Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Ulrik Buchholtz <ulrikbuchholtz@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Re: Foundational question about a large set of small sets
Date: Sat, 27 Feb 2021 15:00:13 -0800
Message-ID: <CAOvivQzawyMqT4uZ7+KbB_PftqbMzB4V8HUvgpJPSoe32p62aw@mail.gmail.com> (raw)
In-Reply-To: <c10d88b0-8856-4a35-b71d-c2f9d36b5f7an@googlegroups.com>

On Sat, Feb 27, 2021 at 2:43 AM Ulrik Buchholtz
<ulrikbuchholtz@gmail.com> wrote:
> Unfortunately, I don't know of a model where there's no set cover of Set U. The counter-model at the nLab for the general statement that sets cover groupoids (https://ncatlab.org/nlab/show/n-types+cover#InModels), using presheaves on the category-join B²ℤ * 1, doesn't work for this purpose, AFAICT. (Using a general 2-group G and presheaves on BG * 1 won't help either, I think.)

I believe a counter-model to "there is a specified set that covers
Set" is presheaves on the 2-category X with two objects a and b,
X(b,a)=0, X(a,a)=1, and X(b,b) = X(a,b) = Bℤ.  One can then get a
counter-model to "there merely exists a set that covers Set" with
presheaves on X * 1.  I worked this out a while ago but never got
around to writing it up; it uses the fact that since these are
"inverse EI categories", there is an explicit description of the
universe (http://arxiv.org/abs/1508.02410).

Mike

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  reply	other threads:[~2021-02-27 23:00 UTC|newest]

Thread overview: 7+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2021-02-26 20:33 [HoTT] " Martin Escardo
2021-02-27 10:43 ` [HoTT] " Ulrik Buchholtz
2021-02-27 23:00   ` Michael Shulman [this message]
2021-02-28 12:45     ` Ulrik Buchholtz
2021-02-28 14:01       ` Ulrik Buchholtz
2021-02-28 15:13       ` Michael Shulman
2021-03-01  7:52         ` Martin Escardo

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Discussion of Homotopy Type Theory and Univalent Foundations

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