stalk_hom_ext — Mathlib

English

Extensionality of stalk morphisms via germs.

Русский

Экстенсиональность морфизмов стэла через зародки.

LaTeX

$$If F is a presheaf on X and x ∈ X, and f1, f2: F.stalk x ⟶ Y satisfy that F.germ U x hx ≫ f1 = F.germ U x hx ≫ f2 for all opens U containing x, then f1 = f2.$$

Lean4

/-- A morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its
composition with the `germ` morphisms.
-/
@[ext]
theorem stalk_hom_ext (F : X.Presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}
    (ih : ∀ (U : Opens X) (hxU : x ∈ U), F.germ U x hxU ≫ f₁ = F.germ U x hxU ≫ f₂) : f₁ = f₂ :=
  colimit.hom_ext fun U => by
    induction U with
    | op U => obtain ⟨U, hxU⟩ := U; exact ih U hxU

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