ΓObjEquivSections_naturality — Mathlib

English

The preceding isomorphism is explicit; it assigns to each F the isomorphism with the composed functor.

Русский

Указанный изоморфизмExplicit сопоставляет каждому шейфу F соответствие, дающее изоморфизм.

LaTeX

$$$\text{explicit isomorphism }\Gamma J (Type w) \cong (\mathrm{sheafToPresheaf } J (Type w)) \circ \mathrm{sectionsFunctor}$$$

Lean4

theorem ΓObjEquivSections_naturality [HasWeakSheafify J (Type w)] [HasGlobalSectionsFunctor J (Type w)]
    {F G : Sheaf J (Type w)} (f : F ⟶ G) (x : (Γ J _).obj F) :
    (ΓObjEquivSections J G) ((Γ J _).map f x) = (Functor.sectionsFunctor _).map f.val ((ΓObjEquivSections J F) x) :=
  by
  dsimp [ΓObjEquivSections]
  exact (congr_arg _ (ΓHomEquiv_naturality_right_symm _ _)).trans (Functor.sectionsEquivHom_naturality_symm _ _ _)

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