| 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 | |