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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 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 77 78 79 80 81 82 | 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
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