On 7 Jun 2017, at 11:40, Peter LeFanu Lumsdaine <p.l.lu...@gmail.com> wrote:
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.)