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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 | import OAI.Algebra.Finitistic.Localization.BoundedHomotopy
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.BoundedHomotopy
variable (C : Type*) [Category* C] [Preadditive C]
/-- An actual bounded representative of each homotopy-category object. -/
def representative (X : K C) : Complex C :=
⟨X.obj.as, (bounded_quotient_obj_iff _ _).mp X.property⟩
@[simp] theorem quotient_representative (X : K C) :
(quotient C).obj (representative C X) = X := by
cases X
rfl
variable {C}
/-- A bounded chain-map representative of a homotopy-category morphism. -/
def representativeMap {X Y : K C} (f : X ⟶ Y) :
representative C X ⟶ representative C Y :=
ObjectProperty.homMk ((HomotopyCategory.quotient C (.up ℤ)).preimage f.hom)
@[simp] theorem quotient_representativeMap {X Y : K C} (f : X ⟶ Y) :
(HomotopyCategory.quotient C (.up ℤ)).map (representativeMap f).hom = f.hom :=
(HomotopyCategory.quotient C (.up ℤ)).map_preimage _
/-- Zero in K means a genuine right-linear null-homotopy of the chain representative. -/
def nullHomotopy {X Y : Complex C} (f : X ⟶ Y) (hf : (quotient C).map f = 0) :
Homotopy f.hom 0 :=
HomotopyCategory.homotopyOfEq _ _ (by
have h := congrArg (fun g : (quotient C).obj X ⟶ (quotient C).obj Y =>
(bounded C).ι.map g) hf
rw [Functor.map_zero] at h
exact h)
end LittleFinitistic.BoundedHomotopy
end
end OAI
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