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216fb31 | 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 | import OAI.Algebra.Finitistic.Localization.BoundedRepresentatives
import OAI.Algebra.Finitistic.Localization.SourceLinear
import OAI.Algebra.Finitistic.Localization.ChainRepresentatives
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.ConcreteChainRepresentatives
open SourceVerdier
universe u
variable {k : Type u} [Field k] {d t : ℕ} (P : QuadraticEncoding.Relations k d t)
abbrev representative (X : K P) : CochainComplex (Proj P) ℤ := X.obj.as
theorem representative_bounded (X : K P) :
BoundedHomotopy.boundedComplex (Proj P) (representative P X) :=
(BoundedHomotopy.bounded_quotient_obj_iff _ _).1 X.property
/-- The lifted source diagram has genuine bounded projective chain representatives
and a specified null-homotopy for every relation; no chain representatives are assumed. -/
theorem quadratic_representatives (na nb nr : ℕ) (X₀ X₁ X₂ : K P)
(a : Fin na → (X₀ ⟶ X₁)) (b : Fin nb → (X₁ ⟶ X₂))
(c : Fin nr → Fin na → Fin nb → k)
(hc : ∀ r, ∑ i, ∑ j, c r i j • (a i ≫ b j) = 0) :
∃ (f : Fin na → (representative P X₀ ⟶ representative P X₁))
(g : Fin nb → (representative P X₁ ⟶ representative P X₂)),
(∀ i, (HomotopyCategory.quotient (Proj P) (.up ℤ)).map (f i) = (a i).hom) ∧
(∀ j, (HomotopyCategory.quotient (Proj P) (.up ℤ)).map (g j) = (b j).hom) ∧
Nonempty (∀ r, Homotopy (∑ i, ∑ j, c r i j • (f i ≫ g j)) 0) := by
let I := (BoundedHomotopy.bounded (Proj P)).ι
apply ChainRepresentatives.quadratic_chain_representatives na nb nr
(representative P X₀) (representative P X₁) (representative P X₂)
(fun i => I.map (a i)) (fun j => I.map (b j)) c
intro r
have h := congrArg I.map (hc r)
simp only [Functor.map_sum,Functor.map_smul,Functor.map_comp,Functor.map_zero] at h
exact h
end LittleFinitistic.ConcreteChainRepresentatives
end
end OAI
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