Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Martin Hotzel Escardo <escardo.martin@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Injective types
Date: Wed, 1 May 2019 09:55:36 -0700
Message-ID: <CAOvivQwkwn1HDttRoOZk5d8fgEwdkO5x3Y7bpP=fFGbYG5s1BA@mail.gmail.com> (raw)
In-Reply-To: <7104f60a-7dba-4574-8629-860c0bc31967@googlegroups.com>

On Tue, Apr 30, 2019 at 11:25 PM <escardo.martin@gmail.com> wrote:
> It is nice, in practice, that universes are uniquely
> determined. Insted of thousands of level constraints, we have a
> unique, small, (polymorphic) level assignment.
> ...
> I think it is fair to the
> reader to let them know what the universe level assignments are rather
> than just say that they exist. In some cases, as injectivity, the
> universe levels are crucial, and if we ignore them we get false
> claims.

Yes, this is a good point in favor of Agda-style non-cumulative
Russell universes over Coq-style cumulative Russell universes.

But isn't there a middle ground with Tarski universes?  Suppose we
have explicit lifting operators Lift : U_i -> U_{i+1}, so that as in
Agda we have unique small polymorphic level assignments.  But then
suppose we have *definitional* equalities El(Lift(A)) == El(A).  Then
on the (rare) occasions when we do have to explicitly lift types from
one universe to another, we would get stronger cumulativity behavior.
(And I could imagine a proof assistant that implements this and sugars
away the El to look like Russell universes to the user.)

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Thread overview: 11+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-04-30 23:05 escardo.martin
2019-05-01  2:50 ` Michael Shulman
2019-05-01  6:25   ` escardo.martin
2019-05-01 16:55     ` Michael Shulman [this message]
2019-05-02 20:46       ` escardo.martin
2019-05-03 11:45         ` Michael Shulman
2019-05-03 13:25           ` Kenji Maillard
2019-05-03 18:23             ` Thierry Coquand
2019-05-07 12:42         ` Nils Anders Danielsson
2019-05-07 13:51           ` Andreas Nuyts
2019-05-07 22:06             ` Martín Hötzel Escardó

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Discussion of Homotopy Type Theory and Univalent Foundations

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