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2c7d975 | 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 | import OAI.Algebra.AuslanderReiten.Coordinates.MapExtend
namespace OAI
noncomputable section
open CategoryTheory Category Limits ZeroObject HomologicalComplex
namespace ArExplicit.ResolutionProducts
universe u v u' v'
variable {C : Type u} [Category.{v} C] [Abelian C]
variable {D : Type u'} [Category.{v'} D] [Abelian D]
variable (F : C ⥤ D) [F.Additive] [PreservesFiniteLimits F] [PreservesFiniteColimits F]
variable [F.PreservesProjectiveObjects]
variable {X : C} (P : ProjectiveResolution X)
omit [Abelian C] [Abelian D] [F.Additive] [PreservesFiniteLimits F]
[PreservesFiniteColimits F] [F.PreservesProjectiveObjects] in
private theorem map_composite_cancel {U V W Z : C} {U' Z' : D}
(a : U' ⟶ F.obj U) (b : U ⟶ V) (f : V ⟶ W) (e : W ≅ Z) (t : F.obj W ⟶ Z') :
a ≫ F.map (b ≫ f ≫ e.hom) ≫ (F.map e.inv ≫ t) =
(a ≫ F.map b) ≫ F.map f ≫ t := by
simp only [Functor.map_comp, Category.assoc]
rw [← F.map_comp_assoc e.hom e.inv t, e.hom_inv_id, F.map_id, id_comp]
def mapResolutionCochainIso : (mapResolution F P).cochainComplex ≅
(F.mapHomologicalComplex (.up ℤ)).obj P.cochainComplex :=
mapExtendIso F P.complex ComplexShape.embeddingDownNat
theorem mapResolutionCochainIso_coordinate (n : ℤ) (j : ℕ) (h : -(j : ℤ) = n) :
(mapResolutionCochainIso F P).hom.f n ≫ F.map (P.cochainComplexXIso n j h).hom =
((mapResolution F P).cochainComplexXIso n j h).hom :=
mapExtendXIso_hom_XIso F P.complex (ComplexShape.embeddingDownNat.r_eq_some h)
theorem mapResolution_π_zero : (mapResolution F P).π.f 0 = F.map (P.π.f 0) := by
change F.map (P.π.f 0) ≫ ((singleMapHomologicalComplex F (.down ℕ) 0).hom.app X).f 0 = _
rw [singleMapHomologicalComplex_hom_app_self]
change F.map (P.π.f 0) ≫ (F.map (𝟙 X) ≫ 𝟙 (F.obj X)) = _
simp
theorem mapResolutionCochainIso_π :
(mapResolutionCochainIso F P).hom ≫
(F.mapHomologicalComplex (.up ℤ)).map P.π' ≫
(F.mapCochainComplexSingleFunctor 0).hom.app X = (mapResolution F P).π' := by
apply to_single_hom_ext
change (mapResolutionCochainIso F P).hom.f 0 ≫ F.map (P.π'.f 0) ≫
((singleMapHomologicalComplex F (.up ℤ) 0).hom.app X).f 0 = (mapResolution F P).π'.f 0
rw [P.π'_f_zero,
(mapResolution F P).π'_f_zero, mapResolution_π_zero,
singleMapHomologicalComplex_hom_app_self]
have he := map_composite_cancel F ((mapResolutionCochainIso F P).hom.f 0)
(P.cochainComplexXIso 0 0 (by simp)).hom (show P.complex.X 0 ⟶ X from P.π.f 0)
(singleObjXSelf (.up ℤ) 0 X).symm (singleObjXSelf (.up ℤ) 0 (F.obj X)).inv
exact he.trans (congrArg (fun a => a ≫ F.map (P.π.f 0) ≫
(singleObjXSelf (.up ℤ) 0 (F.obj X)).inv) (mapResolutionCochainIso_coordinate F P 0 0 (by simp)))
end ArExplicit.ResolutionProducts
end
end OAI
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