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

We have an interval object I, and write d0 and d1 for the endpoint inclusi=
ons 1 -> I. We want to ensure in any case that for i =3D 0,1 di has the =
enriched/fibred/internal left lifting property against every fibration. Tha=
t 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 g=
oing 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 bec=
ause 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 th=
at we do so in a way that guarantees (1, r) : B -> B x I is a weak equiv=
alence. If I has connections, then it would be easier, but they are not pre=
sent in cartesian cubical sets, so we look for some other way.

One w= ay to do this is to choose the generating trivial cofibrations so that ever= y 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 cofibrat= ions. 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 c= ategory 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.

One w= ay to do this is to choose the generating trivial cofibrations so that ever= y 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 cofibrat= ions. 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 c= ategory 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 structu=
re we also need it to be a cofibration. This means we can't take the fa=
ce lattice to be the (generating) cofibrations.

Th=
erefore we need a way to choose the trivial cofibrations that makes every m=
ap (1, r) : B -> B x I a weak equivalence without adding any new cofibra=
tions. 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 ma=
pping cylinder factorisation of r to get 1 -> T -> I. One can show th=
at the map 1 -> T is a cofibration (assuming endpoint inclusions are dis=
joint and both cofibrations, and cofibrations are closed under pullback). H=
ence if we make 1 -> T a trivial cofibration, it won't add any new c=
ofibrations. Moreover making 1 -> T a weak equivalence promises to be a =
reasonable substitute for making r a weak equivalence, because the map T -&=
gt; I should also be weak equivalence in any case. Now, as before we also c=
lose under pushout product with m, again computed in the slice category ove=
r 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" sha=
pe 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, usi=
ng the boundary inclusion 2 -> I as an example: The codomain is the prod=
uct 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 T=
s. It's a little tricky to show the resulting definition of fibration f=
ollows from Anders and Evan's definition, but I think it works, by usin=
g their observation that they do have Kan composition in the usual sense fo=
r open boxes (pushout products of cofibrations and endpoint inclusions).

=

On Thursday, 1= 4 February 2019 20:05:07 UTC+1, Anders M=C3=B6rtberg wrote:

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

Best,

Andrew

On Thursday, 1= 4 February 2019 20:05:07 UTC+1, Anders M=C3=B6rtberg wrote:

Evan Cavallo and I have worked out a new cart= esian 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/u= nifying- 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=3Ds. Instead we introduce weak cartesian Kan operations that are

only the identity function up to a path when r=3Ds. 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

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