Discussion of Homotopy Type Theory and Univalent Foundations
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From: Ulrik Buchholtz <ulrikbuchholtz@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Re: Foundational question about a large set of small sets
Date: Sun, 28 Feb 2021 04:45:09 -0800 (PST)	[thread overview]
Message-ID: <b3e72012-1aa4-44f7-b2e6-ab83ebe7b9bfn@googlegroups.com> (raw)
In-Reply-To: <CAOvivQzawyMqT4uZ7+KbB_PftqbMzB4V8HUvgpJPSoe32p62aw@mail.gmail.com>


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On Sunday, February 28, 2021 at 12:00:29 AM UTC+1 Mike wrote:

> 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.


Thanks for the hint! Let me see if I can work it out:
 
We let G = Bℤ, considered as a 2-group with delooping BG = B²ℤ. Write * for 
the basepoint.
Take ∂ : BG → U, ∂ t = (* = t). We get the direct category X by attaching 
to the terminal category 1 a new level of objects BG along the boundary 
operator ∂.
(We could make another direct category using ∂ t = (t = t); this also 
works!)
The object a is the unique object at level 0; write b(t), t : BG, for the 
objects at level 1.

Contexts Γ ⊢ :

Γ₀ : U
Γ₁ : (t : BG) → (∂ t → Γ₀) → U

Representable contexts a and b(t), t : BG :

a₀ = 1
a₀ t u = 0

b(t)₀ = ∂ t = (* = t)
b(t)₁ t' u = (q : t' = t) × (u =_q id)

Evaluation of Γ at a is Γ₀, evaluation at b(t), t : BG, is

Γ(t) = (u : ∂ t → Γ₀) × Γ₁ t u

(So Γ evaluates to sets iff Γ₀ and Γ₁ are Set-valued)

Substitutions σ : Γ → Δ :

σ₀ : Γ₀ → Δ₀
σ₁ : (t : BG) → (u : ∂ t → Γ₀) → Γ₁ t u → Δ₁ t (σ₀ ∘ u)

Action of σ on values at b(t), t : BG :

σ(t) : Γ(t) → Δ(t)
σ(t) (u , γ) = (σ₀ ∘ u , σ₁ t u γ)

Families Γ ⊢ A :

A₀ : Γ₀ → U
A₁ : (t : BG) → (u : ∂ t → Γ₀) → Γ₁ t u → ((x : ∂ t) → A₀ (u x)) → U

Universe of sets, Set ⊢ :

Set₀ = Set
Set₁ t u = ((x : ∂ t) → u x) → Set

hence value over b(t) is:

Set(t) = (u : ∂ t → Set) × (((x : ∂ t) → u x) → Set)

(I think this is correct; I'm skipping the entire calculation of the 
interpretation of (A : U) × is-set A. We really only need the level zero 
component, and that's definitely Set.)

Now assume there is a set context Γ and a levelwise surjection σ : Γ → Set :

σ₀ : Γ₀ → Set  (i.e., a cover of Set at the meta-level)
σ₁ : (t : BG) → (u : ∂ t → Γ₀) → Γ₁ t u → ((x : ∂ t) → σ₀ (u x)) → Set

so we assume that

σ(t) : Γ(t) → (v : ∂ t → Set) × (((x : ∂ t) → v x) → Set)
σ(t) (u , γ) = (σ₀ ∘ u , σ₁ t u γ)

is surjective for each t : BG.

This is a proposition, so it holds iff it holds at the basepoint * : BG.
We have ∂ * = (* = *) = G = Bℤ, so v : Bℤ → Set amounts to a set with an 
automorphism.
However, any u : Bℤ → Γ₀ is constant, since Γ₀ is a set, so a non-trivial v 
cannot be written as σ₀ ∘ u for any such u. Nice!

One can then get a 
> counter-model to "there merely exists a set that covers Set" with 
> presheaves on X * 1.


This I didn't check. But couldn't we here instead appeal to homotopy 
canonicity? Since Set is a closed type, if there is a proof of the mere 
existence of a set cover of Set, we could extract a specific term for one, 
and then obtain a contradiction in the above model.

Cheers,
Ulrik

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  reply	other threads:[~2021-02-28 12:45 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
2021-02-28 12:45     ` Ulrik Buchholtz [this message]
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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