| import OAI.Algebra.Finitistic.Rectification.TotalFunctorial |
| import OAI.Algebra.Finitistic.Rectification.ExactProducts |
| import Mathlib.Algebra.Category.ModuleCat.Biproducts |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex |
| noncomputable section |
| namespace LittleFinitistic.RectificationTotal |
| universe u |
| variable {S : Type u} [Ring S] |
|
|
| /-- The canonical binary sum/product identification respects the supplied maps. -/ |
| theorem prodIso_naturality {X Y X' Y' : ModuleCat.{u} S} (f : X ⟶ X') (g : Y ⟶ Y') : |
| biprod.map f g ≫ (ModuleCat.biprodIsoProd X' Y').hom = |
| (ModuleCat.biprodIsoProd X Y).hom ≫ ModuleCat.ofHom (f.hom.prodMap g.hom) := by |
| apply (cancel_epi (ModuleCat.biprodIsoProd X Y).inv).1 |
| apply (cancel_mono (ModuleCat.biprodIsoProd X' Y').inv).1 |
| simp only [Category.assoc, Iso.hom_inv_id, Category.comp_id, Iso.inv_hom_id_assoc] |
| apply biprod.hom_ext |
| · simp only [Category.assoc, biprod.map_fst, ModuleCat.biprodIsoProd_inv_comp_fst] |
| rw [← Category.assoc, ModuleCat.biprodIsoProd_inv_comp_fst] |
| rfl |
| · simp only [Category.assoc, biprod.map_snd, ModuleCat.biprodIsoProd_inv_comp_snd] |
| rw [← Category.assoc, ModuleCat.biprodIsoProd_inv_comp_snd] |
| rfl |
|
|
| variable {L₀ L₁ L₂ M₀ M₁ M₂ N₀ N₁ N₂ : CochainComplex (ModuleCat.{u} S) ℤ} |
| variable {p : L₁ ⟶ L₀} {q : L₂ ⟶ L₁} {h : Homotopy (q ≫ p) 0} |
| variable {p' : M₁ ⟶ M₀} {q' : M₂ ⟶ M₁} {h' : Homotopy (q' ≫ p') 0} |
| variable {p'' : N₁ ⟶ N₀} {q'' : N₂ ⟶ N₁} {h'' : Homotopy (q'' ≫ p'') 0} |
|
|
| /-- The native total's term as the actual product of the three input modules. -/ |
| def totalTermIso (p : L₁ ⟶ L₀) (q : L₂ ⟶ L₁) (h : Homotopy (q ≫ p) 0) (n : ℤ) : |
| (total p q h).X n ≅ ModuleCat.of S ((L₂.X (n+2) × L₁.X (n+1)) × L₀.X n) := |
| termIso p q h n ≪≫ |
| biprod.mapIso (ModuleCat.biprodIsoProd _ _) (Iso.refl _) ≪≫ |
| ModuleCat.biprodIsoProd _ _ |
|
|
| namespace MapData |
| variable (a : MapData p q h p' q' h') |
|
|
| /-- The explicit total map acts componentwise in the concrete term product. -/ |
| theorem totalMap_product (n : ℤ) : |
| a.totalMap.f n ≫ (totalTermIso p' q' h' n).hom = |
| (totalTermIso p q h n).hom ≫ ModuleCat.ofHom |
| (((a.f₂.f (n+2)).hom.prodMap (a.f₁.f (n+1)).hom).prodMap (a.f₀.f n).hom) := by |
| have hb : |
| biprod.map (biprod.map (a.f₂.f (n+2)) (a.f₁.f (n+1))) (a.f₀.f n) ≫ |
| biprod.map (ModuleCat.biprodIsoProd (M₂.X (n+2)) (M₁.X (n+1))).hom (𝟙 (M₀.X n)) = |
| biprod.map (ModuleCat.biprodIsoProd (L₂.X (n+2)) (L₁.X (n+1))).hom (𝟙 (L₀.X n)) ≫ |
| biprod.map (ModuleCat.ofHom ((a.f₂.f (n+2)).hom.prodMap (a.f₁.f (n+1)).hom)) (a.f₀.f n) := by |
| rw [biprodMap_comp, biprodMap_comp, Category.comp_id, Category.id_comp, |
| prodIso_naturality] |
| dsimp only [totalTermIso, Iso.trans_hom, biprod.mapIso_hom, Iso.refl_hom] |
| rw [← Category.assoc, a.totalMap_term] |
| simp only [Category.assoc] |
| erw [biprod.mapIso_hom, biprod.mapIso_hom, Iso.refl_hom, Iso.refl_hom] |
| rw [reassoc_of% hb] |
| rw [prodIso_naturality] |
|
|
| /-- A short exact triple of column maps induces short exactness of the actual |
| supplied total maps, retaining the inclusion and quotient maps themselves. -/ |
| theorem totalMap_shortExact (b : MapData p' q' h' p'' q'' h'') |
| (ex₀ : ∀ n, Function.Exact (a.f₀.f n) (b.f₀.f n)) |
| (ex₁ : ∀ n, Function.Exact (a.f₁.f n) (b.f₁.f n)) |
| (ex₂ : ∀ n, Function.Exact (a.f₂.f n) (b.f₂.f n)) |
| (inj₀ : ∀ n, Function.Injective (a.f₀.f n)) |
| (inj₁ : ∀ n, Function.Injective (a.f₁.f n)) |
| (inj₂ : ∀ n, Function.Injective (a.f₂.f n)) |
| (surj₀ : ∀ n, Function.Surjective (b.f₀.f n)) |
| (surj₁ : ∀ n, Function.Surjective (b.f₁.f n)) |
| (surj₂ : ∀ n, Function.Surjective (b.f₂.f n)) (n : ℤ) : |
| Function.Injective (a.totalMap.f n) ∧ |
| Function.Exact (a.totalMap.f n) (b.totalMap.f n) ∧ |
| Function.Surjective (b.totalMap.f n) := by |
| let inner := shortExact_prod (ex₂ (n+2)) (ex₁ (n+1)) |
| (inj₂ (n+2)) (inj₁ (n+1)) (surj₂ (n+2)) (surj₁ (n+1)) |
| let all := shortExact_prod inner.2.1 (ex₀ n) inner.1 (inj₀ n) inner.2.2 (surj₀ n) |
| apply shortExact_transport (totalTermIso p q h n).toLinearEquiv.toEquiv |
| (totalTermIso p' q' h' n).toLinearEquiv.toEquiv |
| (totalTermIso p'' q'' h'' n).toLinearEquiv.toEquiv (by simp) |
| (fun x => congrArg (fun f => f x) (a.totalMap_product n)) |
| (fun x => congrArg (fun f => f x) (b.totalMap_product n)) all.1 all.2.1 all.2.2 |
| end MapData |
| end LittleFinitistic.RectificationTotal |
|
|
| end |
|
|
| end OAI |
|
|
