Discussion of Homotopy Type Theory and Univalent Foundations
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From: Andrew Swan <wakelin.swan@gmail.com>
Subject: [HoTT] Re: A unifying cartesian cubical type theory
Date: Mon, 18 Feb 2019 06:05:54 -0800 (PST)	[thread overview]

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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
>

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Thread overview: 18+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-02-14 19:04 [HoTT] " Anders Mortberg
2019-02-14 20:06 ` Andrew Pitts
2019-02-15 15:38   ` Anders Mörtberg
2019-02-15  8:16 ` Bas Spitters
2019-02-15 16:32   ` Anders Mörtberg
2019-02-16  0:01     ` Michael Shulman
2019-02-16  0:14       ` Steve Awodey
2019-02-16 12:30         ` streicher
2019-02-16 19:51           ` Thomas Streicher
2019-02-16 22:27             ` Steve Awodey
2019-02-17  9:43               ` Thomas Streicher
2019-02-17 14:14                 ` Licata, Dan
2019-02-16 21:58           ` Richard Williamson
2019-02-17  9:15             ` Thomas Streicher
2019-02-17 13:49               ` Richard Williamson
2019-02-18 14:05 ` Andrew Swan [this message]
2019-02-18 15:31   ` [HoTT] " Anders Mörtberg
2019-06-16 16:04     ` Anders Mörtberg

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