From: Michael Shulman <shu...@sandiego.edu>
To: "HomotopyT...@googlegroups.com" <homotopyt...@googlegroups.com>
Subject: cubical type theory with UIP
Date: Sun, 23 Jul 2017 15:54:16 -0700 [thread overview]
Message-ID: <CAOvivQyFLkhoGhFLVSA9uSsitXJszOXouxDih2Ph0e-1HLNxsw@mail.gmail.com> (raw)
I am wondering about versions of cubical type theory with UIP. The
motivation would be to have a type theory with canonicity for
1-categorical semantics that can prove both function extensionality
and propositional univalence. (I am aware of Observational Type
Theory, which I believe has UIP and proves function extensionality,
but I don't think it proves propositional univalence -- although I
would be happy to be wrong about that.)
Presumably we obtain a cubical type theory that's compatible with
axiomatic UIP if in CCHM cubical type theory we postulate only a
single universe of propositions. But I wonder about some possible
refinements, such as:
1. In this case do we still need *all* the Kan composition and gluing
operations? If all types are hsets then it seems like it ought to be
unnecessary to have these operations at all higher dimensions.
2. Can it be enhanced to make UIP provable, such as by adding a
computing K eliminator?
Mike
next reply other threads:[~2017-07-23 22:54 UTC|newest]
Thread overview: 20+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-07-23 22:54 Michael Shulman [this message]
2017-07-29 1:47 ` Matt Oliveri
2017-07-29 2:25 ` [HoTT] " Jon Sterling
2017-07-29 7:29 ` Matt Oliveri
2017-07-29 6:19 ` Michael Shulman
2017-07-29 7:23 ` Matt Oliveri
2017-07-29 8:07 ` Michael Shulman
2017-07-29 10:19 ` Matt Oliveri
2017-07-29 11:08 ` Matt Oliveri
2017-07-29 21:19 ` Michael Shulman
[not found] ` <8f052071-09e0-74db-13dc-7f76bc71e374@cs.bham.ac.uk>
2017-07-31 3:49 ` Matt Oliveri
2017-07-31 15:50 ` Michael Shulman
2017-07-31 17:36 ` Matt Oliveri
2017-08-01 9:14 ` Neelakantan Krishnaswami
2017-08-01 9:20 ` Michael Shulman
2017-08-01 9:34 ` James Cheney
2017-08-01 16:26 ` Michael Shulman
2017-08-01 21:27 ` Matt Oliveri
2017-07-31 4:19 ` Matt Oliveri
2017-08-02 9:40 ` [HoTT] " Andrea Vezzosi
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