| import Mathlib.CategoryTheory.Abelian.Projective.Resolution |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace ArExplicit.ResolutionProducts |
| universe u v |
| variable {C : Type u} [Category.{v} C] [Abelian C] |
| variable {X Y Z : C} |
| variable (P : ProjectiveResolution X) (Q : ProjectiveResolution Y) |
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|
| omit [Abelian C] in |
| private theorem compose_square {A B D X Y Z : C} |
| (p : A ⟶ X) (q : B ⟶ Y) (r : D ⟶ Z) |
| (f : A ⟶ B) (g : B ⟶ D) (a : X ⟶ Y) (b : Y ⟶ Z) (c : X ⟶ Z) |
| (hp : p ≫ a = f ≫ q) (hq : q ≫ b = g ≫ r) (hc : p ≫ c = (f ≫ g) ≫ r) : |
| p ≫ c = p ≫ a ≫ b := by |
| calc |
| p ≫ c = (f ≫ g) ≫ r := hc |
| _ = f ≫ g ≫ r := Category.assoc _ _ _ |
| _ = f ≫ q ≫ b := congrArg (fun z => f ≫ z) hq.symm |
| _ = (f ≫ q) ≫ b := (Category.assoc _ _ _).symm |
| _ = (p ≫ a) ≫ b := congrArg (fun z => z ≫ b) hp.symm |
| _ = p ≫ a ≫ b := Category.assoc _ _ _ |
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| def resolutionEndpointMap (f : P.complex ⟶ Q.complex) : X ⟶ Y := |
| P.isColimitCokernelCofork.desc (CokernelCofork.ofπ (f.f 0 ≫ Q.π.f 0) (by |
| rw [← Category.assoc, ← f.comm 1 0, Category.assoc, |
| Q.complex_d_comp_π_f_zero, comp_zero])) |
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| @[reassoc] theorem resolutionEndpointMap_augmentation (f : P.complex ⟶ Q.complex) : |
| P.π.f 0 ≫ resolutionEndpointMap P Q f = f.f 0 ≫ Q.π.f 0 := |
| Cofork.IsColimit.π_desc P.isColimitCokernelCofork |
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| theorem resolutionEndpointMap_id : resolutionEndpointMap P P (𝟙 _) = 𝟙 X := by |
| apply (cancel_epi (P.π.f 0)).1 |
| erw [resolutionEndpointMap_augmentation] |
| exact (Category.id_comp _).trans (Category.comp_id _).symm |
|
|
| theorem resolutionEndpointMap_comp (T : ProjectiveResolution Z) |
| (f : P.complex ⟶ Q.complex) (g : Q.complex ⟶ T.complex) : |
| resolutionEndpointMap P T (f ≫ g) = resolutionEndpointMap P Q f ≫ resolutionEndpointMap Q T g := by |
| apply (cancel_epi (P.π.f 0)).1 |
| exact compose_square (show P.complex.X 0 ⟶ X from P.π.f 0) |
| (show Q.complex.X 0 ⟶ Y from Q.π.f 0) (show T.complex.X 0 ⟶ Z from T.π.f 0) |
| (f.f 0) (g.f 0) _ _ _ (resolutionEndpointMap_augmentation P Q f) |
| (resolutionEndpointMap_augmentation Q T g) (resolutionEndpointMap_augmentation P T (f ≫ g)) |
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| def resolutionEndpointIso (e : P.complex ≅ Q.complex) : X ≅ Y where |
| hom := resolutionEndpointMap P Q e.hom |
| inv := resolutionEndpointMap Q P e.inv |
| hom_inv_id := by erw [← resolutionEndpointMap_comp, e.hom_inv_id, resolutionEndpointMap_id] |
| inv_hom_id := by erw [← resolutionEndpointMap_comp, e.inv_hom_id, resolutionEndpointMap_id] |
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| @[reassoc] theorem resolutionEndpointIso_augmentation (e : P.complex ≅ Q.complex) : |
| P.π.f 0 ≫ (resolutionEndpointIso P Q e).hom = e.hom.f 0 ≫ Q.π.f 0 := |
| resolutionEndpointMap_augmentation P Q e.hom |
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| theorem resolutionEndpointIso_hom_unique (e : P.complex ≅ Q.complex) (a : X ⟶ Y) |
| (ha : P.π.f 0 ≫ a = e.hom.f 0 ≫ Q.π.f 0) : |
| (resolutionEndpointIso P Q e).hom = a := by |
| apply (cancel_epi (P.π.f 0)).1 |
| erw [resolutionEndpointIso_augmentation, ha] |
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| end ArExplicit.ResolutionProducts |
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| end |
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| end OAI |
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