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.