import Mathlib.CategoryTheory.Idempotents.Biproducts import Mathlib.CategoryTheory.Triangulated.Subcategory import Mathlib.Tactic.Abel namespace OAI namespace LittleFinitistic.OddDouble open CategoryTheory CategoryTheory.Category CategoryTheory.Limits open CategoryTheory.Preadditive CategoryTheory.Pretriangulated open CategoryTheory.Idempotents variable {C : Type*} [Category C] [Preadditive C] [HasShift C ℤ] noncomputable def sourceProjector (T : Triangle C) (u : T.obj₂ ⟶ T.obj₁) := 𝟙 T.obj₁ - T.mor₁ ≫ u noncomputable def targetProjector (T : Triangle C) (u : T.obj₂ ⟶ T.obj₁) := 𝟙 T.obj₂ - u ≫ T.mor₁ variable (T : Triangle C) (u : T.obj₂ ⟶ T.obj₁) variable (hu : T.mor₁ ≫ u ≫ T.mor₁ = T.mor₁) include hu theorem sourceProjector_idem : sourceProjector T u ≫ sourceProjector T u = sourceProjector T u := by unfold sourceProjector simp only [comp_sub,sub_comp,id_comp,comp_id,assoc,reassoc_of% hu] abel theorem targetProjector_idem : targetProjector T u ≫ targetProjector T u = targetProjector T u := by unfold targetProjector simp only [comp_sub,sub_comp,id_comp,comp_id,assoc,hu] abel theorem sourceProjector_comp : sourceProjector T u ≫ T.mor₁ = 0 := by simp only [sourceProjector,sub_comp,id_comp,assoc,hu,sub_self] theorem comp_targetProjector : T.mor₁ ≫ targetProjector T u = 0 := by simp only [targetProjector,comp_sub,comp_id,hu,sub_self] variable [HasZeroObject C] [∀ n : ℤ, (shiftFunctor C n).Additive] [Pretriangulated C] local instance : HasFiniteBiproducts (Karoubi C) := Karoubi.karoubi_hasFiniteBiproducts local instance : HasBinaryBiproducts (Karoubi C) := hasBinaryBiproducts_of_finite_biproducts (Karoubi C) variable (hT : T ∈ distTriang C) include hT omit hu in theorem targetProjector_comp_second : targetProjector T u ≫ T.mor₂ = T.mor₂ := by simp only [targetProjector,sub_comp,id_comp,assoc,comp_distTriang_mor_zero₁₂ T hT, comp_zero,sub_zero] omit hu in theorem third_comp_sourceProjector : T.mor₃ ≫ (sourceProjector T u)⟦(1 : ℤ)⟧' = T.mor₃ := by simp only [sourceProjector,Functor.map_sub,Functor.map_comp,comp_sub, ← assoc,comp_distTriang_mor_zero₃₁ T hT,zero_comp,sub_zero] rw [(shiftFunctor C (1 : ℤ)).map_id,comp_id] /-- Splitting data for the cone of a morphism admitting a generalized inverse. These are the five biproduct equations in the idempotent completion. -/ theorem cone_splitting : ∃ (s : T.obj₁⟦(1 : ℤ)⟧ ⟶ T.obj₃) (t : T.obj₃ ⟶ T.obj₂), (sourceProjector T u)⟦(1 : ℤ)⟧' ≫ s = s ∧ s ≫ T.mor₃ = (sourceProjector T u)⟦(1 : ℤ)⟧' ∧ t ≫ targetProjector T u = t ∧ T.mor₂ ≫ t = targetProjector T u ∧ s ≫ t = 0 ∧ t ≫ T.mor₂ + T.mor₃ ≫ s = 𝟙 T.obj₃ := by let p := sourceProjector T u let q := targetProjector T u have hp : p ≫ p = p := sourceProjector_idem T u hu have hq : q ≫ q = q := targetProjector_idem T u hu have hpf : p ≫ T.mor₁ = 0 := sourceProjector_comp T u hu have hfq : T.mor₁ ≫ q = 0 := comp_targetProjector T u hu have hqg : q ≫ T.mor₂ = T.mor₂ := targetProjector_comp_second T u hT have hhp : T.mor₃ ≫ p⟦(1 : ℤ)⟧' = T.mor₃ := third_comp_sourceProjector T u hT have hp1 : p⟦(1 : ℤ)⟧' ≫ p⟦(1 : ℤ)⟧' = p⟦(1 : ℤ)⟧' := by rw [← Functor.map_comp,hp] obtain ⟨s₀,hs₀⟩ := T.coyoneda_exact₁ hT (p⟦(1 : ℤ)⟧') (by rw [← Functor.map_comp,hpf,Functor.map_zero]) let s := p⟦(1 : ℤ)⟧' ≫ s₀ have hps : p⟦(1 : ℤ)⟧' ≫ s = s := by dsimp [s]; rw [← assoc,hp1] have hsh : s ≫ T.mor₃ = p⟦(1 : ℤ)⟧' := by dsimp [s]; rw [assoc,← hs₀,hp1] have hdelta : (𝟙 T.obj₃ - T.mor₃ ≫ s) ≫ T.mor₃ = 0 := by rw [sub_comp,id_comp,assoc,hsh,hhp,sub_self] obtain ⟨t₀,ht₀⟩ := T.coyoneda_exact₃ hT _ hdelta let t := t₀ ≫ q have htq : t ≫ q = t := by dsimp [t]; rw [assoc,hq] have htg : t ≫ T.mor₂ = 𝟙 T.obj₃ - T.mor₃ ≫ s := by dsimp [t]; rw [assoc,hqg,← ht₀] have hgt : T.mor₂ ≫ t = q := by have hz : (T.mor₂ ≫ t - q) ≫ T.mor₂ = 0 := by rw [sub_comp,assoc,htg,comp_sub,comp_id,← assoc, comp_distTriang_mor_zero₂₃ T hT,zero_comp,hqg,sub_zero,sub_self] obtain ⟨v,hv⟩ := T.coyoneda_exact₂ hT _ hz have he := congrArg (fun z => z ≫ q) hv simp only [sub_comp,assoc,htq,hq,hfq,comp_zero] at he exact sub_eq_zero.mp he have hst : s ≫ t = 0 := by have hz : (s ≫ t) ≫ T.mor₂ = 0 := by rw [assoc,htg,comp_sub,comp_id,← assoc,hsh,hps,sub_self] obtain ⟨v,hv⟩ := T.coyoneda_exact₂ hT _ hz have he := congrArg (fun z => z ≫ q) hv simpa only [assoc,htq,hfq,comp_zero] using he exact ⟨s,t,hps,hsh,htq,hgt,hst,by rw [htg,sub_add_cancel]⟩ noncomputable def formalKernel : Karoubi C where X := T.obj₁⟦(1 : ℤ)⟧ p := (sourceProjector T u)⟦(1 : ℤ)⟧' idem := by rw [← Functor.map_comp,sourceProjector_idem T u hu] noncomputable def formalCokernel : Karoubi C where X := T.obj₂ p := targetProjector T u idem := targetProjector_idem T u hu /-- The cone of a regular morphism is the direct sum of its formal cokernel and the shifted formal kernel in the idempotent completion. -/ theorem regular_cone_iso : Nonempty ((toKaroubi C).obj T.obj₃ ≅ formalCokernel T u hu ⊞ formalKernel T u hu) := by obtain ⟨s,t,hps,hsh,htq,hgt,hst,htotal⟩ := cone_splitting T u hu hT let Q := formalCokernel T u hu let P := formalKernel T u hu let S := (toKaroubi C).obj T.obj₃ let gi : Q ⟶ S := ⟨T.mor₂,by change targetProjector T u ≫ T.mor₂ ≫ 𝟙 _ = T.mor₂ rw [comp_id,targetProjector_comp_second T u hT]⟩ let tr : S ⟶ Q := ⟨t,by change 𝟙 _ ≫ t ≫ targetProjector T u = t rw [id_comp,htq]⟩ let hr : S ⟶ P := ⟨T.mor₃,by change 𝟙 _ ≫ T.mor₃ ≫ (sourceProjector T u)⟦(1 : ℤ)⟧' = T.mor₃ rw [id_comp,third_comp_sourceProjector T u hT]⟩ let si : P ⟶ S := ⟨s,by change (sourceProjector T u)⟦(1 : ℤ)⟧' ≫ s ≫ 𝟙 _ = s rw [comp_id,hps]⟩ let bc : BinaryBicone Q P := { pt := S fst := tr snd := hr inl := gi inr := si inl_fst := by apply Karoubi.hom_ext; exact hgt inl_snd := by apply Karoubi.hom_ext; exact comp_distTriang_mor_zero₂₃ T hT inr_fst := by apply Karoubi.hom_ext; exact hst inr_snd := by apply Karoubi.hom_ext; exact hsh } have hb : bc.IsBilimit := isBinaryBilimitOfTotal bc (by apply Karoubi.hom_ext exact htotal) exact ⟨biprod.uniqueUpToIso Q P hb⟩ omit hu hT noncomputable def formalShift (P : Karoubi C) (n : ℤ) : Karoubi C where X := P.X⟦n⟧ p := P.p⟦n⟧' idem := by rw [← Functor.map_comp,P.idem] /-- The first odd double is the cone of the complementary idempotent. -/ theorem odd_double_one (U : Karoubi C) : ∃ S : C, Nonempty ((toKaroubi C).obj S ≅ U ⊞ formalShift U 1) := by let f := 𝟙 U.X - U.p have hf : f ≫ f = f := by dsimp [f] simp only [comp_sub,sub_comp,id_comp,comp_id,U.idem] abel obtain ⟨S,g,h,hT⟩ := distinguished_cocone_triangle f let T := Triangle.mk f g h have hu : T.mor₁ ≫ f ≫ T.mor₁ = T.mor₁ := by change f ≫ f ≫ f = f; rw [hf,hf] have hq : formalCokernel T f hu = U := by refine Karoubi.ext (P := formalCokernel T f hu) (Q := U) rfl ?_ change (𝟙 U.X - f ≫ f) ≫ 𝟙 U.X = 𝟙 U.X ≫ U.p rw [hf,comp_id,id_comp] dsimp [f] abel have hp : formalKernel T f hu = formalShift U 1 := by refine Karoubi.ext (P := formalKernel T f hu) (Q := formalShift U 1) rfl ?_ change (𝟙 U.X - f ≫ f)⟦(1 : ℤ)⟧' ≫ 𝟙 _ = 𝟙 _ ≫ U.p⟦(1 : ℤ)⟧' rw [comp_id,id_comp,hf] congr 1 dsimp [f] abel refine ⟨S,?_⟩ have hc := regular_cone_iso T f hu hT rw [hq,hp] at hc exact hc noncomputable def formalShiftFunctor (n : ℤ) : Karoubi C ⥤ Karoubi C where obj P := formalShift P n map {P Q} f := ⟨f.f⟦n⟧',by change P.p⟦n⟧' ≫ f.f⟦n⟧' ≫ Q.p⟦n⟧' = f.f⟦n⟧' simp only [← Functor.map_comp,f.comm]⟩ map_id _ := by apply Karoubi.hom_ext; rfl map_comp _ _ := by apply Karoubi.hom_ext; exact Functor.map_comp _ _ _ instance (n : ℤ) : (formalShiftFunctor (C := C) n).Additive where map_add := by intros; apply Karoubi.hom_ext; exact Functor.map_add _ noncomputable def retractProjector {A P : Karoubi C} (i : P ⟶ A) (r : A ⟶ P) (hi : i ≫ r = 𝟙 P) : Karoubi C where X := A.X p := r.f ≫ i.f idem := by have h : i.f ≫ r.f = P.p := congrArg Karoubi.Hom.f hi calc (r.f ≫ i.f) ≫ (r.f ≫ i.f) = (r.f ≫ (i.f ≫ r.f)) ≫ i.f := by simp only [assoc] _ = (r.f ≫ P.p) ≫ i.f := by rw [h] _ = r.f ≫ i.f := by rw [Karoubi.comp_p] noncomputable def retractProjectorIso {A P : Karoubi C} (i : P ⟶ A) (r : A ⟶ P) (hi : i ≫ r = 𝟙 P) : retractProjector i r hi ≅ P := by have h : i.f ≫ r.f = P.p := congrArg Karoubi.Hom.f hi exact { hom := ⟨r.f,by change (r.f ≫ i.f) ≫ r.f ≫ P.p = r.f simp only [assoc,h,Karoubi.comp_p]⟩ inv := ⟨i.f,by change P.p ≫ i.f ≫ (r.f ≫ i.f) = i.f rw [← assoc,Karoubi.p_comp,← assoc,h,Karoubi.p_comp]⟩ hom_inv_id := by apply Karoubi.hom_ext; rfl inv_hom_id := by apply Karoubi.hom_ext; exact h } /-- The second cone realizes the third odd double, with three successive shifts. -/ theorem odd_double_three_nested (U : Karoubi C) : ∃ Z : C, Nonempty ((toKaroubi C).obj Z ≅ U ⊞ formalShift (formalShift (formalShift U 1) 1) 1) := by classical obtain ⟨S,⟨e⟩⟩ := odd_double_one U let A := (toKaroubi C).obj S let F := formalShiftFunctor (C := C) 1 let P := F.obj U let i₀ : U ⟶ A := biprod.inl ≫ e.inv let r₀ : A ⟶ U := e.hom ≫ biprod.fst let i₁ : P ⟶ A := biprod.inr ≫ e.inv let r₁ : A ⟶ P := e.hom ≫ biprod.snd have h₀ : i₀ ≫ r₀ = 𝟙 U := by simp only [i₀,r₀,assoc,Iso.inv_hom_id_assoc,biprod.inl_fst] have h₁ : i₁ ≫ r₁ = 𝟙 P := by change (biprod.inr ≫ e.inv) ≫ e.hom ≫ biprod.snd = 𝟙 (formalShift U 1) rw [assoc, e.inv_hom_id_assoc] exact biprod.inr_snd (X := U) (Y := formalShift U 1) have htotal : r₀ ≫ i₀ + r₁ ≫ i₁ = 𝟙 A := by change (e.hom ≫ biprod.fst) ≫ (biprod.inl ≫ e.inv) + (e.hom ≫ biprod.snd) ≫ (biprod.inr ≫ e.inv) = 𝟙 A calc _ = e.hom ≫ (biprod.fst ≫ biprod.inl + biprod.snd ≫ biprod.inr) ≫ e.inv := by simp only [comp_add,add_comp,assoc] _ = 𝟙 A := by rw [biprod.total,id_comp,e.hom_inv_id] let δ : F.obj A ⟶ A := F.map r₀ ≫ i₁ let γ : A ⟶ F.obj A := r₁ ≫ F.map i₀ have hδγ : δ ≫ γ = F.map (r₀ ≫ i₀) := by dsimp only [δ,γ] rw [assoc,← assoc i₁ r₁,h₁,id_comp,← F.map_comp] have hγδ : γ ≫ δ = r₁ ≫ i₁ := by dsimp only [δ,γ] rw [assoc,← assoc (F.map i₀) (F.map r₀),← F.map_comp,h₀,F.map_id,id_comp] have hreg : δ ≫ γ ≫ δ = δ := by rw [← assoc,hδγ] dsimp only [δ] rw [F.map_comp,assoc,← assoc (F.map i₀) (F.map r₀),← F.map_comp,h₀,F.map_id,id_comp] obtain ⟨Z,g,h,hT⟩ := distinguished_cocone_triangle δ.f let T := Triangle.mk δ.f g h have hu : T.mor₁ ≫ γ.f ≫ T.mor₁ = T.mor₁ := congrArg Karoubi.Hom.f hreg have hq : formalCokernel T γ.f hu = retractProjector i₀ r₀ h₀ := by refine Karoubi.ext (P := formalCokernel T γ.f hu) (Q := retractProjector i₀ r₀ h₀) rfl ?