \ --to="p.l.lu..."@gmail.com \ --cc="Thierry..."@cse.gu.se \ --cc="awo..."@cmu.edu \ --cc="homotopyt..."@googlegroups.com \ --cc="shu..."@sandiego.edu \ /path/to/YOUR_REPLY https://kernel.org/pub/software/scm/git/docs/git-send-email.html * If your mail client supports setting theIn-Reply-Toheader via mailto: links, try the mailto: link

This is a public inbox, see mirroring instructions for how to clone and mirror all data and code used for this inbox; as well as URLs for NNTP newsgroup(s).