Thanks Dan and Andrew for analyzing our work further!

I find Dan's reformulation of our Kan condition quite illuminating:

A : I → Set has Kan composition iff Π r:I, the map (λ f → f r) : (Π (x : I) → A x) → (A r) is an equivalence

My intuition is that this says that A is fibrant iff for any r : I the type A r can be extended to all of I in a uniform way.

I believe that we can reformulate the Kan condition we had in CCHM as:

A : I → Set has Kan composition iff the map (λ f → f 0) : (Π (x : I) → A x) → (A 0) is an equivalence

In the presence of a meet connection these two formulations are path-equal by moving along "i /\ r" (this is what motivates the use of connections in CCHM).

What our note shows is that this natural generalization of CCHM is closed under all of the cubical type formers and hence form a model of univalent type theory even in the absence of connections. In particular it is not necessary to further require the strict fibers as in AFH/ABCFHL when generalizing CCHM. This is what lets us drop the assumption that the diagonal I -> I x I is a cofibration (what we referred to as "diagonal cofibrations" above) in order to construct univalent fibrant universes.

I haven't yet had time to analyze Andrew's definition, but if it works then I would be very interested in knowing if the Sattler model structure construction works. If I understand Christian's work correctly the construction of the WFS's require very few assumptions and the 2-out-of-3 property relies on the equivalence extension property which follows from the existence of fibrant Glue types (which is in our note).

Anders

On Monday, February 18, 2019 at 9:05:54 AM UTC-5, Andrew Swan wrote:

I decided to have a go at translating the ideas over to lifting problems and model structures. Dan's remarks were quite helpful and possibly some of this is a rephrasing of those ideas.

We have an interval object I, and write d0 and d1 for the endpoint inclusions 1 -> I. We want to ensure in any case that for i = 0,1 di has the enriched/fibred/internal left lifting property against every fibration. That is, for every object B, we want that the maps (1, di) : B -> B x I are trivial cofibrations. Now if the (trivial) (co)fibrations we defined are going to form part of a model structure, we will need that for any map r : B -> I, the map (1, r) : B -> B x I is a weak equivalence. This is because the projection B x I -> B is a weak equivalence by applying 3-for-2 and using that (1, d0) is a trivial cofibration, and then applying 3-for-2 again the other way, it follows that (1, r) is a weak equivalence.

Therefore when we define fibrations, we want to ensure that we do so in a way that guarantees (1, r) : B -> B x I is a weak equivalence. If I has connections, then it would be easier, but they are not present in cartesian cubical sets, so we look for some other way.

One way to do this is to choose the generating trivial cofibrations so that every map (1, r) is a trivial cofibration. For some other arguments to work, we include not just these maps, but close under pushout product with cofibrations. Therefore we take the generating trivial cofibrations to be every map generated as follows: Given a map r : B -> I, and a cofibration m : A -> B, we note that m and (1, r) can both be viewed as maps in the slice category C/B. We construct the pushout product of (1, r) and m in the slice category, and take this to be a generating trivial cofibration. This gives the ABCFHL definition of fibration.

However, this has the disadvantage that as a special case we have made the map I -> I x I a trivial cofibration, so if we want this to be part of a model structure we also need it to be a cofibration. This means we can't take the face lattice to be the (generating) cofibrations.

Therefore we need a way to choose the trivial cofibrations that makes every map (1, r) : B -> B x I a weak equivalence without adding any new cofibrations. We again work in the slice category over B. Since we are now working in the slice category, the terminal object 1, is the identity on B, and we have a cofibrant subobject A of 1, and a map r : 1 -> I. We take the mapping cylinder factorisation of r to get 1 -> T -> I. One can show that the map 1 -> T is a cofibration (assuming endpoint inclusions are disjoint and both cofibrations, and cofibrations are closed under pullback). Hence if we make 1 -> T a trivial cofibration, it won't add any new cofibrations. Moreover making 1 -> T a weak equivalence promises to be a reasonable substitute for making r a weak equivalence, because the map T -> I should also be weak equivalence in any case. Now, as before we also close under pushout product with m, again computed in the slice category over B.

Unfolding the definition of mapping cylinder, we get a concrete description of T. It is the pushout of two copies of I, along the maps d0 : 1 -> I and r : 1 -> I, making a "T" shape where the end of one interval is joined to the other at point r. We can also illustrate what the pushout product with a cofibration looks like, using the boundary inclusion 2 -> I as an example: The codomain is the product T x I and the domain is the subobject consisting of two copies of T on each end of the cylinder together with a line connecting the bases of the Ts. It's a little tricky to show the resulting definition of fibration follows from Anders and Evan's definition, but I think it works, by using their observation that they do have Kan composition in the usual sense for open boxes (pushout products of cofibrations and endpoint inclusions).

It seems reasonable to conjecture then that the Mortberg-Cavallo definition of fibration and trivial fibration form part of a model structure, and moreover we might also conjecture that if we define fibration to be "right lifting property against open box inclusion" and cofibration to be given by the face lattice it does not extend to a model structure on cartesian cubical sets.

Best,
Andrew

On Thursday, 14 February 2019 20:05:07 UTC+1, Anders Mörtberg wrote:
Evan Cavallo and I have worked out a new cartesian cubical type theory
that generalizes the existing work on cubical type theories and models
based on a structural interval:

http://www.cs.cmu.edu/~ecavallo/works/unifying-cartesian.pdf

The main difference from earlier work on similar models is that it
depends neither on diagonal cofibrations nor on connections or
reversals. In the presence of these additional structures, our notion
of fibration coincides with that of the existing cartesian and De
Morgan cubical set models. This work can therefore be seen as a
generalization of the existing models of univalent type theory which
also clarifies the connection between them.

The key idea is to weaken the notion of fibration from the cartesian
Kan operations com^r->s so that they are not strictly the identity
when r=s. Instead we introduce weak cartesian Kan operations that are
only the identity function up to a path when r=s. Semantically this
should correspond to a weaker form of a lifting condition where the
lifting only satisfies some of the eqations up to homotopy. We verify
in the note that this weaker notion of fibration is closed under the
type formers of cubical type theory (nat, Sigma, Pi, Path, Id, Glue,
U) so that we get a model of univalent type theory. We also verify
that the circle works and we don't expect any substantial problems
with extending it to more complicated HITs (like pushouts).

--
Anders and Evan