English
If F is a sheaf for J, then the sheafification of G is a sheaf for J.
Русский
Если F является шарфом для J, то шейфайфикация G является шарфом для J.
LaTeX
$$$ \\mathrm{IsSheaf}(J, (G.sheafify J).toPresheaf) $$$
Lean4
theorem sheafify_isSheaf (hF : Presieve.IsSheaf J F) : Presieve.IsSheaf J (G.sheafify J).toPresheaf :=
by
refine (isSeparated _ hF.isSeparated).isSheaf fun U S hS x hx ↦ ?_
let S' := Sieve.bind S fun Y f hf => G.sieveOfSection (x f hf).1
have := fun (V) (i : V ⟶ U) (hi : S' i) => hi
choose W i₁ i₂ hi₂ h₁ h₂ using this
dsimp [-Sieve.bind_apply] at *
let x'' : Presieve.FamilyOfElements F S' := fun V i hi => F.map (i₁ V i hi).op (x _ (hi₂ V i hi))
have H : ∀ s, x''.IsAmalgamation s.1 → x.IsAmalgamation s :=
by
intro s H V i hi
refine Subtype.ext ?_
apply (hF _ (x i hi).2).isSeparatedFor.ext
intro V' i' hi'
have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩
have := H _ hi''
rw [op_comp, F.map_comp] at this
exact this.trans (congr_arg Subtype.val (hx _ _ (hi₂ _ _ hi'') hi (h₂ _ _ hi'')))
have : x''.Compatible := by
intro V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e
rw [← FunctorToTypes.map_comp_apply, ← FunctorToTypes.map_comp_apply]
exact
congr_arg Subtype.val
(hx (g₁ ≫ i₁ _ _ S₁) (g₂ ≫ i₁ _ _ S₂) (hi₂ _ _ S₁) (hi₂ _ _ S₂) (by simp only [Category.assoc, h₂, e]))
obtain ⟨t, ht, ht'⟩ := hF _ (J.bind_covering hS fun V i hi => (x i hi).2) _ this
refine ⟨⟨t, _⟩, H ⟨t, ?_⟩ ht⟩
refine J.superset_covering ?_ (J.bind_covering hS fun V i hi => (x i hi).2)
intro V i hi
dsimp
rw [ht _ hi]
exact h₁ _ _ hi
