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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import OAI.Algebra.Finitistic.Rectification.VertexZeroTotal
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex
open scoped TensorProduct ModuleCat.Algebra
namespace LittleFinitistic.FirstVertex
open ExternalTensor.Section4
section General
variable {C D : Type*} [Category C] [Category D] [Preadditive C] [Preadditive D]
[HasBinaryBiproducts C] [HasBinaryBiproducts D]
def biprodInlIso {X Y : D} (hY : IsZero Y) : X ≅ X ⊞ Y where
hom := biprod.inl
inv := biprod.fst
hom_inv_id := by simp
inv_hom_id := by
apply biprod.hom_ext
· simp
· exact hY.eq_of_tgt _ _
theorem mapped_inl_isIso (F : C ⥤ D) [F.Additive] (X Y : C) (hY : IsZero (F.obj Y)) :
IsIso (F.map (biprod.inl : X ⟶ X ⊞ Y)) := by
have := preservesBinaryBiproducts_of_preservesBiproducts F
rw [← isIso_comp_right_iff _ (F.mapBiprod X Y).hom]
have he : F.map (biprod.inl : X ⟶ X ⊞ Y) ≫ (F.mapBiprod X Y).hom = biprod.inl := by
apply biprod.hom_ext
· erw [Functor.mapBiprod_hom, Category.assoc, biprod.lift_fst,
← Functor.map_comp, biprod.inl_fst, F.map_id, biprod.inl_fst]
· erw [Functor.mapBiprod_hom, Category.assoc, biprod.lift_snd,
← Functor.map_comp, biprod.inl_snd, F.map_zero, biprod.inl_snd]
rw [he]
exact (biprodInlIso hY).isIso_hom
end General
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : QuadraticEncoding.Relations k d t)
local notation "B" => QuadraticEncoding.B P
local notation "s" => ConcreteQuadraticDiagram.quadratic P
abbrev vertexComplex := (VertexZeroTotal.vertexFunctor P).mapHomologicalComplex (.up ℤ)
variable (D₀ D₁ D₂ : RightComplex (QuadraticEncoding.B P))
variable (f : Fin (d+1) → (D₀ ⟶ D₁)) (g : Fin (d+1) → (D₁ ⟶ D₂))
variable (H : ∀ r, Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g r) 0)
/-- The original first induced column includes into the actual zero column. -/
def columnInclusion : (columnComplex k B (s).e₀).obj D₀ ⟶ L0Prime s D₀ D₁ D₂ f g := biprod.inl
theorem columnInclusion_isIso :
IsIso ((vertexComplex P).map (columnInclusion P D₀ D₁ D₂ f g)) := by
have := preservesBinaryBiproducts_of_preservesBiproducts (vertexComplex P)
apply mapped_inl_isIso
apply ((vertexComplex P).mapBiprod _ _).isZero_iff.mpr
rw [biprod_isZero_iff]
exact ⟨VertexColumnComplex.reverse_isZero (QuadraticEncoding.composition P) 0 1 (by decide) D₁,
VertexColumnComplex.reverse_isZero (QuadraticEncoding.composition P) 0 2 (by decide) D₂⟩
/-- The manuscript's original first-vertex insertion into the constructed total. -/
def iota : D₀ ⟶ (vertexComplex P).obj (InducedRectification.total s D₀ D₁ D₂ f g H) :=
(VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 0 D₀).hom ≫
(vertexComplex P).map (columnInclusion P D₀ D₁ D₂ f g) ≫
(vertexComplex P).map (InducedRectification.inclusion s D₀ D₁ D₂ f g H)
/-- At the first vertex the original insertion is already a chain isomorphism. -/
theorem iota_isIso : IsIso (iota P D₀ D₁ D₂ f g H) := by
have := columnInclusion_isIso P D₀ D₁ D₂ f g
have := VertexZeroTotal.inclusion_isIso P D₀ D₁ D₂ f g H
let e : D₀ ≅ (vertexComplex P).obj ((columnComplex k B (s).e₀).obj D₀) :=
VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 0 D₀
change IsIso (e.hom ≫ (vertexComplex P).map (columnInclusion P D₀ D₁ D₂ f g) ≫
(vertexComplex P).map (InducedRectification.inclusion s D₀ D₁ D₂ f g H))
infer_instance
end LittleFinitistic.FirstVertex
end
end OAI
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