PS. We don't actually need to endoequivalences at level 1. A simpler counter-model is obtained via the 2-level direct category with two objects a,b, and Hom(a,b) = Bℤ. Ulrik On Sunday, February 28, 2021 at 1:45:09 PM UTC+1 Ulrik Buchholtz wrote: > 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 > > -- 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 HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/848b3f3b-045b-46e4-b661-f639abbafff7n%40googlegroups.com.