Discussion of Homotopy Type Theory and Univalent Foundations
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From: Anders Mortberg <andersmortberg@gmail.com>
To: Homotopy Type Theory <homotopytypetheory@googlegroups.com>
Subject: [HoTT] A unifying cartesian cubical type theory
Date: Thu, 14 Feb 2019 14:04:54 -0500	[thread overview]
Message-ID: <CAMWCppkw1yGyey0rDGnUawyiVN7TQ2cL6GKNfSXV__zuJvvONA@mail.gmail.com> (raw)

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:


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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             reply	other threads:[~2019-02-14 19:05 UTC|newest]

Thread overview: 18+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-02-14 19:04 Anders Mortberg [this message]
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 ` [HoTT] " Andrew Swan
2019-02-18 15:31   ` Anders Mörtberg
2019-06-16 16:04     ` Anders Mörtberg

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