English
For extensive topology, a morphism of presheaves is locally surjective iff each component is surjective on objects.
Русский
Для экстенсивной топологии морфизм презашифра локально сюръективен тогда и только тогда, когда каждая компонента сюръективна на объектах.
LaTeX
$$$\mathrm{IsLocallySurjective}(\mathrm{extensiveTopology}(C), f) \iff \forall X,\; \text{surjective on } f(X).$$$
Lean4
theorem isLocallySurjective_iff [Preregular C] {F G : Cᵒᵖ ⥤ D} (f : F ⟶ G) :
Presheaf.IsLocallySurjective (regularTopology C) f ↔
∀ (X : C) (y : ToType (G.obj ⟨X⟩)),
(∃ (X' : C) (φ : X' ⟶ X) (_ : EffectiveEpi φ) (x : ToType (F.obj ⟨X'⟩)), f.app ⟨X'⟩ x = G.map ⟨φ⟩ y) :=
by
constructor
· intro ⟨h⟩ X y
specialize h y
rw [regularTopology.mem_sieves_iff_hasEffectiveEpi] at h
obtain ⟨X', π, h, h'⟩ := h
exact ⟨X', π, h, h'⟩
· intro h
refine ⟨fun y ↦ ?_⟩
obtain ⟨X', π, h, h'⟩ := h _ y
rw [regularTopology.mem_sieves_iff_hasEffectiveEpi]
exact ⟨X', π, h, h'⟩
