exists_hom_from_galois_of_fiber_nonempty

English

Every connected object X admits a morphism from a Galois object.

Русский

Каждый связный объект X допускает морфизм from Галуа-объекта.

LaTeX

$$$\exists A\;\exists f:\, A \to X,\ IsGalois A$$$

Lean4

/-- Any object with non-empty fiber admits a hom from a Galois object. -/
theorem exists_hom_from_galois_of_fiber_nonempty (X : C) (h : Nonempty (F.obj X)) : ∃ (A : C) (_ : A ⟶ X), IsGalois A :=
  by
  obtain ⟨x⟩ := h
  obtain ⟨A, f, a, h1, _⟩ := exists_hom_from_galois_of_fiber F X x
  exact ⟨A, f, h1⟩

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