isSheaf_yoneda' — Mathlib

English

For every α, the Yoneda presheaf yoneda.obj α is a sheaf for the topology typesGrothendieckTopology.

Русский

Для каждого α предшефенка Yoneda yoneda.obj α является шарфом для топологии typesGrothendieckTopology.

LaTeX

$$$ Presieve.IsSheaf typesGrothendieckTopology (yoneda.obj α) $$$

Lean4

/-- The sheaf condition for `yoneda'`. -/
theorem isSheaf_yoneda' {α : Type u} : Presieve.IsSheaf typesGrothendieckTopology (yoneda.obj α) := fun β _ hs x hx =>
  ⟨fun y => x _ (hs y) PUnit.unit, fun γ f h =>
    funext fun z => by convert congr_fun (hx (𝟙 _) (fun _ => z) (hs <| f z) h rfl) PUnit.unit using 1, fun f hf =>
    funext fun y => by convert congr_fun (hf _ (hs y)) PUnit.unit⟩

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