From: Michael Shulman <firstname.lastname@example.org> To: Ulrik Buchholtz <email@example.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: <firstname.lastname@example.org> On Sat, Feb 27, 2021 at 2:43 AM Ulrik Buchholtz <email@example.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 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheoryfirstname.lastname@example.org. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQzawyMqT4uZ7%2BKbB_PftqbMzB4V8HUvgpJPSoe32p62aw%40mail.gmail.com.
next prev parent 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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