Re: [HoTT] Semantics of higher inductive types

[Michael Shulman <shu...@sandiego.edu>]

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.
>>
>>
>>
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