_ change (𝟙 A.X - γ.f ≫ δ.f) ≫ 𝟙 A.X = 𝟙 A.X ≫ (r₀.f ≫ i₀.f) rw [comp_id,id_comp] have he : 𝟙 A - γ ≫ δ = r₀ ≫ i₀ := by rw [hγδ,← htotal]; abel have he' := congrArg (Karoubi.inclusionHom A A) he rw [map_sub] at he' exact he' have h₁' : F.map (F.map i₁) ≫ F.map (F.map r₁) = 𝟙 (F.obj (F.obj P)) := by rw [← F.map_comp,← F.map_comp,h₁,F.map_id,F.map_id] have hp : formalKernel T γ.f hu = retractProjector (F.map (F.map i₁)) (F.map (F.map r₁)) h₁' := by refine Karoubi.ext (P := formalKernel T γ.f hu) (Q := retractProjector (F.map (F.map i₁)) (F.map (F.map r₁)) h₁') rfl ?_ change (𝟙 (F.obj A).X - δ.f ≫ γ.f)⟦(1 : ℤ)⟧' ≫ 𝟙 _ = 𝟙 _ ≫ ((F.map (F.map r₁)).f ≫ (F.map (F.map i₁)).f) rw [comp_id,id_comp] have he : 𝟙 (F.obj A) - δ ≫ γ = F.map (r₁ ≫ i₁) := by rw [hδγ,← F.map_id,← F.map_sub] congr 1 rw [← htotal] abel have hA : (F.obj A).p = 𝟙 (F.obj A).X := (shiftFunctor C (1 : ℤ)).map_id S have he' := congrArg (Karoubi.inclusionHom (F.obj A) (F.obj A)) he rw [map_sub] at he' change (F.obj A).p - δ.f ≫ γ.f = (F.map (r₁ ≫ i₁)).f at he' rw [hA] at he' rw [he'] change (shiftFunctor C (1 : ℤ)).map ((shiftFunctor C (1 : ℤ)).map (r₁.f ≫ i₁.f)) = _ simp only [Functor.map_comp] rfl obtain ⟨econe⟩ := regular_cone_iso T γ.f hu hT rw [hq,hp] at econe refine ⟨Z,⟨econe ≪≫ biprod.mapIso (retractProjectorIso i₀ r₀ h₀) (retractProjectorIso (F.map (F.map i₁)) (F.map (F.map r₁)) h₁')⟩⟩ noncomputable def formalIso {P Q : Karoubi C} (e : P.X ≅ Q.X) (he : P.p ≫ e.hom = e.hom ≫ Q.p) : P ≅ Q := by have hi : Q.p ≫ e.inv = e.inv ≫ P.p := by apply (cancel_epi e.hom).mp rw [← assoc,← he,assoc,e.hom_inv_id,comp_id,e.hom_inv_id_assoc] exact { hom := ⟨P.p ≫ e.hom,by simp only [assoc,reassoc_of% he,he,Q.idem]⟩ inv := ⟨Q.p ≫ e.inv,by simp only [assoc,reassoc_of% hi,hi,P.idem]⟩ hom_inv_id := by apply Karoubi.hom_ext change (P.p ≫ e.hom) ≫ (Q.p ≫ e.inv) = P.p rw [assoc,hi,← assoc e.hom e.inv,e.hom_inv_id,id_comp,P.idem] inv_hom_id := by apply Karoubi.hom_ext change (Q.p ≫ e.inv) ≫ (P.p ≫ e.hom) = Q.p rw [assoc,he,← assoc e.inv e.hom,e.inv_hom_id,id_comp,Q.idem] } noncomputable def formalShiftAdd (P : Karoubi C) (n m : ℤ) : formalShift (formalShift P n) m ≅ formalShift P (n+m) := formalIso ((shiftFunctorAdd C n m).app P.X).symm ((shiftFunctorAdd C n m).inv.naturality P.p) /-- Both odd doubles in the manuscript are represented by actual objects. -/ theorem odd_double_three (U : Karoubi C) : ∃ Z : C, Nonempty ((toKaroubi C).obj Z ≅ U ⊞ formalShift U 3) := by obtain ⟨Z,⟨e⟩⟩ := odd_double_three_nested U exact ⟨Z,⟨e ≪≫ biprod.mapIso (Iso.refl U) (formalShiftAdd (formalShift U 1) 1 1 ≪≫ formalShiftAdd U 1 2)⟩⟩ end LittleFinitistic.OddDouble end OAI