| import Mathlib.Data.Fintype.Sigma |
| import OAI.Algebra.Finitistic.Iteration.BinomialIndex |
| import OAI.Algebra.Finitistic.Iteration.BinomialBiproduct |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits |
| noncomputable section |
| namespace ExternalTensor.Binomial |
| universe v u w |
| variable {C : Type u} [Category.{v} C] [Preadditive C] [HasFiniteBiproducts C] [HasBinaryBiproducts C] |
| variable [HasShift C ℤ] [∀ n : ℤ, (shiftFunctor C n).Additive] |
| variable {S : Type w} (Phi : C ⥤ C) [Phi.Additive] [Phi.CommShift ℤ] |
| variable (M : S → C) (H : S → S) (b d : ℤ) |
|
|
| / |
| def branchIteration (split : ∀ Y, Phi.obj (M Y) ≅ (M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+d⟧) |
| (j : ℕ) (Y : S) : |
| (Phi.obj^[j]) (M Y) ≅ |
| ⨁ fun x : Branch j => (M ((H^[j]) Y))⟦(j : ℤ)*b + d*(weight x : ℤ)⟧ := by |
| induction j with |
| | zero => |
| letI : Unique (Branch 0) := inferInstanceAs (Unique Unit) |
| simpa only [Function.iterate_zero_apply, Nat.cast_zero, zero_mul, mul_zero, zero_add, weight] using |
| (shiftZero ℤ (M Y)).symm ≪≫ |
| (biproductUniqueIso (fun _ : Branch 0 => (M Y)⟦(0 : ℤ)⟧)).symm |
| | succ j ih => |
| rw [Function.iterate_succ_apply'] |
| refine Phi.mapIso ih ≪≫ Phi.mapBiproduct _ ≪≫ ?_ |
| let X := M ((H^[j+1]) Y) |
| let f : Branch j → C := fun x => X⟦((j+1 : ℕ) : ℤ)*b + d*(weight x : ℤ)⟧ |
| let g : Branch j → C := fun x => X⟦((j+1 : ℕ) : ℤ)*b + d*((weight x+1 : ℕ) : ℤ)⟧ |
| have step (x : Branch j) : |
| Phi.obj ((M ((H^[j]) Y))⟦(j : ℤ)*b+d*(weight x : ℤ)⟧) ≅ f x ⊞ g x := by |
| simpa only [f, g, X, Function.iterate_succ_apply', Nat.cast_add, Nat.cast_one, |
| add_mul, mul_add, mul_one, one_mul, add_assoc, add_left_comm, add_comm] using |
| splitShift Phi (M ((H^[j]) Y)) (M (H ((H^[j]) Y))) b d |
| ((j : ℤ)*b+d*(weight x : ℤ)) (split ((H^[j]) Y)) |
| refine biproduct.mapIso step ≪≫ biproductBiprod f g ≪≫ biproduct.mapIso (J := Branch j ⊕ Branch j) |
| (f := Sum.elim f g) |
| (g := fun x : Branch j ⊕ Branch j => |
| (M ((H^[j+1]) Y))⟦((j+1 : ℕ) : ℤ)*b + d*(weight (n := j+1) x : ℤ)⟧) ?_ |
| intro x |
| cases x with |
| | inl x => exact Iso.refl _ |
| | inr x => exact Iso.refl _ |
|
|
| / |
| def binomialObject (j : ℕ) (X : C) : C := |
| ⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) => |
| X⟦(j : ℤ)*b + d*(q.val : ℤ)⟧ |
|
|
| / |
| def groupBiproduct (j : ℕ) (X : C) : |
| (⨁ fun x : Branch j => X⟦(j : ℤ)*b + d*(weight x : ℤ)⟧) ≅ |
| binomialObject b d j X := |
| biproduct.whiskerEquiv |
| (f := fun x : Branch j => X⟦(j : ℤ)*b + d*(weight x : ℤ)⟧) |
| (g := fun p : Σ q : Fin (j+1), Fin (Nat.choose j q.val) => |
| X⟦(j : ℤ)*b + d*(p.1.val : ℤ)⟧) |
| (groupEquiv j) (fun _ => Iso.refl _) ≪≫ |
| (biproductBiproductIso (fun q : Fin (j+1) => Fin (Nat.choose j q.val)) |
| (fun q _ => X⟦(j : ℤ)*b + d*(q.val : ℤ)⟧)).symm |
|
|
| / |
| def binomialIteration (split : ∀ Y, Phi.obj (M Y) ≅ (M (H Y))⟦b⟧ ⊞ (M (H Y))⟦b+d⟧) |
| (j : ℕ) (Y : S) : |
| (Phi.obj^[j]) (M Y) ≅ binomialObject b d j (M ((H^[j]) Y)) := |
| branchIteration Phi M H b d split j Y ≪≫ groupBiproduct b d j _ |
|
|
| / |
| def section4Iteration (split : ∀ Y, Phi.obj (M Y) ≅ M (H Y) ⊞ (M (H Y))⟦(3 : ℤ)⟧) |
| (j : ℕ) (Y : S) : |
| (Phi.obj^[j]) (M Y) ≅ |
| ⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) => |
| (M ((H^[j]) Y))⟦(3 : ℤ)*(q.val : ℤ)⟧ := by |
| have h (Y : S) : Phi.obj (M Y) ≅ (M (H Y))⟦(0 : ℤ)⟧ ⊞ (M (H Y))⟦(0+3 : ℤ)⟧ := |
| split Y ≪≫ biprod.mapIso (shiftZero ℤ (M (H Y))).symm (Iso.refl _) |
| simpa only [binomialObject, mul_zero, zero_add] using binomialIteration Phi M H 0 3 h j Y |
| end ExternalTensor.Binomial |
|
|
| end |
|
|
| end OAI |
|
|
