isConstant_iff_mem_essImage — Mathlib

English

If an isomorphism exists between F and a constant sheaf at some X, then F is constant.

Русский

Если существует изоморфизм между F и константным шейфом в некотором X, тогда F константен.

LaTeX

$$$\\text{isConstant\_of\_iso}: F \\cong (\\text{constantSheaf } J D).obj X \\Rightarrow \\text{IsConstant } J F$$$

Lean4

theorem isConstant_iff_mem_essImage {L : D ⥤ Sheaf J D} {T : C} (hT : IsTerminal T)
    (adj : L ⊣ (sheafSections J D).obj ⟨T⟩) (F : Sheaf J D) : IsConstant J F ↔ L.essImage F :=
  by
  rw [essImage_eq_of_natIso (adj.leftAdjointUniq (constantSheafAdj J D hT))]
  exact ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩

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