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February 27, 2022 13:00
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| import tactic | |
| variables (P : Type) [partial_order P] | |
| def presheaf : Type := | |
| { s : P → Prop // ∀ a b, a ≤ b → s b → s a } | |
| instance : has_coe_to_fun (presheaf P) (λ _, P → Prop) := | |
| ⟨subtype.val⟩ | |
| @[simp] lemma presheaf.coe_mk (s : P → Prop) (hs : ∀ a b, a ≤ b → s b → s a) : | |
| @coe_fn (presheaf P) _ _ (⟨s, hs⟩ : presheaf P) = s := rfl | |
| instance : partial_order (presheaf P) := | |
| { le := λ A B, ∀ x, A x → B x, | |
| le_trans := λ A B C hAB hBC x hAx, hBC _ (hAB _ hAx), | |
| le_refl := λ A x, id, | |
| le_antisymm := λ A B hAB hBA, subtype.val_injective (funext $ λ x, propext ⟨hAB _, hBA _⟩) } | |
| lemma presheaf.le_def {A B : presheaf P} : A ≤ B = ∀ x, A x → B x := rfl | |
| instance : has_Inf (presheaf P) := | |
| { Inf := λ s, ⟨λ p, ∀ A : presheaf P, A ∈ s → A p, | |
| λ a b hab h A hAs, A.2 _ _ hab (h _ hAs)⟩ } | |
| instance : complete_lattice (presheaf P) := | |
| complete_lattice_of_Inf _ | |
| (λ s, begin | |
| split, | |
| { dsimp [Inf, lower_bounds], | |
| intros A hAs p h, | |
| apply h, | |
| exact hAs }, | |
| { dsimp [Inf, upper_bounds, lower_bounds], | |
| intros A h p hAp B hBs, | |
| apply h, | |
| exact hBs, | |
| exact hAp } | |
| end) | |
| lemma infi_def {ι : Sort*} (A : ι → presheaf P) : | |
| infi A = ⟨λ p, ∀ i, A i p, λ x y hxy h i, (A i).2 _ y hxy (h i)⟩ := | |
| le_antisymm | |
| (infi_le_iff.2 (λ B h p hBp i, h i _ hBp)) | |
| (le_infi (λ i p h, h _)) | |
| variable {P} | |
| def yoneda (a : P) : presheaf P := | |
| ⟨λ b, b ≤ a, λ b c, le_trans⟩ | |
| def yoneda_le_iff (A : presheaf P) (p : P) : yoneda p ≤ A ↔ A p := | |
| begin | |
| simp [yoneda, presheaf.le_def], | |
| split, | |
| { intro h, apply h, exact le_rfl }, | |
| { intros h x hxp, | |
| apply A.2, | |
| apply hxp, | |
| exact h } | |
| end | |
| @[simp] lemma yoneda_mono {a b : P} : yoneda a ≤ yoneda b ↔ a ≤ b := | |
| begin | |
| rw yoneda_le_iff, refl, | |
| end | |
| lemma eq_supr (A : presheaf P) : A = ⨆ (p : P) (h : A p), yoneda p := | |
| begin | |
| apply le_antisymm; simp only [le_supr_iff, supr_le_iff, yoneda_le_iff], | |
| { intros B hB p, | |
| exact hB p }, | |
| { exact λ _, id } | |
| end | |
| variables {A : Type*} [complete_lattice A] (f : P → A) (hf : monotone f) | |
| include hf | |
| def ump : presheaf P → A := | |
| λ A, ⨆ (p : P) (h : A p), f p | |
| lemma ump_supr {ι : Sort*} (a : ι → presheaf P) : ump f hf (supr a) = ⨆ i, ump f hf (a i) := | |
| begin | |
| apply le_antisymm, | |
| simp only [ump, le_supr_iff, supr_le_iff], | |
| { intros x hx y hy, | |
| } | |
| end |
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