English
Naturality of eval with respect to a morphism f between presheaves S1 and S2.
Русский
Единство вычисления eval относительно морфизма f между пресшефами S1 и S2.
LaTeX
$$$ eval_{S_2}(f, α, s) = eval_{S_1}(s) $ с учётом натуральности $f$$$
Lean4
/-- Given a sheaf `S` on the category of types, construct a map
`(α → S(*)) → S(α)` that is inverse to `eval`. -/
noncomputable def typesGlue (S : Type uᵒᵖ ⥤ Type u) (hs : IsSheaf typesGrothendieckTopology S) (α : Type u)
(f : α → S.obj (op PUnit)) : S.obj (op α) :=
(hs.isSheafFor _ _ (generate_discretePresieve_mem α)).amalgamate
(fun _ g hg => S.map (↾fun _ => PUnit.unit).op <| f <| g <| Classical.choose hg) fun β γ δ g₁ g₂ f₁ f₂ hf₁ hf₂ h =>
(hs.isSheafFor _ _ (generate_discretePresieve_mem δ)).isSeparatedFor.ext fun ε g ⟨x, _⟩ =>
by
have : f₁ (Classical.choose hf₁) = f₂ (Classical.choose hf₂) :=
Classical.choose_spec hf₁ (g₁ <| g x) ▸ Classical.choose_spec hf₂ (g₂ <| g x) ▸ congr_fun h _
simp_rw [← FunctorToTypes.map_comp_apply, this, ← op_comp]
rfl
