app_bijective_of_stalkFunctor_map_bijective

English

In a chain of partial refinements, any element is bounded above by the chain supremum.

Русский

Для любой цепи частичных уточнений любой элемент ограничен сверху верхней гранью цепи.

LaTeX

$$$\forall c: Set(PartialRefinement\ u\ s\ p), hc: IsChain(·≤·\, c), ne: c.Nonempty, hfin, hU, v∈c \Rightarrow v ≤ chainSup c hc ne hfin hU$$$

Lean4

theorem app_bijective_of_stalkFunctor_map_bijective {F G : Sheaf C X} (f : F ⟶ G) (U : Opens X)
    (h : ∀ x ∈ U, Function.Bijective ((stalkFunctor C x).map f.1)) : Function.Bijective (f.1.app (op U)) :=
  ⟨app_injective_of_stalkFunctor_map_injective f.1 U fun x hx => (h x hx).1,
    app_surjective_of_stalkFunctor_map_bijective f U h⟩

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