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import Mathlib.CategoryTheory.Sites.Sieves import Mathlib.CategoryTheory.EffectiveEpi.Basic namespace CategoryTheory open Limits variable {C : Type*} [Category C] def Sieve.EffectiveEpimorphic {X : C} (S : Sieve X) : Prop := Nonempty (IsColimit (S : Presieve X).cocone) abbrev Presieve.EffectiveEpimorphic {X : C} (S : Presieve X) : Prop := (Sieve.generate S).EffectiveEpimorphic def Sieve.generateSingleton {X Y : C} (f : Y ⟶ X) : Sieve X where arrows Z := { g | ∃ (e : Z ⟶ Y), e ≫ f = g } downward_closed := by rintro W Z g ⟨e,rfl⟩ q exact ⟨q ≫ e, by simp⟩ lemma Sieve.generateSingleton_eq {X Y : C} (f : Y ⟶ X) : Sieve.generate (Presieve.singleton f) = Sieve.generateSingleton f := by ext Z g constructor · rintro ⟨W,i,p,⟨⟩,rfl⟩ exact ⟨i,rfl⟩ · rintro ⟨g,h⟩ exact ⟨Y,g,f,⟨⟩,h⟩ def isColimitOfEffectiveEpiStruct {X Y : C} (f : Y ⟶ X) (Hf : EffectiveEpiStruct f) : IsColimit (Sieve.generateSingleton f : Presieve X).cocone := letI D := FullSubcategory fun T : Over X => Sieve.generateSingleton f T.hom letI F : D ⥤ _ := (Sieve.generateSingleton f).arrows.diagram { desc := fun S => Hf.desc (S.ι.app ⟨Over.mk f, ⟨𝟙 _, by simp⟩⟩) <| by intro Z g₁ g₂ h let Y' : D := ⟨Over.mk f, 𝟙 _, by simp⟩ let Z' : D := ⟨Over.mk (g₁ ≫ f), g₁, rfl⟩ let g₁' : Z' ⟶ Y' := Over.homMk g₁ let g₂' : Z' ⟶ Y' := Over.homMk g₂ (by simp [h]) change F.map g₁' ≫ _ = F.map g₂' ≫ _ simp only [S.w] fac := by rintro S ⟨T,g,hT⟩ dsimp nth_rewrite 1 [← hT, Category.assoc, Hf.fac] let y : D := ⟨Over.mk f, 𝟙 _, by simp⟩ let x : D := ⟨Over.mk T.hom, g, hT⟩ let g' : x ⟶ y := Over.homMk g change F.map g' ≫ _ = _ rw [S.w] rfl uniq := by intro S m hm dsimp generalize_proofs h1 h2 apply Hf.uniq _ h2 exact hm ⟨Over.mk f, 𝟙 _, by simp⟩ } noncomputable def effectiveEpiStructOfIsColimit {X Y : C} (f : Y ⟶ X) (Hf : IsColimit (Sieve.generateSingleton f : Presieve X).cocone) : EffectiveEpiStruct f := let aux {W : C} (e : Y ⟶ W) (h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) : Cocone (Sieve.generateSingleton f).arrows.diagram := { pt := W ι := { app := fun ⟨T,hT⟩ => hT.choose ≫ e naturality := by rintro ⟨A,hA⟩ ⟨B,hB⟩ (q : A ⟶ B) dsimp; simp only [← Category.assoc, Category.comp_id] apply h rw [Category.assoc, hB.choose_spec, hA.choose_spec, Over.w] } } { desc := fun {W} e h => Hf.desc (aux e h) fac := by intro W e h dsimp have := Hf.fac (aux e h) ⟨Over.mk f, 𝟙 _, by simp⟩ dsimp at this; rw [this]; clear this nth_rewrite 2 [← Category.id_comp e] apply h generalize_proofs hh rw [hh.choose_spec, Category.id_comp] uniq := by intro W e h m hm dsimp apply Hf.uniq (aux e h) rintro ⟨A,g,hA⟩ dsimp nth_rewrite 1 [← hA, Category.assoc, hm] apply h generalize_proofs hh rwa [hh.choose_spec] }
Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean
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theorem Sieve.effectiveEpimorphic_singleton {X Y : C} (f : Y ⟶ X) : (Presieve.singleton f).EffectiveEpimorphic ↔ (EffectiveEpi f) := by
constructor · intro (h : Nonempty _) rw [Sieve.generateSingleton_eq] at h constructor apply Nonempty.map (effectiveEpiStructOfIsColimit _) h · rintro ⟨h⟩ show Nonempty _ rw [Sieve.generateSingleton_eq] apply Nonempty.map (isColimitOfEffectiveEpiStruct _) h
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