| import Mathlib.Algebra.Homology.HomotopyCategory |
| import Mathlib.Algebra.Homology.HomologicalComplexLimits |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| noncomputable section |
| namespace DerivedBar |
|
|
| section |
| universe u v w |
| variable {C : Type u} [Category.{v} C] [Preadditive C] |
| [HasColimitsOfShape (Discrete ℕ) C] {ι : Type w} {c : ComplexShape ι} |
| (F : Discrete ℕ ⥤ HomologicalComplex C c) (s : Cocone F) (hs : IsColimit s) |
| {Y : HomologicalComplex C c} (f g : s.pt ⟶ Y) |
| (h : ∀ r, Homotopy (s.ι.app r ≫ f) (s.ι.app r ≫ g)) |
|
|
| def coproductHomotopyHom (i j : ι) : s.pt.X i ⟶ Y.X j := |
| (isColimitOfPreserves (HomologicalComplex.eval C c i) hs).desc |
| { pt := Y.X j, ι := Discrete.natTrans (fun r => (h r).hom i j) } |
|
|
| @[reassoc] theorem ι_coproductHomotopyHom (i j : ι) (r : Discrete ℕ) : |
| (s.ι.app r).f i ≫ coproductHomotopyHom F s hs f g h i j = (h r).hom i j := |
| (isColimitOfPreserves (HomologicalComplex.eval C c i) hs).fac _ r |
|
|
| /-- Homotopies out of the summands assemble through the actual degreewise |
| coproduct, so equality after the homotopy quotient is still detected summandwise. -/ |
| def coproductHomotopy : Homotopy f g where |
| hom := coproductHomotopyHom F s hs f g h |
| zero i j hij := by |
| apply (isColimitOfPreserves (HomologicalComplex.eval C c i) hs).hom_ext |
| intro r |
| change (s.ι.app r).f i ≫ coproductHomotopyHom F s hs f g h i j = _ |
| rw [ι_coproductHomotopyHom, CategoryTheory.Limits.comp_zero] |
| exact (h r).zero i j hij |
| comm i := by |
| apply (isColimitOfPreserves (HomologicalComplex.eval C c i) hs).hom_ext |
| intro r |
| change (s.ι.app r).f i ≫ f.f i = (s.ι.app r).f i ≫ |
| (s.pt.d i (c.next i) ≫ coproductHomotopyHom F s hs f g h (c.next i) i + |
| coproductHomotopyHom F s hs f g h i (c.prev i) ≫ Y.d (c.prev i) i + g.f i) |
| simp only [Preadditive.comp_add] |
| erw [(s.ι.app r).comm_assoc i (c.next i)] |
| rw [ι_coproductHomotopyHom, ι_coproductHomotopyHom_assoc] |
| exact (h r).comm i |
|
|
| end |
| universe u v w |
| variable {C : Type u} [Category.{v} C] [Preadditive C] |
| [HasColimitsOfShape (Discrete ℕ) C] {ι : Type w} {c : ComplexShape ι} |
| (F : Discrete ℕ ⥤ HomologicalComplex C c) (s : Cocone F) (hs : IsColimit s) |
| local notation "Q" => HomotopyCategory.quotient C c |
|
|
| def homotopyCoconeLift (t : Cocone (F ⋙ Q)) : Cocone F where |
| pt := t.pt.as |
| ι := Discrete.natTrans (fun r => (t.ι.app r).out) |
|
|
| /-- The actual coproduct survives the homotopy quotient. Both existence and |
| uniqueness are proved; the latter assembles the component homotopies. -/ |
| def homotopyCoproductIsColimit : IsColimit ((Q).mapCocone s) where |
| desc t := (Q).map (hs.desc (homotopyCoconeLift F t)) |
| fac t r := by |
| change (Q).map (s.ι.app r) ≫ (Q).map (hs.desc (homotopyCoconeLift F t)) = _ |
| rw [← Functor.map_comp, hs.fac] |
| exact HomotopyCategory.quotient_map_out _ |
| uniq t m hm := by |
| have h : Homotopy m.out (hs.desc (homotopyCoconeLift F t)) := |
| coproductHomotopy F s hs m.out (hs.desc (homotopyCoconeLift F t)) (fun r => |
| HomotopyCategory.homotopyOfEq _ _ (by |
| erw [hs.fac] |
| have hmr := hm r |
| change (Q).map (s.ι.app r) ≫ m = t.ι.app r at hmr |
| simp only [Functor.map_comp, homotopyCoconeLift, Discrete.natTrans_app, |
| HomotopyCategory.quotient_map_out] |
| exact hmr)) |
| exact (HomotopyCategory.quotient_map_out m).symm.trans |
| (HomotopyCategory.eq_of_homotopy _ _ h) |
| end DerivedBar |
|
|
| end |
|
|
| end OAI |
|
|
