File: Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean

full_name: CategoryTheory.regularTopology.isLocallySurjective_iff
code: lemma regularTopology.isLocallySurjective_iff [Preregular C] {F G : Cᵒᵖ ⥤ D} (f : F ⟶ G) :
    Presheaf.IsLocallySurjective (regularTopology C) f ↔
      ∀ (X : C) (y : G.obj ⟨X⟩), (∃ (X' : C) (φ : X' ⟶ X) (_ : EffectiveEpi φ) (x : F.obj ⟨X'⟩),
        f.app ⟨X'⟩ x = G.map ⟨φ⟩ y) 
goal: CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.regularTopology C) f ↔
  ∀ (X : C) (y : (CategoryTheory.forget D).obj (G.obj (Opposite.op X))),
    ∃ X' φ, ∃ (_ : CategoryTheory.EffectiveEpi φ), ∃ x, (f.app (Opposite.op X')) x = (G.map (Opposite.op φ)) y

full_name: CategoryTheory.extensiveTopology.surjective_of_isLocallySurjective_sheafOfTypes
code: lemma extensiveTopology.surjective_of_isLocallySurjective_sheafOfTypes [FinitaryPreExtensive C]
    {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
      (h : Presheaf.IsLocallySurjective (extensiveTopology C) f) {X : C} :
        Function.Surjective (f.app (op X)) 
goal: Function.Surjective (f.app (Opposite.op X))

full_name: CategoryTheory.extensiveTopology.presheafIsLocallySurjective_iff
code: lemma extensiveTopology.presheafIsLocallySurjective_iff [FinitaryPreExtensive C] {F G : Cᵒᵖ ⥤ D}
    (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
      [PreservesFiniteProducts (forget D)] : Presheaf.IsLocallySurjective (extensiveTopology C) f ↔
        ∀ (X : C), Function.Surjective (f.app (op X)) 
goal: CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.extensiveTopology C) f ↔
  ∀ (X : C), Function.Surjective ⇑(f.app (Opposite.op X))

full_name: CategoryTheory.extensiveTopology.isLocallySurjective_iff
code: lemma extensiveTopology.isLocallySurjective_iff [FinitaryExtensive C]
    {F G : Sheaf (extensiveTopology C) D} (f : F ⟶ G)
      [PreservesFiniteProducts (forget D)] : IsLocallySurjective f ↔
        ∀ (X : C), Function.Surjective (f.val.app (op X)) 
goal: CategoryTheory.Sheaf.IsLocallySurjective f ↔ ∀ (X : C), Function.Surjective ⇑(f.val.app (Opposite.op X))

full_name: CategoryTheory.regularTopology.isLocallySurjective_sheafOfTypes
code: lemma regularTopology.isLocallySurjective_sheafOfTypes [Preregular C] [FinitaryPreExtensive C]
    {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
      (h : Presheaf.IsLocallySurjective (coherentTopology C) f) :
        Presheaf.IsLocallySurjective (regularTopology C) f 
goal: CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.regularTopology C) f

full_name: CategoryTheory.coherentTopology.presheafIsLocallySurjective_iff
code: lemma coherentTopology.presheafIsLocallySurjective_iff {F G : Cᵒᵖ ⥤ D} (f : F ⟶ G)
    [Preregular C] [FinitaryPreExtensive C] [PreservesFiniteProducts F] [PreservesFiniteProducts G]
      [PreservesFiniteProducts (forget D)] :
        Presheaf.IsLocallySurjective (coherentTopology C) f ↔
          Presheaf.IsLocallySurjective (regularTopology C) f 
goal: CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.coherentTopology C) f ↔
  CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.regularTopology C) f

full_name: CategoryTheory.coherentTopology.isLocallySurjective_iff
code: lemma coherentTopology.isLocallySurjective_iff [Preregular C] [FinitaryExtensive C]
    {F G : Sheaf (coherentTopology C) D} (f : F ⟶ G) [PreservesFiniteProducts (forget D)] :
      IsLocallySurjective f ↔ Presheaf.IsLocallySurjective (regularTopology C) f.val 
goal: CategoryTheory.Sheaf.IsLocallySurjective f ↔
  CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.regularTopology C) f.val