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0dd3af4 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Homology.TotalSingle
namespace OAI
section
open CategoryTheory Limits SimplicialObject Simplicial Opposite AlgebraicTopology
open HomologicalComplex HomologicalComplex₂
namespace ModelAssembly
open DiagonalResolution
variable {X Y : Dᵒᵖ ⥤ Type} (φ : X ⟶ Y)
noncomputable def mapCocone (F : D ⥤ A) : Cocone (projection X ⋙ F) where
pt := (assembly Y).obj F
ι := {
app e := colimit.ι (projection Y ⋙ F) (φ.mapElements.op.obj e)
naturality e e' f := by
simpa [projection,NatTrans.mapElements,assembly] using
colimit.w (projection Y ⋙ F) (φ.mapElements.op.map f) }
noncomputable def assemblyMapF (F : D ⥤ A) : (assembly X).obj F ⟶ (assembly Y).obj F :=
colimit.desc (projection X ⋙ F) (mapCocone φ F)
@[reassoc (attr := simp)] lemma ι_assemblyMapF (F : D ⥤ A) (e : X.Elementsᵒᵖ) :
colimit.ι (projection X ⋙ F) e ≫ assemblyMapF φ F =
colimit.ι (projection Y ⋙ F) (φ.mapElements.op.obj e) :=
colimit.ι_desc _ _
noncomputable def assemblyMap : assembly X ⟶ assembly Y where
app := assemblyMapF φ
naturality F G α := by
apply colimit.hom_ext
intro e
change colimit.ι (projection X ⋙ F) e ≫
colim.map (Functor.whiskerLeft (projection X) α) ≫ assemblyMapF φ G =
colimit.ι (projection X ⋙ F) e ≫ assemblyMapF φ F ≫
colim.map (Functor.whiskerLeft (projection Y) α)
erw [colimit.ι_map_assoc,ι_assemblyMapF,←Category.assoc,ι_assemblyMapF,colimit.ι_map]
rfl
@[reassoc (attr := simp)] lemma ι_assemblyMap (F : D ⥤ A) (e : X.Elementsᵒᵖ) :
colimit.ι (projection X ⋙ F) e ≫ (assemblyMap φ).app F =
colimit.ι (projection Y ⋙ F) (φ.mapElements.op.obj e) :=
ι_assemblyMapF φ F e
@[reassoc] lemma map_forward (d : D) :
(assemblyMap φ).app (free d) ≫ forward Y d =
forward X d ≫ (sigmaConst.obj Z).map (φ.app (op d)) := by
apply colimit.hom_ext
intro e
apply Sigma.hom_ext
intro f
change Sigma.ι (fun _ : d ⟶ e.unop.1.unop => Z) f ≫ colimit.ι (projection X ⋙ free d) e ≫ _ = _
erw [← Category.assoc (colimit.ι (projection X ⋙ free d) e)
((assemblyMap φ).app (free d)) (forward Y d), ι_assemblyMap, ι_to]
change Sigma.ι (fun _ : d ⟶ e.unop.1.unop => Z) f ≫
(freeCocone Y d).ι.app (φ.mapElements.op.obj e) =
Sigma.ι (fun _ : d ⟶ e.unop.1.unop => Z) f ≫
colimit.ι (projection X ⋙ free d) e ≫ forward X d ≫ _
erw [← Category.assoc (colimit.ι (projection X ⋙ free d) e) (forward X d)
((sigmaConst.obj Z).map (φ.app (op d))), ι_to]
dsimp only [freeCocone]
erw [Sigma.ι_comp_desc, Sigma.ι_comp_desc_assoc]
dsimp only [sigmaConst]
erw [Sigma.ι_comp_map', Category.id_comp]
congr 1
exact (ConcreteCategory.congr_hom (φ.naturality f.op) e.unop.2).symm
end ModelAssembly
end
open CategoryTheory Limits HomologicalComplex HomologicalComplex₂
namespace TotalFunctor
universe u v u' v'
variable {C : Type u} [Category.{v} C] [Preadditive C] [HasCoproducts.{0} C]
{E : Type u'} [Category.{v'} E] [Preadditive E] [HasCoproducts.{0} E]
open TotalHomotopy
variable {F G : C ⥤ E} [F.Additive] [G.Additive] (α : F ⟶ G)
noncomputable abbrev nat₂ (K : HomologicalComplex₂ C c c) : map₂ F K ⟶ map₂ G K :=
((α.mapHomologicalComplex c).mapHomologicalComplex c).app K
@[reassoc] lemma comparison_natural (K : HomologicalComplex₂ C c c) :
total.map (nat₂ α K) c ≫ comparison G K =
comparison F K ≫ (α.mapHomologicalComplex c).app (K.total c) := by
apply HomologicalComplex.Hom.ext
funext n
apply total.hom_ext
intro p q hpq
change (map₂ F K).ιTotal c p q n hpq ≫ (total.map (nat₂ α K) c).f n ≫ comparisonF G K n =
(map₂ F K).ιTotal c p q n hpq ≫ comparisonF F K n ≫ α.app _
rw [ιTotal_map_assoc,ι_comparisonF,ι_comparisonF_assoc]
exact (α.naturality _).symm
@[reassoc] lemma iso_natural [PreservesColimitsOfSize.{0,0} F] [PreservesColimitsOfSize.{0,0} G]
(K : HomologicalComplex₂ C c c) :
(α.mapHomologicalComplex c).app (K.total c) ≫ (iso G K).hom =
(iso F K).hom ≫ total.map (nat₂ α K) c := by
apply (cancel_mono (comparison G K)).mp
simp only [Category.assoc]
change _ ≫ inv (comparison G K) ≫ comparison G K =
inv (comparison F K) ≫ total.map (nat₂ α K) c ≫ comparison G K
rw [IsIso.inv_hom_id,Category.comp_id,comparison_natural,←Category.assoc,
IsIso.inv_hom_id,Category.id_comp]
end TotalFunctor
end OAI
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