ΓObjEquivSections — Mathlib

English

For sheaves of types, the global sections functor is isomorphic to the composition of the sheaf-to-presheaf functor with the sections functor.

Русский

Для шейфов типов глобальные секциицы изоморфны композиции: шейф к префешуф конструируется через секции.

LaTeX

$$$\Gamma J (Type w) \cong (\mathrm{sheafToPresheaf } J (Type w)) \;\circ\; \mathrm{sectionsFunctor} (\mathrm{Opposite} C)$$$

Lean4

/-- Global sections of a sheaf of types correspond to sections of the underlying presheaf. -/
noncomputable def ΓObjEquivSections [HasWeakSheafify J (Type w)] [HasGlobalSectionsFunctor J (Type w)]
    (F : Sheaf J (Type w)) : (Γ J (Type w)).obj F ≃ F.val.sections :=
  (Equiv.trans (by exact (Equiv.funUnique PUnit _).symm) ΓHomEquiv.symm).trans (F.val.sectionsEquivHom PUnit).symm

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