Re: [HoTT] Re: Foundational question about a large set of small sets

[Michael Shulman <shulman@sandiego.edu>]

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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